Steven Mathey thesis electronic version

Steven Mathey thesis electronic version
Dissertation
submitted to the
Combined Faculties of the Natural Sciences and Mathematics
of the Ruperto-Carola-University of Heidelberg. Germany
for the degree of
Doctor of Natural Sciences
Put forward by
Steven, Mathey
born in: Geneva, Switzerland
Oral examination: 15th October 2014
Functional renormalisation approach
to driven dissipative dynamics
Referees:
Prof. Dr. Thomas Gasenzer
Prof. Dr. Jürgen Berges
Zusammenfassung
In der vorliegenden Arbeit werden die getrieben-dissipativen stationären skaleninvarianten Zustände der Burgers- und Gross–Pitaevskii-Gleichungen (GPE) untersucht.
Die Pfadintegral-Darstellung des stationären Zustands der stochastischen BurgersGleichung wird verwendet, um skaleninvariante Lösungen des Systems an Fixpunkten der
Renormierungsgruppe zu studieren. Die funktionale Renormierungsgruppe wird genutzt,
um die Physik in einer nicht-perturbativen Näherung zu beschreiben. Eine Approximation, die Galileiinvarianz berücksichtigt und die die Frequenz- und Impulsabhängigkeit der
zwei-Punkt Geschwindigkeits-Korrelationsfunktion beschreiben kann, wird konstruiert.
Ein System von Fixpunktgleichungen der Renormierungsgruppe für beliebige Frequenz
und Impulsabhängigkeiten des inversen Propagators wird aufgestellt. In allen betrachteten Raumdimensionen ergeben diese ein Kontinuum von Fixpunkten und einen isolierten
Fixpunkt. Diese Ergebnisse weisen nur für d=1 eine gute. . . Werten auf. In der Literatur
werden jedoch fast ausschliesslich wirbelfreie Lösungen behandelt, während die in der
vorliegenden Arbeit verwendete Näherung exklusiv für Lösungen mit Vortizität anwendbar ist. Dadurch ist diese ähnlicher zur Navier–Stokes-Turbulenz.
Stationäre Nichtgleichgewichtszustände ultrakalter Bose-Gase gekoppelt an externe
Energie- und Teilchenreservoirs, wie zum Beispiel Exziton-Polariton Kondensate, stehen mit der stochastischen Kardar–Parisi–Zhang (KPZ) Gleichung durch die Dichteund Phasenzerlegung der gemittelten komplexen Wellenfunktion in Beziehung. Unsere
Ergebnisse legen nahe, dass die skaleninvarianten Lösungen, die wir in diesem Kontext
finden, auch gültig sind für quasi-stationäre Zustände des konservativen Systems fern
des Gleichgewichts (nicht-thermische Fixpunkte), welche mit Hilfe der GPE beschrieben
werden können. Für das KPZ-Modell bekannte Ergebnisse werden auf ultrakalte BoseGase angewandt. Auf diese Weise läßt sich eine neue Skalenrelation herleiten, welche
dazu verwendet werden kann, Kolmogorovs Skalenexponent 5/3 der turbulenten Energieverteilung in einer inkompressiblen Flüssigkeit zu bestimmen. Darüberhinaus erhält
man eine anomale Korrektur zum inkompressiblen wie auch zum kompressiblen Anteil
des Energiespektrums eines verdünnten Bose-Gases.
iii
Abstract
In this thesis we investigate driven-dissipative stationary scaling states of Burgers’ and
Gross–Pitaevskii equations (GPE).
The path integral representation of the steady state of the stochastic Burgers equation
is used in order to investigate the scaling solutions of the system at renormalisation group
fixed points. We employ the functional renormalisation group in order to access the nonperturbative regime. We devise an approximation that respects Galilei invariance and is
designed to resolve the frequency and momentum dependence of the two-point velocity
correlation function. We establish a set of renormalisation group fixed point equations
for effective inverse propagators with an arbitrary frequency and momentum dependence.
In all spatial dimensions they yield a continuum of fixed points as well as an isolated
one. These results are fully compatible with the existing literature for d = 1 only. For
d ̸= 1 however results of the literature focus almost exclusively on irrotational solutions
while the solutions that our approximation can capture contain necessarily vorticity and
are closer to Navier-Stokes turbulence.
Non-equilibrium steady states of ultra-cold Bose gases coupled to external reservoirs of
energy and particles such as exciton–polariton condensates are related to the stochastic
KPZ equation by the density and phase decomposition of the average complex wave
function. We postulate that the scaling that we obtain in this context applies as well
to far-from-equilibrium quasi-stationary steady states (non-thermal fixed points) of the
corresponding closed system described by the GPE. We translate results found in the
KPZ literature to their corresponding dual in the ultra-cold Bose gas set-up. We find
that this provides a new scaling relation which can be used to analytically identify
the classical Kolmogorov −5/3 exponent and its anomalous correction. Moreover we
estimate the anomalous correction to the scaling exponent of the compressible part of
the kinetic energy spectrum of the Bose gas which is confirmed by numerical simulations
of the GPE.
iv
Steven Mathey
Institut für Theoretische Physik
Philosophenweg 16
D-69120 Heidelberg
Deutschland
Primary advisor
Prof. Dr. Thomas Gasenzer
Institut für Theoretische Physik
Philosophenweg 16
D-69120 Heidelberg
Deutschland
Secondary advisor
Prof. Dr. Jan M. Pawlowski
Institut für Theoretische Physik
Philosophenweg 16
D-69120 Heidelberg
Deutschland
v
Publication
This thesis contains discussions and results from the following paper which is currently
under review at Physical Review A.
• S. Mathey, T. Gasenzer, J. M. Pawlowski, Anomalous Scaling at Non-thermal
Fixed Points of Burgers’ and Gross–Pitaevskii Turbulence, arXiv:1405.7652 [condmat.quant-gas]
I conducted this work myself under the supervision of T. Gasenzer and J.M. Pawlowski.
Declaration by author
This thesis is composed of my original work, and contains no material previously published or written by another person except where due reference has been made in the
text. I have clearly stated the contribution by other authors to jointly-authored works
that I have included in my thesis. The content of my thesis is the result of work I have
carried out since the commencement of my graduate studies at the Heidelberg Graduate
School of Fundamental Physics, Institut für Theoretische Physik, Universität Heidelberg
and does not include material that has been submitted by myself to qualify for the award
of any other degree or diploma in any university or other tertiary institution.
vi
Acknowledgments
My first and biggest thanks go to my advisors Thomas Gasenzer and Jan M. Pawlowski.
They placed me at the fascinating interface of turbulence and renormalisation. They
assembled enough funds to send me to conferences all over Europe. They offered challenging guidance that made me improve by myself and straightforward advice when it
was needed. They questioned my work when I was not able to recognize that it was
questionable. They where patient and helped me improve. They made me feel welcome
in their respective research groups.
I would like to acknowledge the Heidelberg Graduate School for Fundamental Physics
for holding together an excellent graduate program and for financing my month in Les
Houches, the University of Heidelberg for providing all the necessary infrastructure and
the Landesgraduirtenförderungsgesetz for the financial support.
I would like to thank as well Sebastian Bock, Léonie Canet, Isara Chantesana, Sebastian Erne, Thomas Gasenzer, Markus Karl, Alexander Liluashvili, David Mesterházy,
Mario Mitter, Boris Nowak, Jan M. Pawlowski, Nikolai Philipp, Andreas Samberg, Martin Trappe, Gilles Tarjus and Nicolas Wschebor for stimulating scientific discussions.
Thank you to Sebastian Bock, Isara Chantesana, Nicolai Christiansen, Martin Gärttner, Sebastian Heupts, Markus Karl, Kevin Falls, Boris Nowak and Andreas Samberg,
for enduring and even trying to answer my random questions, for saving me when my
computer was not being cooperative and/or for the warm welcome in the cellar of the
institute.
Thanks to my special proofreading team: Sebanstian Bock, Isara Chantesana, Markus
Karl, Gabriela Loza, David Mesterházy, Mario Mitter and Andreas Samberg. Be it by
pointing out typos or telling me to redo whole sections, they made me improve this work
more than I could ever have by myself. Of course, any remaining errors or mistakes are
solely due to my shortcoming. I thank as well Boris Nowak and Jan Schole who made
their numerical data easily available to me.
I thank my parents who supported me during my studies and encouraged me along a
road that I could choose for myself.
I thank my Ph.D. examination committee for taking an interest in my work and the
time to learn about it. Thomas Gasenzer and Jürgen Berges who will read this thesis
and Markus Oberthaler and Karlheinz Meier who will take part in the oral examinations.
I thank the staff of the Institute for theoretical physics. They are always helpful and
in a good mood. They are as well very patient with my poor German skills.
I thank as well the very wise users of the LATEX Stack Exchange. They saved my life
countless times.
My final thanks go to Gabriela Loza for the continued encouragement, support, patience and advice during the years that lead the conclusion of this work. Thank you for
filling my life with more than equations.
vii
Para mi Amorcita
viii
Contents
1. Introduction
1.1. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
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2. The Functional Renormalisation Group
2.1. Effective action . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1. Flowing effective action . . . . . . . . . . . . . . . . . . .
2.1.2. Flowing Schwinger functional . . . . . . . . . . . . . . . .
2.1.3. Flow equation . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.4. One particle irreducible effective action . . . . . . . . . .
2.2. Functional renormalisation group fixed point equations . . . . . .
2.2.1. Local potential approximation . . . . . . . . . . . . . . .
2.2.1.1. Renormalisation group flow equation . . . . . .
2.2.1.2. Renormalisation group fixed point . . . . . . . .
2.2.1.3. Critical exponents . . . . . . . . . . . . . . . . .
2.2.2. Frequency and momentum dependent inverse propagator
2.2.2.1. Ultraviolet divergent fixed points . . . . . . . . .
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3. Burgers Turbulence
3.1. Scaling and correlation functions . . . . . . . . . . . .
3.2. Functional treatment . . . . . . . . . . . . . . . . . . .
3.3. Functional renormalisation group calculation . . . . .
3.3.1. Approximation scheme . . . . . . . . . . . . . .
3.3.2. Flow equations . . . . . . . . . . . . . . . . . .
3.3.3. Fixed point equations . . . . . . . . . . . . . .
3.3.4. Computing observable quantities . . . . . . . .
3.3.5. Asymptotic properties of the flow integrals . .
3.3.5.1. Scaling limit (p ≪ k) . . . . . . . . .
3.3.5.2. Scaling limit (p ≫ k) . . . . . . . . .
3.3.5.3. Implications for driving and turbulent
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4. Ultra-cold Bose Gases
4.1. Super-fluid turbulence and non-thermal fixed points
4.2. Driven-dissipative Gross–Pitaevskii equation . . . . .
4.3. Strong wave and quantum turbulence . . . . . . . .
4.3.1. Galilei invariance and Kolmogorov scaling . .
4.3.2. Acoustic turbulence in a super-fluid . . . . .
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Contents
5. Conclusions and Outlook
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Appendices
90
A. Local Potential Approximation, Fixed Point Coefficients
91
B. Flow Integrals
94
C. Re-scaled Flow Integrals
98
C.1. Scaling limit (p ≪ k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
C.2. Scaling limit (p ≫ k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
D. Equations for Ai
105
E. Energy spectrum decomposition
108
F. Notation and conventions
109
G. List of Abbreviations
114
G.1. Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
G.2. Short-hand notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
Bibliography
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2
1. Introduction
"A process cannot be understood by stopping it. Understanding must move
with the flow of the process, must join it and flow with it."
(The First Law of Mentat, [1])
Physical systems may look very different when observed at different scales [2]. In most
cases it appears that processes which occur on very large scales are decoupled from those
happening on small ones and can be described almost independently from each other.
Even if it is technically true that a butterfly flapping its wings can affect distant weather
patterns, this is never included in tropical cyclone forecast models. There are however
systems realised in nature where correlations can propagate across a large range of
scales. This typically leads to scale invariance of observables. It is well known that such
phenomena occur for specific values of the thermodynamic parameters of equilibrium
systems. These are critical points of the phase diagram where fluctuations occur on
all spatial scales and the correlation length is infinite [3, 4]. In thermal equilibrium
critical states are not the norm though. Indeed, the parameters of the system have to
be precisely tuned in order to observe criticality.
The situation is however different outside of thermal equilibrium. There, many systems
spontaneously evolve to a critical steady state (see e.g. [5–8]). One of the most famous
examples of such a far-from-equilibrium critical state is turbulence [9–14]. Indeed, it
seems to appear almost spontaneously in many fundamentally different systems ranging
from classical hydrodynamics to high-energy heavy-ion collisions all the way to ultra-cold
Bose gas dynamics. It can either be sustained by an appropriate driving mechanism or be
a transient before the thermalisation of the system. One of the hallmarks of turbulence is
that, when it is realised, conserved charges undergo cascades. The charge is transported
either from large to small scales or in the other direction in a way that is local in Fourier
space. A stationary, unidirectional and local transport of charge is established. For
example in the case of a direct cascade of energy dissipation happens at small spatial
scales while energy is injected at the large ones. Then a steady state is established with
energy flowing from large to small scales while keeping its spectral distribution constant.
Because they involve many different scales such processes depend on the dynamics of
many interacting degrees of freedom and are still poorly understood and under heavy
investigation [15].
Considerable progress in our understanding of turbulence in quantum field theories
has been made recently because of the use [16–18] of non-perturbative methods such
as 1/N re-summations of 2 Particle Irreducible (2PI) effective field equations [19–27]
3
1. Introduction
Figure 1.1.: Self-similar turbulence on the Neckar (Heidelberg)
and semi-classical methods [16, 25, 28–49] as well as recent experimental realisations of
quantum turbulence [50–53]. Applications of Functional Renormalisation Group (FRG)
methods to dynamical evolution, non-thermal fixed points, and turbulence have received
increasing attention recently [17, 54–61]. In particular the dynamics of ultra-cold Bose
gases [28, 30–33, 35, 36, 39–43, 45–49] and gauge systems such as relativistic heavy-ion
collisions [62–72] have received a great deal of attention.
In this work we investigate stationary classical and quantum turbulence. We focus
on driven-dissipative dynamics which provide a natural mechanism to establish outof-equilibrium steady states. It is well known from experimental [73–84], numerical
[78, 85–89] as well as analytical [10, 11, 90–100] studies that both classical and quantum
turbulence exhibits scale invariant observables. Moreover in the case of classical hydrodynamic turbulence this is as well evident form everyday life (see Figure (1.1)). Based on
this observation many authors have proposed analogies between classical turbulence and
critical systems (see e.g. [75, 80, 101, 102]). This points towards the Renormalisation
Group (RG) [103–106] and the idea that turbulence is realised as one of its fixed points.
Although it was introduced as a method to handle the divergences appearing in quantum field theory (see [107, 108] and references therein), the RG has proven to be a
tremendously effective tool in statistical physics. It relates different effective descriptions of a system under increasing levels of coarse-graining of the degrees of freedom. It
has enabled us to understand the reason for the universality of critical physics. Why
critical systems only come in a small number of different kinds even though the underlying microscopic physics can be infinitely varied. Moreover it enables the calculation of
the critical properties of the system such as scaling exponents and dimensionless ratios
and explains the infinite correlation length at a critical point as being the only nonzero correlation length that one can find at a fixed point of the RG transformation (see
e.g. [4, 109, 110] and references therein for overviews).
The RG has also been applied very successfully outside of thermal equilibrium. Examples of such applications can be found in e.g. [90, 111–114] and in [90, 93, 115, 116]
in the case of stationary hydrodynamic turbulence. See also [92] for an overview and
references to the older literature and [17, 54–61] for applications of the FRG. In the case
of hydrodynamic turbulence its applications are however limited because the system is
4
1. Introduction
fundamentally non-perturbative. Indeed, at an RG fixed point one finds perfect scale
invariance. The cascade dynamics are realised from infinitely large to infinitely small
scales and all the degrees of freedom take part in them. Then the corresponding Reynolds
number (or coupling constant) which determines the extent of the scaling range must be
infinite.
In this work we turn to the FRG which is a non-perturbative version of the RG. It
expresses the RG directly in terms the flow of a functional (typically a coarse-grained
effective action or a generating functional) and contains by construction the full interacting theory. Note that the (perturbative) RG is defined in terms of the flow of a few
parameters of the system which makes it only effective close to fixed points. The FRG
was first introduced directly as a flow equation for the effective Lagrangian of the theory
in [117]. Many FRG flow equations have been written since then (see e.g. [118, 119] for
overviews). The flow equation of Wetterich [120, 121] which acts on the Infrared (IR)
coarse-grained 1 Particle Irreducible (1PI) effective action has proven to be particularly
useful. It provides an intuitive framework to compute RG transformations because it
relates directly physical observables and bare quantities by a continuous tuning of the
cut-off scale. See Refs. [119, 122, 123] for general and [118, 124–133] for more specific
reviews. Wetterich’s FRG has proven to be a powerful and versatile tool in many areas
of physics. It has recently gained a lot of interest in a wide range of non-equilibrium
phenomena [54, 56], including reaction-diffusion processes [134], quantum decoherence
[135, 136], open quantum systems [137, 138], critical dynamics [17, 139, 140], transport
in quantum systems [141–143], cosmology [144], strongly correlated transport in solids
and Kondo physics [145–147], disordered systems [148] and driven-dissipative dynamics
[59, 60].
The FRG has been applied to classical steady state Navier–Stokes (NS) turbulence
in [91, 94, 96]. The results of the classical perturbative approaches [90, 92, 115, 116]
where recovered but the truly non-perturbative turbulent fixed point is still out of reach.
However the work of the authors of [55, 57, 58, 61] who apply the FRG to the similar
although simpler problem of the stochastic Kardar–Parisi–Zhang (KPZ) equation has
been very successful and gives hope that the FRG can be of use in the context of
hydrodynamic turbulence.
1.1. Summary
In this work we study stationary driven-dissipative dynamics. As an intermediate step
in between KPZ and NS dynamics we study the stochastic Burgers equation [149] in
arbitrary number of spatial dimension. Burgers’ equation is formally similar to NS
apart from the fact that it does not contain a pressure term. This simplifies the problem
greatly as compared to the usual incompressible fluid dynamics because in the latter
case the pressure is used to enforce the incompressibility condition. It is a slave to the
dynamics of the velocity field and is determined by an inhomogeneous Laplace equation.
This introduces a non-local component to the equations of motion of the velocity field
when this equation is inverted to express pressure in terms of the velocity field [150].
5
1. Introduction
Burgers’ equation has received a lot of attention over the years because, for a given
set of initial conditions, is can be solved implicitly in the limit of vanishing viscosity
with the method of characteristics (see [151, 152] and references therein). Despite this
apparent simplicity Burgers’ equation still contains interesting and complex features. Its
solutions may acquire discontinuities i.e. shocks, after a finite time evolution and these
in turn produce a rich scaling behaviour when stochastic components are included in
the dynamics. Moreover for spatial dimension d > 1 the literature almost exclusively
concentrates on solutions where the velocity field is the gradient of a potential v = ∇θ.
To the best of our knowledge only [153, 154] does not make this simplification. This case
is equivalent to KPZ equation and can even be linearised by expressing the potential as1
θ = −2ν log(Z).
Section 2 - The Functional Renormalisation Group
The second part of this work is about the RG and its application to critical phenomena.
In particular we focus on the FRG. We emphasise the properties of RG fixed point and
explain how to compute them.
We start with a short qualitative introduction to the RG and FRG. Precise definitions
are given in Section (2.1). We turn to FRG fixed point equations in Section (2.2).
We first report on a calculation that was made as an introductory exercise to the FRG,
Section (2.2.1). We reproduce results that can be found in Ref. [155] and explain in detail
how the they are obtained. We consider the critical λ4 scalar field theory [156–158] and
apply the local potential approximation [120, 159, 160] to find the non-Gaussian RG
fixed point of the theory and compute its scaling exponents for d = 3. Section (2.2.1)
serves as well as an introduction to the RG method since it is relatively simple and
contains the most important ideas.
Section (2.2.2) represents original research work. We establish a set of FRG fixed
point equations that can be used when the space and time dependence of the two-point
correlation function is an essential property of the system. Indeed, the advective nonlinearity of fluid mechanics contains a spatial derivative which produces a rich momentum
dependence of correlation functions. For this reason the local potential approximation
does not provide good results in the context of hydrodynamic turbulence. This section
generalises the method first used in [161] in the context of thermal equilibrium YangMills theory [156, 158] to an out-of-equilibrium set-up.
Section 3 - Burgers Turbulence
The third part of this work is devoted to the stationary state of the stochastic Burgers
equation. The homogeneous Burgers equation is supplemented with a Gaussian random
forcing which exhibits a stationary two-point correlation function. Then even though
individual solutions of the forced hydrodynamic equation have a strong time dependence
1
ν is the kinematic viscosity of the fluid. The equation of motion of Z is linear.
6
1. Introduction
the problem becomes stationary when the average over the different realisations of the
forcing is taken.
We start with a brief introduction of the physics that we wish to describe and set up its
mathematical formulation. In Section (3.1) we review the results of the literature with
a focus on the scaling properties of different correlation functions. Next we establish a
path integral formulation of the steady state generating functional in Section (3.2). In
particular we compute its action (see Eqs. (3.42) or (3.43)).
Section (3.3) is the heart of the present work and contains exclusively original research. We apply the FRG and the fixed point equations established in Section (2.2.2)
to compute the two-point correlation function of the stochastic Burgers equation. We
establish an approximation scheme in Section (3.3.1) and write RG flow equations for its
parameters in Section (3.3.2). Next we adapt the fixed point equations of Section (2.2.2)
to our truncation in Section (3.3.3). The different quantities that we have introduced
in Sections (3.3.1) and (3.3.3) are related to physically observable quantities in Section
(3.3.4). Finally the RG fixed point equations are analysed in detail in Section (3.3.5)
and fixed points as well as their properties are discussed.
In Section (3.3.5.1) we solve the RG flow equations in the asymptotic regime of vanishing rescaled momentum. We find in all dimensions a continuum of fixed points supplemented with an additional isolated fixed point (see Figure (3.2)). Along the continua
of fixed points the value of the scaling exponents are extracted (see Eq. (3.101)) from
the asymptotic form of the RG fixed point equations. In Section (3.3.5.2) we study the
opposite asymptotic regime of infinite rescaled momentum. We identify a range of values
of the scaling exponents for which the fixed point theories are Ultraviolet (UV) convergent (see Figure (3.4)). Finally in Section (3.3.5.3) we give a physical interpretation for
the boundaries of this range in terms of the locality properties of the applied forcing
and the transport of kinetic energy. In particular we find that the fixed point theories
become UV divergent when a cascade of energy to the UV (direct cascade) sets in.
For d = 1 our results are in good agreement with known results. Based on the values
of the scaling exponents the fixed point that appear in the perturbative calculation of
[112] are contained within our continuum of fixed points. Moreover, in accord with [61]
we find that the continuum of fixed points extends further into the region where the
perturbative calculation breaks down. For d ̸= 1 the perturbative calculation can only
access the Gaussian fixed points which are trivially contained in our calculation since we
write fixed point equations without analysing their stability. We try however to compare
our results to those of [61] which is an FRG calculation of the scaling properties of the
stochastic KPZ equation. Based on the comparison of the obtained scaling exponents the
authors of [61] find more fixed points than we do. We attribute this discrepancy to the
fact that for d ̸= 1 Burgers’ and KPZ equations are only equivalent when the velocity
field of Burgers’ equation is the gradient of a potential and conclude that the fixed
points that we find describe physics that is closer that of NS equation which contains
vorticity. The isolated fixed point that we find in all dimensions appears no where else
in the literature. Assuming that it is not an artefact of our approximation scheme we
postulate that it is relevant in the case of forcing mechanisms that are strongly non local.
7
1. Introduction
Section 4 - Ultra-cold Bose Gases
In the fourth part of this work we change gears and consider the dynamics of dilute
Bose gases. We start by summarising relevant results concerning out-of-equilibrium
steady states in closed systems. In particular in the regime where occupation numbers are
large and the Gross–Pitaevskii equation (GPE) and the truncated Wigner approximation
apply. We introduce the concept of Non-Thermal Fixed Points (NTFP) and their scaling
properties in Section (4.1). In Section (4.2) we introduce the Driven-Dissipative Gross–
Pitaevskii Equation (DDGPE) as a model for a dilute gas of Bosons in contact with
external reservoirs of particles and energy. The inclusion of driving and dissipation into
the closed system enables its mapping to the stochastic Burgers equation which was
introduced in the Part 3.
We now go on to the original research.In Section (4.3) we assume that both approaches, the closed GPE and the open DDGPE, describe the same out-of-equilibrium
steady state and employ the mapping in between the DDGPE and Burgers’ equation
to apply the results of Section (3.3) and extract non-trivial scaling relations in the context of out-of-equilibrium Bose gases at a NTFP. This analysis produces two important
results.
In the context of classical hydrodynamics a scaling relation in between the two exponents of the stochastic Burgers equation (3.16) emerges because of the Galilei invariance
of the hydrodynamic theory. The first important result, which is detailed in Section
(4.3.1), comes from the translation of this scaling relation to a dual relation in between
the exponents of the Bose gas (see Eq. (4.22)). In conjuncture with the scaling relations
of [18] this reduces the number of unknown exponents characterising the NTFP of the
ultra-cold Bose gas to one. In particular this makes it possible to identify the scaling
exponent of the kinetic energy spectrum of the energy cascade as containing a canonical
Kolmogorov scaling exponent of −5/3 with an unknown anomalous correction.
Section (4.3.2) contains the second important result. It comes from translating the
values of the scaling exponents of Burgers’ equation computed in the KPZ literature
for d = 1, 2 and 3 to the ultra-cold Bose gas set-up. We find an interesting difference
between the dynamics of the phase of the Bose gas wave function and the traditional KPZ
equation. Indeed, even though they are formally identical, the latter can not be applied
to the former because it describes an unbounded field which can not represent a phase
angle. We compute anomalous corrections to the scaling of the compressible kinetic
energy spectrum of the Bose gas. See Eqs. (4.27) where these exponents are shown and
Figure (4.2) where they are compared to the results of a numerical simulation.
Appendices A to G
Most of the technical details are given in the appendices. Appendix A contains the
derivation of a recursion relation (2.57) which is necessary to compute the fixed point
properties of the λ4 scalar field theory in Section (2.2.1.2). Appendices B to D provide
many details on the fixed point analysis of Burgers’ equation. The derivation of the RG
8
1. Introduction
flow equations is discussed in Appendix B. Their rescaled form as well as the computation
of their asymptotic behaviour is given in detail in Appendix C. Appendix D contains the
derivation of sets of equations which are discussed in detail in Section (3.3.5.1) and used
to constrain the scaling exponents of the fixed points. In Appendix E we give details on
the decomposition of the kinetic energy spectrum of the ultra-cold Bose gas in order to
give a precise definition to the compressible kinetic energy spectrum which is discussed
in Section (4.3.2). Appendix F contains a list notation and conventions that are use
through out this work and may be a little fuzzy in the main text. Finally Appendix
G contains a reminder of the different acronyms and short-hand notations that we use
through out this work.
9
2. The Functional Renormalisation Group
In this section we briefly review the Renormalisation Group (RG) and its application to
critical phenomena [103–106]. In particular we focus on the Functional Renormalisation
Group (FRG) [120, 121]. We emphasise the properties of RG fixed points and explain
how to compute them.
We start with a short qualitative introduction to the RG and FRG. Precise definitions
are kept for Section (2.1). We turn to FRG fixed point equations in Section (2.2). We
start by reporting on a calculation that was made as an introductory exercise to the
FRG, Section (2.2.1). We reproduce results that can be found in Ref. [155] and explain
in detail how the they are obtained. Section (2.2.1) serves as well as an introduction
to the RG method since it is relatively simple and contains the most important ideas.
Finally we establish in Section (2.2.2), a set of FRG fixed point equations that can be
used when the space and time dependence of the propagator is an essential property of
the fixed point and can not be neglected as it is with the local potential approximation.
This generalises the method first used in [161] in the case of thermal equilibrium YangMills theory [156, 158] to an out-of-equilibrium set-up.
In most of this work we consider the theory of stochastic hydrodynamics. The dynamical variables are described by a real field of d components v(t, x) and depending on d+1
space-time variables (t, x). d is the dimension of space. We consider a particular unit
system where time has the dimension of space squared1 . Then the canonical dimension
of the velocity field v(t, x), is one over space.
RG transformations provide a link between effective descriptions of our system at
different levels of coarse-graining. We consider here field theories that are characterised
by a generating functional that can be expressed as a path integral
Z
R
Z[J ] = D [v] e−S[v]+ J ·v ,
(2.1)
with a given action functional S[v]. We use the notation D [v] = Πω,p dd v(ω, p). As usual
correlation functions of the velocity field are obtained by taking derivatives of Z[J ] and
setting J = 0,
⟨vi1 (t1 , x1 ) · .. · vin (tn , xn )⟩ =
δ n Z[J = 0]
.
δJi1 (t1 , x1 )..δJin (tn , xn )
(2.2)
Note that more precise definitions are given in Appendix F. We introduce a momentum
cut-off scale k and write
Z
R
Z[J ] =
Π dd v(ω, p) e−Sk [v]+ J ·v .
(2.3)
ω,p<k
1
This implies that the kinematic viscosity is dimensionless. See Section (3).
10
2. The Functional Renormalisation Group
The functional integration over momentum scales larger than the cut-off k is absorbed
into the exponential of the Wilson effective action, Sk [v] [104]. The cut-off dependence
of Sk [v] is defined by Eq. (2.3) in such a way that the integration over momentum scales
larger than k does not have to be performed any more. By definition the RG transformation does not affect the observable physics. In order for Z[J ] to be independent of
the cut-off scale we need to change the action with k in just the right way. Sk [v] can be
interpreted physically as an effective action for the theory coarse-grained at the cut-off
scale. For momentum scales smaller than k correlation functions are computed as in
Eq. (2.2) except that the momenta that enter the calculation are restricted to be smaller
than k and Sk [v] is used instead of S[v]. On the other hand, for momentum scales
larger than k no functional integration is necessary any more. Correlation functions are
directly given by the derivatives of Sk [v].
Note that we do not cut off the frequency dependence. Spatial fluctuations of the
velocity field are truncated if they are smaller than 1/k while temporal fluctuations over
time are left completely free. Since the coarse-graining is done under the functional
integration the physical quantities do not depend on the way in which we implement the
cut-off. This does however make a difference when approximations are made. One great
advantage of the RG technique is that fluctuations are included gradually in such a way
that we do not encounter divergences at as we go from one level of coarse-graining to
another. It was found that only cutting off the spatial fluctuations does not spoil this
property (see e.g. [55, 57–61, 91, 94, 96, 112, 162–164]). We will see in Section (3.3)
that in the case of stochastic hydrodynamics described by Burgers’ equation this is not
necessary as well.
Before we discuss how Sk [v] is computed in practice, let us introduce the re-scaled
dimensionless variables
x̂ = x k,
p̂ = p k −1 ,
t̂ = t k 2 Z1 ,
ω̂ = ω /(k 2 Z1 ),
p
Ĵ = J k −d−1 Z2 .
p
v̂ = v 1/(k Z2 ),
(2.4)
The k-dependent rescaling factors Zi will be introduced shortly. With these our generating functional takes the form2
Z
R
(2.5)
Z[J ] =
Π dd v̂(ω̂, p̂) e−Ŝk [v̂]+ t̂,x̂ Ĵ (t̂,x̂)·v̂(t̂,x̂) .
ω̂,p̂<1
With the re-scaled variables the dependence on k disappears from the integral measure
and is entirely confined to Ŝk [v̂]. Then there is no reason to single out S[v] as compared
to all the other effective theories given by Ŝk [v̂] for different values of k. Ŝk [v̂] can be
used instead of S[v] as an effective action and leads to the same generating functional
Z[J ].
Note that there is a certain freedom in the re-scaling of Sk [v]. Indeed, since we use a
momentum cut-off space must be re-scaled with the cut-off in order for p̂ to be always
2
We use the short-hand notations
R
t,x
=
R
dt dd x and
11
R
ω,p
= (2π)−d−1
R
dω dd p.
2. The Functional Renormalisation Group
smaller than one, but the time and the field can be re-scaled with arbitrary cut-off
dependent factors. This is natural and usually implicit in the case of the original action
S[v]. Whatever the context S[v] always contains some operators with no associated
coupling. These have been set to one. Compare e.g. the terms containing time derivatives
in Eqs. (3.42) and (4.1) with the terms containing a Laplace operator. These couplings
are set to one because if they where not we could simply go to variables where they are.
In other words theories defined by S[v] and S̃[v] = S[λv] describe the same physics.
If we want to have equivalent theories to be all represented by a single Ŝk [v̂] along
the RG flow we need to impose RG conditions. Typically we set to one the first two
terms of the Taylor expansion of the momentum dependent inverse propagator of Sk [v],
(2)
(2)
i.e. Sk (0, 0) = 1 and ∂p2 Sk (0, 0) = 1. Such conditions make it possible to define Z1
and Z2 and re-scale the field and the time variables unambiguously. Once this is done
the RG flow is a property of the re-scaled quantities Sk [v̂] only. The re-scaling factors
Zi are "spectators" and can be expressed in terms of Sk [v̂]. Their logarithmic cut-off
derivative k∂k log(Zi ) = ηi (k) are the anomalous dimensions of the field and time. They
account for the anomalous scaling of correlation functions at fixed points.
In the theory of renormalisation, fixed points of the RG flow play a special role.
Indeed, if Ŝk [v̂] reaches a fixed point it looses its dependence on k. Then a change of
the cut-off scale only affects the dimension-full quantities of Sk [v] by re-scaling them.
A typical correlation function O(t, x; g) will depend on the space-time variables as well
as on the different parameters (or couplings) of the system g. By construction when
it is expressed in terms of re-scaled variables the re-scaled expression cancels out the
k-dependence with cut-off dependent couplings,
O(t, x; g) = k dO (Z2 )n/2 Ô(ω̂, p̂; ĝ(k)).
(2.6)
k dO carries the dimension of O(t, x, g) and n is the number of fields that it contains.
At an RG fixed point the parameters are numbers ĝ(k) = ĝ ∗ and the re-scaling factors
are simple scaling functions of the cut-off scale3 Zi = Zi0 k ηi . Then we find that the
correlation functions satisfy
O(t, x; g) = (Z20 )n/2 k dO +nη2 /2 Ô(tk 2+η1 , xk; ĝ ∗ ),
for all k > 0.
(2.7)
The freedom that we have to choose the cut-off scale is expressed as a scale invariance
of correlation functions. One can see this as a theory which describes a system that is
invariant under simultaneous coarse-graining and re-scaling of the microscopic degrees
of freedom. One averages over the finest details and tries to cancel this by scaling them
down to a size where they would not be observable any way. If one can do this perfectly,
then the system contains details on all scales which are all identical to each other.
The computation of the RG flow of Sk [v] gives access to all the information contained
in the generating functional. Indeed, it is apparent from Eq. (2.3) that in the limit
k → ∞ the flowing effective action reduces to the original action S∞ [v] = S[v]. On the
other hand, in the limit k → 0 all the fluctuations have been included. This feature
3
This is because the re-scaled equations for Zi only depend on ĝ(k), k∂k Zi /Zi = ηi (ĝ(k)).
12
2. The Functional Renormalisation Group
is particularly explicit when the RG flow is expressed in terms of the flowing effective
action Γk [v] which can be computed fro Sk [v] by a Legendre transform of the short
scale variable features of Sk [v]. See Section (2.1.1) for details. Γk [v] represents physical
observables on scales smaller than the inverse cut-off 1/k and acts as an effective action
on larger scales. As k is lowered it changes continuously from being equal to the original
action for k → ∞ to the 1 Particle Irreducible (1PI) effective action once the cut-off is
sent to zero. It can be defined through the Legendre transform of the coarse-grained
Schwinger functional Wk [J ] = log(Zk [J ]),
Z
Γk [v] = sup − log (Zk [J ]) + J · v − ∆Sk [v].
(2.8)
J
Zk [J ] is computed in the same way as Z[J ] in Eq. (2.1) with the difference that the
action is supplemented with a cut-off term,
Z
1
S[v] → S[v] + ∆Sk [v]
∆Sk [v] =
v(ω, p) · v(−ω, −p) Rk (p).
(2.9)
2 ω,p
Rk (p) is a positive function that is very large for p ≪ k and very small otherwise. See
section 2.1.4 or [165] for more details.
In practice, the exact computation of an RG transformation is almost always impossible. An appropriate approximation scheme must be devised. Typical RG transformations project Sk [v] on a finite set of flowing parameters that characterise the theory
at different scales. In its FRG formulation this projection is done rather late in the
calculation. Indeed, an exact differential equation [121] can be written for Γk [v],
−1
1
(2)
k∂k Γk [v] = Tr Γk [v] + Rk
k∂k Rk .
(2.10)
2
Eq. (2.10) is then used to project the RG flow on an appropriate set of parameters and
can be used to include as much non-perturbative effects as possible. Rk (p) is a positive
function that is zero in the limit k/p → 0 and diverges when k/p → ∞. It acts as a
momentum dependent mass term that cuts off field fluctuations with momentum smaller
than k. See Section (2.1.3) for precise definitions.
The RG approach to scaling and critical phenomena was first introduced in [103–
106]. See e.g. [4, 109, 110] for introductory texts. The FRG provides a non-perturbative
framework to implement the coarse-graining inherent to the RG. As opposed to the usual
field theoretical perturbative RG it takes into account irrelevant operators and provides
physical information far away from fixed points. The flow equation for the effective
action Eq. (2.10) was introduced in [120, 121] and provides an intuitive framework to
compute RG transformations since the derivatives of Γk→0 [v] are directly related to
physical observables while they can be interpreted as effective couplings at a finite cut-off
value. Moreover the flow of Γk [v] is reversible. Even though the small scale fluctuations
are coarse-grained infinitely high order correlation functions are taken into account such
that no memory is lost as k is decreased.
13
2. The Functional Renormalisation Group
2.1. Effective action
In this section we give some details on the definition, interpretation and use of Γk [v]
and its k → 0 limit. We relate Wk [J ] and Γk [v] to Sk [v] through different Legendre
transformations. This clarifies the use of Wk [J ] and Γk [v] as coarse-grained quantities.
As in the previous section we use a sharp cut-off because it makes the coarse-graining
procedure more intuitive. We will however arrive at expressions that do not depend
on the particular choice of the cut-off and are valid for smooth cut-off’s as well. See
e.g. [119, 166] and references therein for overviews.
We start by giving a precise meaning to Eq. (2.3). Our starting point is the generating
functional of velocity correlation functions defined in Eq. (2.1). In order to implement
the coarse-graining of small scale fluctuations, we introduce the cut-off scale k and
break the functional integration in two parts. The small scale (as compared to 1/k)
velocity fluctuations are integrated first while the large scale features of v(t, x) are kept
as parameters. Then the integration over the remaining field variables is performed,
Z
R
Z[J ] =
Π dd v(ω, p) Π dd v(ω, p) e−S[v]+ J ·v
ω,p<k
ω,p>k
Z
R
≡
Π dd v(ω, p) e−Sk [v< ,J > ]+ J < ·v< .
(2.11)
ω,p<k
We have used the notation X >,< defined in Fourier space,
X > (ω, p) = θ(p − k) X(ω, p),
X < (ω, p) = θ(k − p) X(ω, p).
(2.12)
It denotes the Fourier truncated field. X < is a coarse-grained version of X. It is identical
to X on large scales but is smooth on scales smaller than 1/k. X > is the opposite. It
contains all the fine details of X but none of its overall features. Sk [v < , J > ] depends
the smooth (over scales smaller than k) features of v and the sharp details of J . It is
an effective action where all the fluctuations over scales smaller than 1/k are already
included,
Z
R
(2.13)
e−Sk [v< ,J > ] =
Π dd v(ω, p) e−S[v]+ J > ·v> .
ω,p>k
Since v and J are conjugate variables, we can take the Legendre transformation of
Sk [v < , J > ] with respect to one of its field variables and recover a functional of the full,
not truncated, other. We show in the following two Sections that the Legendre transform
of Sk [v < , J > ] with respect to J > is the flowing effective action which interpolates from
the original action S[v] for k → ∞ and the 1PI effective action for k → 0 and that
its Legendre transform with respect to v < is closely related to the flowing Schwinger
functional.
14
2. The Functional Renormalisation Group
2.1.1. Flowing effective action
We start by changing variables from J > to v > . We take the Legendre transform of
Sk [v < , J > ] with respect to J > and define the flowing effective action Γk [v],
Z
(2.14)
Γk [v] = sup Sk [v < , J > ] +
J > · v> ,
J>
When this is inserted in Eq. (2.11) we get,
Z
R
Z[J ] =
Π dd v(ω, p) e−Γk [v]+
ω,p<k
J ·v
.
(2.15)
Note that the functional integration is only performed on v < because of the cut-off. On
the other hand Γk [v] depends on the full v field. On the right hand side of Eq. (2.15) Γk [v]
is actually an implicit function of J > through the relation δΓk [v]/δv > = J > . Eq. (2.15)
can be reformulated in such a way that the integration measure is not restricted,
"
#
2
Z
Z
Z 1
δΓ
[v]
k
J(ω, p) −
R̃k (p) . (2.16)
J ·v−
Z[J ] = D[v] exp −Γk [v] +
2 ω,p δv(ω, p) We have introduced R̃k (p) which is zero for p < k and very large otherwise. It makes it
possible to enforces the cut-off without restricting the integration measure through the
identity
"
#
2
Z δΓk [v]
1
1
δΓ
[v]
k
J(ω, p) −
θ(p − k) . (2.17)
δ J> −
exp − 2
= lim
σ→0 N (σ)
δv >
2σ ω,p δv(ω, p) and the identification R̃k (p) = limσ→0 θ(p − k)/σ 2 . N (σ) is a normalisation factor
which can be reabsorbed into the path integral measure. Note that the integrand inside
the exponential of Eq. (2.17) actually only contains the Fourier truncated quantities
J> − δΓk [v]/δv > since it is multiplied by θ(p − k).
One can see from Eq. (2.16) what happens with Γk [v] in both limits k → 0 and k → ∞.
If the cut-off k is sent to infinity, we have R̃k→∞ (p) = 0. Then all the coarse-graining is
gone and we recover Eq. (2.1). We have Γk→∞ [v] = S[v]. On the other hand, if k → 0
we have R̃k→0 (p) → ∞ for all p and the delta distribution of Eq. (2.17) acts on the full
field v(t, x) instead of its short scale features only. Then there are no fluctuations of
v(t, x) left in the path integral and we have instead δΓk→0 [v]/δv = J for all momenta.
Only the average velocity field remains. Γk→0 [v] = Γ[v] is the 1PI effective action.
Eq. (2.16) shows as well that all the fluctuations of v < are present in the path integral
while v > is fixed to its average value v > = δSk [v < , J > ]/δJ > . Γk [v] generates physical
correlation functions on scales smaller than 1/k. On large scales however, it acts as
an effective action and can be used to replace S[v] in Eq. (2.1) if the fluctuations with
momentum larger than k are cut off.
15
2. The Functional Renormalisation Group
2.1.2. Flowing Schwinger functional
We now go back to Eq. (2.13). This time we change variables from v < to J < ,
Z
W̃k [J ] = sup −Sk [v < , J > ] +
J < · v< .
(2.18)
When this is inserted in Eq. (2.13) we get
Z
R
W̃k [J ]
e
=
Π dd v(ω, p) e−S[v]+
(2.19)
v<
ω,p>k
J ·v
.
This time it is v < in the integrand that implicitly depends on J < through δ W̃k [J ]/δJ < =
v < . Note that the Legendre transform is not defined with the same sign as the transform
with respect to J > . We make this choice of the definition of W̃k [J ] so that it is the
Legendre of Γk [v]. As before we can shift the cut-off from the integration measure to
the integrand


2
Z
Z
Z 1
δ W̃k eW̃k [J ] = D[v] exp −S[v] +
J ·v−
v(ω, p) −
Rk (p) . (2.20)
2 ω,p δJ (ω, p) Rk (p) is zero for p > k and infinite otherwise. It is an Infrared (IR) cut-off. Note the
difference with the Ultraviolet (UV) cut-off, R̃k (p).
The flowing Schwinger functional is defined by subtracting the term
Z
1
v(ω, p) · v(−ω, −p) Rk (p),
(2.21)
∆Sk [v] =
2 ω,p
to the action S[v] in Eq. (2.11) and taking the logarithm of the obtained coarse-grained
generating functional,
Z
R
Wk [J ]
e
= D[v] e−S[v]−∆Sk [v]+ J ·v .
(2.22)
We see that W̃k [J ], the functional that we obtain through the Legendre transform of
Sk [v < , J > ], is slightly different from Wk [J ]. It is however clear that W̃k [J ] is the Legendre transform of Γk [v]. Then it is no surprise that W̃k [J ] ̸= Wk [J ]. Indeed, in order to
recover the Schwinger functional from the flowing effective action an additional factor
of ∆Sk [v] must be added to Γk [v] before its Legendre transformation. See Eqs. (2.25)
and (2.26). One can write W̃k [J ] as a double Legendre transform of Wk [J ] with a factor of ∆Sk [v] inserted in between the two transformations and recover Eq. (2.20) from
Eq. (2.22),
Z
W̃k [J ] = sup −Γk [v] + v · J
v
Z
Γk [v] = sup −Wk [J ] + v · J − ∆Sk [v].
(2.23)
J
16
2. The Functional Renormalisation Group
Γk [v] and W̃k [J ] where both defined with a sharp cut-off. Note that the cut-off
procedure has been entirely shifted into the definitions of R̃k (p) and Rk (p) in Eqs. (2.16)
and (2.20). We are actually free to choose smooth functions instead of R̃k (p) and Rk (p).
It is only important that they are positive, that R̃k (p < k) and Rk (p > k) be very small
and R̃k (p > k) and Rk (p < k) be very large.
The sharp cut-off makes the coarse-graining procedure more intuitive and helps to
understand what is going on. Since it was introduced as an intermediate step in the
computation of Z[J ], it will in principle not affect the outcome of the computation of
physical quantities. However it does not interact well approximation schemes because
it is highly non-analytic. When an approximation is made in the computation of Z[J ]
the error depends on the cut-off that we choose and may be large with a sharp cut-off.
In practice it is better to use smooth cut-off functions instead of R̃k (p) and Rk (p). In
the following we will simply keep Rk (p) and R̃k (p) as free parameters until it becomes
necessary to choose a specific form for one of them in Sections (2.2.1.1) and (3.3).
2.1.3. Flow equation
We have used Eqs. (2.13) and (2.14) to define Γk [v]. A somewhat simpler, although
equivalent, definition is written in terms of the flowing Schwinger functional Wk [J ].
We introduce a positive cut-off function Rk (p) that satisfies with Rk (p ≫ k) = 0 and
Rk (p ≪ k) = ∞ and define Wk [J ] as
Z
Z
R
1
Wk [J ]
−S[v]−∆Sk [v]+ J ·v
e
= D[v] e
, ∆Sk [v] =
v(ω, p) · v(−ω, −p) Rk (p).
2 ω,p
(2.24)
As we discussed in the previous section, Γk [v] is not exactly the Legendre transform of
Wk [J ]. Instead we have
Z
Γ̃k [v] = sup −Wk [J ] + J · v ,
(2.25)
J
and the flowing effective action is
Γk [v] = Γ̃k [v] − ∆Sk [v].
(2.26)
On can check from this definition of Γk [v] that Γk→∞ [v] = S[v] and Γk→0 [v] = Γ[v].
Moreover we have seen in Section (2.1.1) that Γk [v] is a coarse-grained effective action.
This definition can be used to write a differential equation for the flowing effective
action. Indeed, Γk [v] changes continuously with k. The cut-off derivative of Wk [J ] can
be computed directly form Eq. (2.24). Then this can be inserted into Eq. (2.26) in order
to write,
−1
1
(2)
k∂k Γk [v] = Tr Γk [v] + Rk
k∂k Rk .
(2.27)
2
17
2. The Functional Renormalisation Group
(2)
See e.g. [119, 121–123] for a detailed derivations and overviews. Γk [v]+Rk is the inverse
of the propagator computed from Wk [J ]. It satisfies
(2)
(2)
Wk [J ] Γk [v] + Rk = 1,
(2.28)
with
(2)
Wk,ij [J ](t, x; t′ , x′ ) =
δ 2 Wk [J ]
,
δJi (t, x)δJj (t′ , x′ )
(2)
Γk,ij [v](t, x; t′ , x′ ) =
δ 2 Γk [v]
,
δvi (t, x)δvj (t′ , x′ )
(2.29)
and
(2)
Rk,ij (t, x; t′ , x′ )
′
Z
′
ei[ω(t−t )−p·(x−x)] Rk (p),
= δij
ω,p
′
1ij (t, x; t , x ) = δij δ(t − t′ )δ(x − x′ ).
(2.30)
The product and the trace of two operators are defined straightforwardly as
Z
Z
′
′
′′
′′
′′
′′ ′
′
AB(t, x; t , x ) =
A(t, x; t , x )B(t , x ; t , x ), Tr (A) =
A(t, x; t, x). (2.31)
t′′ ,x′′
t,x
Eq. (2.27) is a functional differential equation for Γk [v]. It contains the full dependence
of Γk [v] on v(t, x) and can be used to extract equations for all of the derivatives of the
flowing effective action. Note that it couples Γk [v] to its second field derivative. The
n-th derivative of Eq. (2.27) will contains derivatives of Γk [v] up to order n + 2. We
(n)
have an infinite hierarchy of equations for all the Γk .
In practice Eq. (2.27) can only be approximately solved. Typically one writes an
ansatz for Γk [v] in terms of unknown parameters which can be extracted by taking
appropriate derivatives of the ansatz (see e.g. Eqs. (2.43) and (3.54)). Then one can
obtain differential equations for the parameters by taking the same derivative on both
sides of Eq. (2.27). See Eqs. (2.49) and (3.60) (with Eqs. (3.61) and (3.62)). The different
parameters of the ansatz are couples to each other through the coupling of the different
derivatives of Γk [v] and obey complex non-linear differential equations because of the
structure of Eq. (2.27). In most cases these equations need to be treated numerically.
2.1.4. One particle irreducible effective action
We can see from Eq. (2.24) that Γk→0 [v] ≡ Γ[v] is the Legendre transform of the
Schwinger functional, W [J ] which generates all the connected correlation functions of
the velocity field. This information is equivalently contained in Γ[v]. Here, we briefly
show how correlation functions are computed from Γ[v]. See e.g. [165] for a detailed
exposition.
Once one has obtained a good estimation for Γ[v] the first step is to determine the
average field, ⟨v(t, x)⟩ by solving the equation, δΓ[v = ⟨v⟩]/δv = 0. This is equivalent to
18
2. The Functional Renormalisation Group
⟨v⟩ = δW [J = 0]/δJ . In the case of stochastic hydrodynamics, we get ⟨v⟩ = 0 because
of rotational symmetry.
The two-point correlation function is the inverse of the second derivative of Γ[v] evaluated at ⟨v⟩,
−1
δ 2 Γ[v = ⟨v⟩]
′
′
⟨vi (t, x)vj (t , x )⟩c =
.
(2.32)
δvi (t, x)δvj (t′ , x′ )
Note that this is a equivalent to Eq. (2.28) in the limit k → 0. We include a c index to differentiate the disconnected correlation functions, generated by Z[J ], from the
connected ones which are generated by W [J ].
Finally, higher order correlation functions can be computed from higher order derivatives of Γ[v]. Indeed, we can express n-point correlation functions using n − 2 functional
derivatives of the two-point function with respect to J ,
⟨v i1 (t1 , x1 ) v i2 (t2 , x2 ) . . . v in (tn , xn )⟩c
δ n−2
δ 2 W [J = 0]
δJi1 (t1 , x1 )..δJin−2 (tn−2 , xn−2 ) δJin−1 (tn−1 , xn−1 )δJin (tn , xn )
−1
δ n−2
δ 2 Γ[v = ⟨v⟩]
=
.
δJi1 (t1 , x1 )..δJin−2 (tn−2 , xn−2 ) δvin−1 (tn−1 , xn−1 )δvin (tn , xn )
=
(2.33)
This can be expressed in terms of the derivatives of Γ[v] by using the chain rule for
derivatives,
δA−1 (t, x; t′ , x′ )
δA
−1
−1
=− A
A
(t, x; t′ , x′ ).
(2.34)
δJ(t′′ , x′′ )
δJ(t′′ , x′′ )
Note that the last line of Eq. (2.33) contains an additional internal derivative. Indeed,
Γ[v] depends on J (t, x) only implicitly through v(t, x) and the relation δΓ[v]/δv = J .
This produces an additional propagator multiplying the whole equation. In particular
we have
⟨v i1 (t1 , x1 ) v i2 (t2 , x2 )v i3 (t3 , x3 )⟩c
−1
Z
δ 3 Γ[v = ⟨v⟩]
δ 2 Γ[v = ⟨v⟩]
=−
δvi1 (t1 , x1 )δvj1 (τ1 , z 1 )
τ1 ,z 1 ;τ2 ,z 2 ;τ3 ,z 3 δvj1 (τ1 , z 1 )δvj2 (τ2 , z 2 )δvj3 (τ3 , z 3 )
−1
−1
δ 2 Γ[v = ⟨v⟩]
δ 2 Γ[v = ⟨v⟩]
×
.
(2.35)
δvi2 (t2 , x2 )δvj2 (τ2 , z 2 )
δvi3 (t3 , x3 )δvj3 (τ3 , z 3 )
One can apply the same procedure to Γk [v] and extract scale dependent correlation
functions. These already represent physics on spatial scales larger than the inverse cutoff. We append a k index to such correlation functions,
2
−1
δ Γk [v = ⟨v⟩k ]
′
′
.
(2.36)
⟨vi (t, x)vj (t , x )⟩k,c =
δvi (t, x)δvj (t′ , x′ )
Note that because of the additional cut-off term these do not correspond to correlation
functions computed from Wk [J ] (see Eq. (2.26)).
19
2. The Functional Renormalisation Group
2.2. Functional renormalisation group fixed point equations
The greatest strength of the FRG is that its starting point Eq. (2.27), is exact and can
be used as a starting point to make non-perturbative approximations. In practice the
flowing effective action must be truncated since Eq. (2.27) generates an infinite hierarchy
of differential equations for all of the derivatives of Γk [v]. Typically one uses physical
intuition and symmetry constraints to write an ansatz for the flowing effective action,
Γk [v]. Such an ansatz should contain unknown parameters which depend on the cut-off
scale, k. Then the exact functional equation Eq. (2.27) is projected on the subset of
effective actions that the ansatz is capable of generating. This procedure leads to a set
of differential equations for the parameters of the ansatz which can be handled with
standard techniques [167–169].
It is clear that the quality of the outcome of any FRG calculation will strongly depend
on the quality of the chosen ansatz. Γk [v] should not explicitly break the symmetries
of the problem at hand. If this is the case the symmetry breaking terms will lead to
either trivial or wrong flow equations since they do not appear in the exact solution.
However within the symmetry constraints Γk [v] should be as general as possible. Then
the ansatz is usually simplified based on the physical process one is modelling. The
truncated flowing effective action should be completely general (at least) in the physical
sector which one hopes to resolve.
In choosing an appropriate ansatz, there are two different paths one can take. If one is
mainly interesting in a precise evaluation of the correlation functions at equal positions
and times,
λn = ⟨vi1 (t, x)vi2 (t, x) . . . vin (t, x)⟩,
(2.37)
the momentum dependence of the parameters of Γk [v] is not so important. One can
make a derivative expansion. I.e. the derivatives of the truncated effective action are
assumed to be polynomials in the momentum and only the smallest powers of p are kept.
Within this approximation it is possible to model correlation functions with very large
number of fields by including very large powers of v(t, x) into the ansatz. This will be
discussed in Section 2.2.1.
On the other hand, if one is more interested in the dependence of the correlation
functions on space and time,
F (t − t′ , x − x′ ) = ⟨vi (t, x)vj (t′ , x′ )⟩,
(2.38)
such an approximation will not produce good results. In this case it is better to use an
ansatz with as much momentum dependence as possible. It then becomes necessary to
consider only a small numbers of powers of v(t, x) and the approximation becomes bad
when high order correlation functions are computed. This is discussed in Section 2.2.2.
RG fixed point can of course be studied within both approaches. We show how
this can be done the two following sections. The Blaizot–Mendez–Wschebor (BMW)
approximation scheme [170–173] tries to bridge these two approaches by truncating the
momentum dependence of correlation functions of order larger than a given order s while
20
2. The Functional Renormalisation Group
considering the full momentum dependence of lower order order correlation functions.
The RG flow equations are evaluated at a an arbitrary constant field v(t, x) = w. They
are closed by equating the field derivatives of order s + 1 and s + 2 of the effective action
to the derivative with respect to the constant field of the derivative of order s,
∂
(s)
Γk [w](ω1 , p1 ; ..; ωs−1 , ps−1 ),
∂wis+1
(s+1)
[w](ω1 , p1 ; ..; ωs , ps ) →
(s+2)
[w](ω1 , p1 ; ..; ωs+1 , ps+1 ) →
Γk
Γk
∂2
(s)
Γk [w](ω1 , p1 ; ..; ωs−1 , ps−1 ).
∂wis+1 ∂wis+2
(2.39)
Such an identification is exact when the additional momentum variables on the lefthand side are set to zero. Functions of at most s different momentum variables are
taken into account. One extracts RG flow equations for correlation functions of order
smaller than s which are fully momentum dependent while the higher order derivatives
of Γk [v] are taken into account by the fact that the lower order derivatives still depend
on a constant field. Such a procedure has been applied successfully for O(N) models at
thermal equilibrium and s = 2. See [173, 174] for detailed discussions. To the best of
our knowledge it has not been generalised to a non-equilibrium set-up yet.
2.2.1. Local potential approximation
The local potential approximation [120, 159, 160] is the outcome of lowest order truncation of the derivative expansion of the effective action with the additional approximation
of setting the anomalous dimension of the field to zero. In situations where it is acceptable to neglect the full momentum dependence of the field derivatives of the effective
action, these can be Taylor expanded in their momentum variables around p = 0. The
flowing effective action can be expanded in powers of the velocity field,
Γk [v] =
∞ Z
X
n=1 ω1 ,p1 ;..;ωn ,pn
vi1 (ω1 , p1 )..vin (ωn , pn ) (n)
Γk,i1 i2 ..in (ω1 , p1 ; ..; ωn , pn ).
n!
(2.40)
(n)
Γk,i1 i2 ..ii (ω1 , p1 ; ..; ωn , pn ) are the field derivatives of Γk [v] evaluated at zero field and
multiplied by (2π)n(d+1) . They contain, if computed exactly, all the information contained in Γk [v]. In a system that is invariant under translations we have
(n)
Γk,i1 i2 ..in (ω1 , p1 ; ..; ωn , pn ) = (2π)d+1 δ (p1 + .. + pn ) δ (ω1 + .. + ωn )
(n)
× Γk,i1 i2 ..in (ω1 , p1 ; ..; ωn−1 , pn−1 ).
(2.41)
(n)
The derivative expansion consists in approximating Γk,i1 i2 ..in (ω1 , p1 ; ..; ωn−1 , pn−1 ) by
their truncated Taylor expansion in frequency and momentum around (ω, p) = 0.
In this section we show how to compute fixed point properties to lowest order in the
derivative expansion. This work was performed as an introduction to the FRG. The
results of [155] were reproduced as an exercise. Extensive discussions on the derivative
21
2. The Functional Renormalisation Group
expansion as well as very precise calculations can be found in [109, 120, 122, 159, 160,
175].
Here we focus on the equilibrium critical properties of a scalar field in d dimensions
φ(x). We leave aside for a while our space and time dependent velocity field because,
as will be discussed in Section 3.3.1, the derivative expansion is not the right way to go
in the case of a hydrodynamic theory. We consider the following action,
Z
Z
1
S[φ] =
p2 + m2 φ(p)φ(−p) + λ φ(x)4 .
(2.42)
2 p
x
It corresponds to the φ4 model which is a well known toy model in quantum field theory
[156–158]. It can be used to describe the thermodynamics of the bosonic scalar field
φ(x) and has many physical realisations ranging from the Higgs mechanism to water. It
contains a phase transition from an ordered ⟨φ⟩ = 0 to a disordered ⟨φ⟩ =
̸ 0 state which
is in the same universality class as the Ising model [176]. See for example [157, 158, 177]
and references therein. Because of the symmetry of S[φ] under spatial rotations, the
lowest order in the derivative expansion is the second one. The corresponding ansatz for
the flowing effective action is
Z
1
Γk [φ] =
Zk (φ(x)) ∇φ(x) · ∇φ(x) + Uk (φ(x)).
(2.43)
x 2
Zk (φ) and Uk (φ) are two unknown functions of the field and the cut-off scale which
can be determined by inserting the ansatz into Eq. (2.27). Before we continue we will
make the further approximation that Zk (φ) = 1. This is justified because the derivative
expansion works best in situations where the anomalous dimension of the scalar field, η
is small. We can neglect the changes of Zk (φ) with the cut-off scale since they are of
higher order in η as compared to the changes of Uk (φ) [175].
2.2.1.1. Renormalisation group flow equation
We are now ready to write an RG flow equation for Uk (φ). This is done by evaluating
the flow equation Eq. (2.27) at a constant field, φ(x) = φ. Indeed, the second field
derivative of the ansatz, Eq. (2.43) reads
Z 2
δ 2 Γk
1
1
∂ Uk
2
[φ] =
p
δ(p
+
q)
+
(φ(x)) e−ix·(p+q) .
(2.44)
2φ
d
2d
δφ(p)δφ(q)
∂
(2π)
(2π)
x
Then choosing a field that does not depend on the spatial variable and adding the cut-off
term provides,
∂ 2 Uk
(2)
2
(2.45)
Γk [φ](p, q) + Rk (p)δ(p + q) = p + 2 (φ) + Rk (p) δ(p + q).
∂ φ
We use the Litim cut-off [178, 179],
Rk (p) = k 2 − p2 θ k 2 − p2 ,
22
(2.46)
2. The Functional Renormalisation Group
which minimises the length of the paths that the couplings follow, and therefore the
error they accumulate because of the truncation, as the cut-off scale is sent to zero [118].
θ(x) is the usual step function. It is one if its argument is positive and zero otherwise.
Within the local potential approximation this cut-off function also has the remarkable
property of removing all the momentum dependence from Eq. (2.45) when p ≤ k.
Finally the flow equation reads
Z
Z
(2π)d
k∂k Uk (φ) = δ(0) k 2 θ(k 2 − p2 ) 2
.
(2.47)
k + ∂φ2 Uk (φ)
x
p
Three remarks are in order here. First, the k 2 θ(k 2 − p2 ) term in the
numerator on the
right hand side is a result of taking the cut-off derivative of k 2 − p2 in Rk (p). There is
in principle an additional term coming from taking the derivative of the theta function
in Eq. (2.46). This term however vanishes since it is the product of a delta function
and its argument. Secondly, the trace on the right-hand side of Eq. (2.27) contains a
momentum integration which is regulated by the θ(k 2 − p2 ) term. It leads to a simple
volume factor,
Z
kd
k d π d/2
θ(k 2 − p2 ) ≡ Ωd
=
,
(2.48)
d
(2π)d Γ(d/2 + 1)
p
since by virtue of the Litim cut-off its integrand does not depend on the loop momentum.
Finally, there are two infinities that cancel each other. On the left hand side, evaluating
Γk [φ] at a constant field produces an integrand which is independent of x in the last
term of Eq. (2.43). Then we are left with Uk (φ) multiplied by an integral over all
space. This volume factor is cancelled by the δ(0) on the right hand side. The latter
term arises because of invariance of our system under spatial translations. Indeed, when
such a symmetry holds, any function of multiple momentum variables is proportional to
δ(p1 + .. + pn ). In our case the trace on the right-hand
R side of Eq. (2.27) does not have
−d
any momentum argument and leads to δ(0) = (2π)
p . Taking these two remarks into
account we write
k∂k Uk (φ) =
k d Ωd /d
.
k 2 + ∂φ2 Uk (φ)
(2.49)
We emphasize here that, local potential approximation has enabled us to reduce the functional differential equation Eq. (2.27) to a partial differential equation for an arbitrary
potential Uk (φ).
In order to look for RG fixed points, the next step is to go to re-scaled variables,
p
x̂ = xk,
φ̂(x̂) = k (2−d)/2 φ(x̂/k),
uk (ρ) = k −d Uk (k (d−2)/2 2ρ).
(2.50)
We have introduced the variable ρ = φ̂2 /2 = k 2−d φ2 /2 which will simplify the following
calculations since our system is symmetric under the transformation φ(x) → −φ(x), see
Eq. (2.42). With u′k (ρ) = duk (ρ)/dρ, the RG flow equation for uk (ρ) finally reads
k∂k uk (ρ) = −duk (ρ) − (2 − d)u′k (ρ)ρ +
23
Ωd /d
.
+ u′k (ρ) + 1
2u′′k (ρ)ρ
(2.51)
2. The Functional Renormalisation Group
2.2.1.2. Renormalisation group fixed point
Now that we have written down the RG flow equation for the re-scaled potential uk (ρ)
we can look for fixed points. At an RG fixed point the effective potential must satisfy
k∂k u(ρ) = 0, i.e.
d u(ρ) = (d − 2)u′ (ρ)ρ +
Ωd /d
.
+ u′ (ρ) + 1
(2.52)
2u′′ (ρ)ρ
In order to solve Eq. (2.52) we expand u(ρ) in a power series around its minimum,
X λn
du n
u(ρ) = λ0 +
= 0.
(2.53)
(ρ − ρ0 ) ,
n!
dρ ρ=ρ0
n=2
We have introduced the set of couplings λn , the mean field expectation value ρ0 , as well as
the minimum of the potential λ0 . All of these couplings can be computed from Eq. (2.52)
by taking the appropriate number of derivatives with respect to ρ and evaluating the
equation at ρ = ρ0 . First let us note that λ0 is completely determined by all the other
couplings but does not enter their determination. When Eq. (2.52) is evaluated at ρ = ρ0
one gets
λ0 =
Ωd
1
.
2
d 2λ2 ρ0 + 1
(2.54)
On the other hand, the right hand side of Eq. (2.52) does not depend on λ0 since it
contains only derivatives of u(ρ). Then λ0 drops out when Eq. (2.52) is differentiated.
Equations for λn and ρ0 are written recursively. We start by taking the first and
second derivative of Eq. (2.52) and evaluating them at ρ = ρ0 ,
Ωd 2λ3 ρ0 + 3λ2
,
d (2λ2 ρ0 + 1)2
"
#
Ωd
(2λ3 ρ0 + 3λ2 )2
2λ4 ρ0 + 5λ3
dλ2 = (d − 2) (λ3 ρ0 + 2λ2 ) +
2
−
.
d
(2λ2 ρ0 + 1)3
(2λ2 ρ0 + 1)2
0 = (d − 2)λ2 ρ0 −
(2.55)
(2.56)
The equations for the other couplings take the form,
λn = Cn (ρ0 , λ2 , .., λn ) + An (ρ0 , λ2 , λ3 ) λn+1 + B(ρ0 , λ2 ) λn+2 ,
n ≥ 3,
(2.57)
with Cn , An and Bn being three functions of the couplings which are determined in
Appendix A and given explicitly in Eqs. (A.9), (A.10) and (A.8) with (A.11).
Such a recursive form of the fixed point equation simplifies greatly the search for its
parameters. Indeed, the coordinates of a fixed point are given by ρ0 and as many λn as
one is able to take into account. This means that if one were to search for solutions of
Eq. (2.52) in a straightforward way, one would have to search for the correct values of the
coupling within a very high dimensional space. This is numerically very demanding and
become practically impossible as we take more couplings into account. The recursion
24
2. The Functional Renormalisation Group
relation given in Eq. (2.57) enables us to relate λn>2 to ρ0 and λ2 from the onset. Then
there are only two unknown parameters left and it becomes easy to scan all their possible
values and find the ones which provide a solution to Eq. (2.52).
λn>2 can be related to ρ0 and λ2 in a recursive way. First, one can simply isolate λ3
from Eq. (2.55). Then, this value of λ3 can be inserted in Eq. (2.56) and λ4 is isolated
in the same way. Next the recursion relation (2.57) enters. Setting n = 3 we get
λ5 =
λ3 − C3 (ρ0 , λ2 , λ3 , λ4 ) − A3 (ρ0 , λ2 , λ3 ) λ4
.
B(ρ0 , λ2 )
(2.58)
λ5 is now a function of ρ0 and λ2 only since λ3 and λ4 are already expressed in terms of
these quantities through Eqs. (2.55) and (2.56). The next step is to insert n = 4 in the
Eq. (2.57) and isolate λ6 is the same way as we did with λ5 . We can keep on going in this
way and, in principle express all the λn in terms of ρ0 and λ2 . In practice, however we
need to stop at some value of n. I.e. we need to further approximate u(ρ) by truncating
the sum in Eq. (2.53) at a given power N ,
u(ρ) ∼
= λ0 +
N
X
λn
n=2
n!
(ρ − ρ0 )n .
(2.59)
Then we can go on applying the recursion relation all the way up to n = N − 2. Notice
that the recursion relation evaluated at n provides and expression for λn+1 in terms of
λ2 and ρ0 . The as we increase n we find that λn=3..N are all expressed in terms of ρ0
and λ2 when we have reached n = N − 2. There are still two equations, for n = N − 1
and n = N , which have not been used yet. They fix the last two unknown parameters ρ0
and λ2 . Indeed, with a truncated sum the recursion relation for n = N − 1 and n = N
becomes
λN −1 = CN −1 (ρ0 , λ2 , ..., λN −1 ) + AN −1 (ρ0 , λ2 , λ3 ) λN ,
λN = CN (ρ0 , λ2 , ..., λN ).
(2.60)
The additional terms of Eq. (2.57) which contain λN +1 and λN +2 have been set to zero.
These two remaining equations are now a system of two equations for the two unknowns
ρ0 and λ2 only.
All the steps that we just outlined can be performed numerically for a given set of
values of ρ0 and λ2 . Then it is possible to evaluate Eqs. (2.60) and check if they are
satisfied. If they are not, another set of values of ρ0 and λ2 can be chosen and the whole
procedure can be repeated until a solution of the whole set of equations is found. Such a
procedure was performed with MATLAB up to order N = 9 and for d = 3. The values of
the obtained couplings are shown in the first column of Table (2.1). In order to compare
with the results of [155], which are shown in the third column of Table (2.1), the field
and potential where re-scaled according to
d
d
Ωd
ρ̂ =
ρ,
û(ρ̂) =
u
ρ̂ .
(2.61)
Ωd
Ωd
d
25
2. The Functional Renormalisation Group
Couplings
λ1
ρ0
λ2
λ3
λ4
λ5
λ6
λ7
λ8
λ9
Re-scaled couplings
3.860821804198 · 10−3
3.064754051849 · 10−2
7.471529410168
1.045641876778 · 102
1.301298120673 · 103
−3.349979877176 · 103
−8.124166870475 · 105
−1.327839825548 · 106
2.514985199681 · 109
8.068138754674 · 1010
0.228628703223
1.814874604703
0.126170700576
0.029818169310
0.006266482270
−0.000272419514
−0.001115639433
−0.000030792128
0.000984868989
0.000533538044
Couplings of [155]
1.814898403687
0.126164421218
0.029814964767
0.006262816384
−0.000275905516
Table 2.1.: Coordinates of the fixed point. We compute the coefficients of the Taylor
expansion of the fixed point potential within the local potential approximation up to order O(ρ − ρ0 )9 and for d = 3 (first column). We reproduce the
right-hand side of Table 1 of Ref. [155] where the same quantities where computed (third column) and re-scale our results according to Eq. (2.61) (middle
column).
The rescaled couplings are shown in the second column of Table (2.1). In the calculation
of [155] the value N = 32 was used. Considering the fact that we stopped our calculation
at N = 9, we find a good agreement in between both calculation.
2.2.1.3. Critical exponents
Here we compute the critical exponents of the system and compare our results to [155]
as well. We compute the RG flow asymptotically close to the fixed point where the RG
flow equations can be linearised. Then RG acts on the different couplings simply by
re-scaling them. The re-scaling that is defined in Eq. (2.50) is not restricted to RG fixed
points. It is made fully general by including a cut-off dependence in u(ρ) = uk (ρ). We
define
δρ0 = ρ0 − ρ∗0 ,
δλn = λn − λ∗n ,
δλ = (δλ0 , δρ0 , δλ1 , .., δλN ) ,
(2.62)
with the starred quantities ρ∗0 and λ∗n being the values the couplings assume at the fixed
point which are given in Table (2.1). The right-hand side of the RG flow equation (2.51)
is a non-linear function of δλn and δρ. By definition it vanishes when δλn = δρ = 0. If
δλn and δρ are small enough it can be expanded around δλn = δρ = 0 and all but its
linear part can be neglected. Then the flow equation takes the form
k∂k δλ = M∗ δλ.
26
(2.63)
2. The Functional Renormalisation Group
M∗ is a matrix that depends on the coordinates of the fixed point, ρ∗0 and λ∗n . Then
Eq. (2.63) can be solved by diagonalising M∗ . Indeed, each eigenvector of M∗ , vn , obeys
k∂k vn = M∗ vn = Λn vn .
(2.64)
Λn is the corresponding eigenvalue. We get vnP
(k) = vn (kΛ )(k/kΛ )Λn . Then if δλ is
decomposed into the eigenvectors of M∗ , δλ = N
n=0 αn vn , we get
δλ =
N
X
vn (kΛ )(k/kΛ )Λn .
(2.65)
n=0
We see that only the eigenvectors with negative eigenvalues, Λn < 0 take the flow away
from the fixed point. These are the relevant directions. If the RG flow is initiated close
to the basin of attraction (also called the critical surface for non trivial fixed points)
of a fixed point, the couplings will flow towards it until they are pulled away again
by their component in the relevant direction. If the RG flow is initiated exactly on
the critical surface, the couplings flow towards their fixed point values and reach them
asymptotically as k → 0.
Since everything is re-scaled by the RG flow the correlation length of the effective
ˆ must decrease when the cut-off scale is decreased, ξˆ = kξ. Therefore, any
theory ξ,
theory that lies on the basin of attraction of a fixed point must have either a vanishing
or an infinite correlation length. If this is not the case ξˆ will depend on k. At a nonGaussian fixed point we have ξ = ∞ since if this was not the case we would have ξ = 0
and the fixed point would be Gaussian. We conclude that the correlation length diverges
as we move close to the critical surface. In the case where there is only one relevant
direction, the critical exponent ν is defined as
ξ ∼ (δλrel )−ν ,
as δλrel → 0.
(2.66)
δλrel ≡ δλrel (kΛ ) is the projection on the relevant direction of the shortest vector joining
the critical surface and the point at which the RG flow is initiated (for k = kΛ ). It is
the physical parameter which one can change to tune the system to its phase transition.
This asymptotic form contains an exponent which can be extracted from the linearised
RG flow since we consider the situation where δλrel is very small compared to all the
other couplings.
Note that in this case the RG fixed point is approximately attractive and the couplings
flow towards their fixed point values and only δλrel truly matters. We find that the
critical properties of the system only depend on the relevant couplings. Since most
couplings are irrelevant many different systems share the same fixed point and therefore
have the same critical properties. This is universality.
Let us now see how this works in the context of the local potential approximation.
The first step is to linearise the RG flow equations. The beta functions are computed
27
2. The Functional Renormalisation Group
by taking derivatives of Eq. (2.51) and evaluating them at ρ = ρ0 . They are given by,
k∂k λ0 ≡ βλ0 = −dλ0 +
Ωd
1
,
d 2λ2 ρ0 + 1
k∂k ρ0 ≡ βρ0 = −(d − 2)ρ0 +
2λ3 ρ0
Ωd
λ2 + 3
,
d (2λ2 ρ0 + 1)2
k∂k λn ≡ βλn = λn+1 βρ0 − dλn + (d − 2) (λn+1 ρ0 + nλn ) +
Ωd
X(n, 1).
d
(2.67)
X(n, 1) is defined in Appendix A, in Eqs. (A.11). Then the fixed point equations (2.55),
(2.56) and (2.57) can be written as4 β(λ∗ ) = 0. The linearised flow equations are
obtained by inserting β(λ) ∼
= β(λ∗ ) + M∗ δλ = M∗ δλ. The matrix elements of M∗ are
given by
∂βi ∗
(M )i,j =
.
(2.68)
∂λj λ∗
They are computed in a straightforward way5 from Eqs. (2.67). Using the values given
in Table (2.1) for λ∗ , the matrix elements of M∗ can be computed explicitly and M∗ can
be diagonalised. The eigenvalues that we find are listed in Table (2.2). Note that we find
only one relevant direction. There is also one vanishing eigenvalue which originates from
the inclusion of δλ0 into δλ. Indeed, since this does not represent anything more than a
shift in the total energy of the system, one can move in this direction without changing
the physics of the fixed point. Finally, note that we find two sets of degenerate eigenvalues
which are not in good agreement with the calculation of [155]. Theses degeneracies are
lifted as N is increased.
The correlation length is related to the negative eigenvalue of Table (2.2) by computing
the two-point correlation function,
2
−1
Z
−i[p·(x+r)+q·x] δ Γk→0 [φ = φ0 ]
⟨φ(x + r)φ(x)⟩c =
e
δφ(p)δφ(q)
p,q
Z
1
= e−ip·r
.
(2.69)
2U
∂
p
p2 + ∂ 2k→0
φ φ=φ0
φ0 is defined through δΓk→0 /δφ(x)|φ=φ0 = 0. The definition of the couplings given in
Eq. (2.53) can be used away from the fixed point. We write
√
Z
e− 2ρ0 (k)λ2 (k)kx
e−ip·r
=πp
.
(2.70)
⟨φ(x + r)φ(x)⟩c,k =
2
2
2ρ0 (k)λ2 (k)k
p p + 2ρ0 (k)λ2 (k)k
4
5
We define β = (βλ0 , βρ0 , λ2 , ..λN ).
P
In principle we may define δu(ρ) through u(ρ) = u∗ (ρ) + δu(ρ) and u∗ (ρ) = λ∗0 + s=2 λ∗s /s!(ρ − ρ∗0 )s ,
insert it in Eq. (2.51) and only keep the linear part in δu(ρ). This yields a linear differential equation
for δu(ρ) from which a set of equations for δλ can be extracted by inserting Eq. (2.53). Such an
approach is not wrong, but is not optimal with respect to the convergence of the results as N → ∞.
It is better to expand the effective potential around its flowing minimum, ρ∗0 + δρ0 .
28
2. The Functional Renormalisation Group
Eigenvalues
Eigenvalues of [155]
−1.539601891232131
0
0.6557966422316
3.1827323507884
5.9126210155394
8.7702584075498
13.9842288669691
13.9842288669691
24.6148666337971
24.6148666337971
−1.539499459806173
0.6557459391933
3.1800065120592
5.9122306127477
8.796092825414
11.798087658337
14.896053175688
Table 2.2.: The eigenvalues of the stability matrix M∗ . We insert the couplings of Table
(2.1) into the stability matrix defined in Eq. (2.68) and compute its eigenvalues. As before we go up to order O(ρ − ρ0 )9 and use d = 3 (first column).
The second column are the values computed in Ref. [155].
p
and identify the flowing correlation length ξk = 1/(k 2ρ0 (k)λ2 (k)). If we are on the
critical surface we have ρ0 (k → 0) = ρ∗ and λ2 (k → 0) = λ∗2 . Then the physical
correlation length ξk→∞ is infinite since the divergence coming from the k factor is not
cancelled by the RG flow of ρ0 (k) and λ2 (k). On the other hand, if the RG flow is not
initiated on the critical surface, two possibilitiesparise. If ρ0 is large enough, it grows
to infinity as k → 0 in such a way that ⟨φ⟩k = 2kρ0 (k) saturates to a constant field
expectation value. On the other hand, if ρ0 is smaller than its critical value it flows
to a constant while λ2 (k) flows to infinity
and the asymptotic field expectation value
p
vanishes. In both cases, the product λ1 (k)ρ0 (k) grows to infinity as k goes to zero in
such a way that the k factor is cancelled out and we recover a finite correlation length.
See [180] for an explicit calculation.
If we start close to the critical surface the RG flow first approaches the fixed point and
the irrelevant couplings assume their fixed point values. On the other hand, the relevant
coupling scales according to δλrel ∼ k Λrel and grows as k decreases. The closer to the
critical surface the RG flow is initiated, the "longer" it stays close to the fixed point. ξk
then grows during all that "time" since its dominant k-dependence comes from its 1/k
factor. In this way it can be made arbitrarily large if the initial distance to the critical
surface is arbitrarily small.
Eq. (2.70) only represents the physical correlation function in the limit k → 0. We
can not use it to relate the physical correlation length to the distance of the original
parameters δλ(kΛ ) ≡ δλ from the critical surface. We can however re-scale Eq. (2.69)
with the cut-off scale if we replace the original parameters by their k-dependent ones
⟨φ(x + r)φ(x)⟩c (δλ) = k⟨φ̂(kx + kr)φ̂(kx)⟩c (δλ(k)).
(2.71)
Such an equation is definitely true for k = kΛ . However, the change of δλ(k) with the
29
2. The Functional Renormalisation Group
cut-off scale is precisely defined in such a way that it stays true for all positives values
of k. Then if δλ is close to the critical surface we can decrease k on the right hand side
until all but the relevant coupling acquire their fixed point value. We have
⟨φ(x + r)φ(x)⟩c (δλ) = k⟨φ̂(kx + kr)φ̂(kx)⟩c (λ∗ , δλrel (k/kΛ )Λrel ).
(2.72)
In principle, such an equation only holds as long a k is small enough that all but the
relevant coupling are equal to their fixed point values but large enough that we are still
close to the fixed point. However, the smaller δλrel is, the bigger the range of values of k
where this is true. In the asymptotic limit δλrel → 0, k is completely free. We can then
choose it such that δλrel (k/kΛ )Λrel = 1 and extract
1/Λrel
ξ ∼ δλrel
,
ν = −1/Λrel ∼
= 0.649518557813480.
(2.73)
This concludes the discussion of the local potential approximation. We have shown how
to compute fixed point properties of the critical scalar theory with the local potential
approximation. In the case d = 3 we find a fixed point and compute its scaling exponents.
The scaling exponent ν that was computed is consistent with the Ising universality class.
The results of [155] are recovered.
2.2.2. Frequency and momentum dependent inverse propagator
We will see in Section (3.2) that in the case of stochastic hydrodynamics the advective
derivative of the velocity field leads to a theory with a non-linearity that contains a spatial
derivative. In such a context, we do not expect the derivative expansion to produce
good results since the momentum dependence of the action plays an essential role. The
derivative expansion was applied to study stationary scaling solutions of the stochastic
Kardar–Parisi–Zhang (KPZ) equation in [181]. As for the theory of hydrodynamics KPZ
equation contains a derivative in its vertex. Although the correct scaling exponents where
recovered for d = 1, unphysical values where obtained for d ≥ 2.
In this section we generalise a method first used in Ref. [161] in the context of thermal
equilibrium Yang Mills theory. We write a set of RG fixed point conditions for the
second field derivative of the flowing effective action without making any restriction on
its momentum or frequency dependence. We will assume that some truncation has been
made on the higher order derivatives of the flowing effective action but not specify it
here.
The main idea behind these fixed point equations is relatively straightforward. We
impose that the inverse propagator looses all of it’s dependence on k and assumes a
scaling form once the cut-off is removed, Eq. (2.79). This makes it possible to define the
fixed point scaling exponents η̄i , the scaling function g(a) and the rescaling factors z̄i .
Then we parametrise the inverse propagator in terms of rescaled variables, Eq. (2.75).
In parallel we require that the rescaled propagator and RG flow equations, Eq. (2.85),
depend on the cut-off scale only through the rescaled variables. This implies that the
rescaled theory does not depend on k. It provides a first constraint on the scaling
exponents and makes it possible to define the fixed point coupling constant h. Next we
30
2. The Functional Renormalisation Group
require that there be a qualitative difference in between the form of the propagator when
k → 0 and k → ∞. This constrains the scaling function g(a). Finally the normalisation
of g(a) is used to constrain the second scaling exponent and the fixed point coupling h.
See Table (2.3) where this is summarised.
We return to the velocity field of classical hydrodynamics v(t, x) and assume that
(2)
the RG flow of our theory has reached a fixed point. We define Γk,ij (ω, p) (with i, j =
1, 2, ..., d) such that
(2)
Γk,ij (ω, p)
δ 2 Γk [v = 0]
=
δ(ω + ω ′ )δ p + p′ .
′
′
d+1
δvi (ω , p )δvj (ω, p)
(2π)
(2.74)
Because of Galilei and spatio-temporal translation invariance of the fixed points of the
theory this particular form for the inverse propagator does not contain any restriction.
We have assumed that the effective action has its extremum for v(t, x) = 0 (δΓk [v =
0]/δvi (t, x) = 0) which up to a Galilei boost is always true.
(2)
Since we are looking for fixed points of the RG flow we write Γk,ij (ω, p) in terms of
re-scaled dimensionless variables and extract its non universal parts in such a way that
we recover a scaling form when k → 0. In this section we set d = 1 and drop the
indices in order to make the notation simpler. The generalisation to the case d ̸= 1 is
straightforward. Since we consider a system which invariant under spatial rotations the
different objects of this section do not depend on the sign of p. In order to make the
notation simpler we use the short-hand notation6 p = |p|. We write
p η̄1 p η̄2 ωz̄ ωz̄ p (2)
2
2
Γk (ω, p) = kz̄1
+ δZ
,
.
(2.75)
g
k
k
k2
k2 k
η̄i , z̄i and the scaling function g(a) are defined through the supplementary conditions
lim δZ(ap̂−η̄2 , p̂) = 0,
for a > 0,
p̂→∞
g(0) = 1,
(2.76)
dg = 1,
da a=0
(2.77)
ω̂ = ω z̄2 /k 2 .
(2.78)
and the re-scaled variables
p̂ = p/k,
z̄i are k-dependent dimensionless re-scaling factors which will be discussed shortly.
Eq. (2.76) must be valid for all dimensionless numbers, a and is used to define g(a),
η̄1 and η̄2 . It ensures that we recover a scaling solution in the limit k → 0. Eqs. (2.77)
define the normalisation of the scaling function. They are arbitrary and have no physical
interpretation but are necessary to properly define z̄1 and z̄2 . They are the fixed point
6
Note that this is only a short hand notation for d = 1. When the momentum is a vector p is defined
as its norm.
31
2. The Functional Renormalisation Group
equivalent of the RG conditions that we discussed at the very beginning of this Section.
The k pre-factor is extracted in order to make the rest dimensionless. δZ(ω̂, p̂), g(a),
η̄1 and η̄2 are unknown physical parameters that can be determined by solving the RG
fixed point equations.
At a fixed point z̄i take a particularly simple form. Indeed, taking the limit k → 0
and inserting Eq. (2.76) we obtain
(2)
lim Γk (ω, p) = kz̄1 p̂η̄1 g p̂η̄2 ω̂ .
k→0
(2.79)
In this limit the dependence on the cut-off scale must vanish, see Section (2.1). This is
only possible if z̄i are power laws of the cut-off scale
z̄1 = z̄10 k η̄1 −1 ,
z̄2 = z̄20 k η̄2 +2 .
(2.80)
The only free parameters that they contain are the pre-factors z̄i0 . As anticipated, we
(2)
are left with a scaling form for Γk→0 (ω, p). η̄1 and η̄2 are the scaling exponents of the
fixed point and g(a) is the scaling function. In particular we can identify the dynamical
scaling exponent z = −η̄2 .
The parametrisation made in Eq. (2.75) corresponds to rewriting the inverse propagator in terms of dimensionless variables, p̂ and ω̂ and asking for the re-scaled inverse
(2)
(2)
propagator Γ̂k (ω̂, p̂) ≡ Γk (ω̂, p̂)/(kz̄1 ), to loose its explicit dependence on the cut-off
scale. If we where not at a fixed point, z̄i would have a non-trivial dependence on k and
δZ(ω̂, p̂) would have an additional explicit k-dependence.
We are now ready to insert Eq. (2.75) in the flow equation Eq. (2.27). Since all the
cut-off dependence drops out in the limit k → 0, the pre-factor of Eq. (2.75) as well as the
scaling function contain no dependence on k (see Eq. (2.79)). Then the scale derivative
on the left-hand side of the flow equation only acts on the arguments of δZ(ω̂, p̂),
∂δZ
∂δZ
(2)
η̄1
k∂k Γk (ω, p) = kz̄1 p̂
η̄2 ω̂
− p̂
.
(2.81)
∂ ω̂
∂ p̂
In order to obtain closed RG flow equations one introduces a truncation for the flowing
effective action Γk [v]. Then Eq. (2.27) can be projected onto the flow of the inverse
propagator by inserting the truncated effective action in
−1 1
δ2
(2)
Tr k∂k Rk Γk + Rk
≡ δ(p + p′ )δ(ω + ω ′ ) Ik (ω, p),
2 δv(ω, p)δv(ω ′ , p′ )
v=0
(2.82)
(2)
and require that Γk (ω, p) obey,
(2)
k∂k Γk (ω, p) = Ik (ω, p).
(2.83)
Since we do not restrict the two-point function of Γk [v] the truncated effective action
must contain approximated higher order vertexes. Then additional equations must be
32
2. The Functional Renormalisation Group
extracted for each unknown term that is included in the truncation and the vertexes of
Γk [v] must be parametrised in the same way as its inverse propagator. I.e. they must
depend on the same re-scaled variables, reduce to a scaling form in the limit k → 0 and
loose all of their cut-off dependence when re-scaled with the appropriate power of k and
z̄ pre-factor. Note that we must introduce a new pre-factor for each vertex. Then when
this is inserted into Eq. (2.82), we find that Ik (ω, p) is automatically parametrised in the
same way as its components,
ˆ p̂).
(2.84)
Ik (ω, p) = kz̄3 I(ω̂,
We have introduced a third k-dependent re-scaling factor z̄3 which is a combination of
the re-scaling pre-factors of the inverse propagator z̄i and of the vertexes of Γk [v], z̄. Its
form can be inferred from the particular flow equation that we use and depends on the
truncation of Γk [v]. See Section (3.3.3) and Eq. (3.73) with Eqs. (3.69) for an example.
ˆ p̂) contains no explicit dependence
Note that with the re-scaled truncation inserted I(ω̂,
on k and that all the re-scaling prefactors z̄i as well as z̄ have been extracted and put
ˆ p̂) only contains, g(a), η̄i and δZ(ω̂, p̂).
together in z̄3 . I(ω̂,
Putting everything together we write the flow equation as
ˆ −η̄2 , p̂)
d
z̄3 I(ap̂
δZ(ap̂−η̄2 , p̂) = −
.
(2.85)
dp̂
z̄1 p̂η̄1 +1
The full frequency dependence of δZ(ω̂, p̂) is taken into account by the fact that Eq. (2.85)
is valid for all values of the dimensionless number a. Eq. (2.76) is used as an initial
condition which makes it possible to determine δZ(ap̂−η̄2 , p̂) if η̄i , z̄3 /z̄1 and g(a) are
known. The scaling exponents η̄i , the scaling functions g(a) and the pre-factors zi0 are
still undetermined and must be constrained in some other way.
First, one of η̄i can be related to the other by noting that the only term that explicitly
depends on k in Eq. (2.85) is z̄3 /z̄1 . It is then apparent that z̄3 /z̄1 does not depend
on k. Given Eqs. (2.80) and the fact that there will be a similar equation for the other
components of Γk [v], we can write that z̄3 = z̄30 k η̄3 . The fact that z̄3 is a combination of
z̄i and z̄ implies that η̄3 will be a (typically affine) function of η̄i and of the corresponding
exponents of the vertexes. z̄1 and z̄3 must scale with k in the same way in order for all
of the explicit k-dependence to drop out. Inserting the first of Eqs. (2.80) we get
η̄3 = η̄1 − 1,
(2.86)
which provides a scaling relation in between the different scaling exponents included in
the truncation of Γk [v] and reduces the number of free parameters by one. Note that
an additional re-scaling factor and scaling exponent will be introduced for each vertex
that is taken into account in Γk [v]. However and additional scaling relation will also be
generated for each additional flow equation that is used. Hence whatever the truncation
we use for Γk [v] there will always be only one independent scaling exponent.
Taking into account Eq. (2.86) we find that the right-hand side of (2.85) has a dimensionless k-independent pre-factor,
z̄3
z̄30
h≡
=
.
(2.87)
z̄1
z̄10
33
2. The Functional Renormalisation Group
h can be interpreted as a fixed point coupling constant. Indeed, if h = 0 we trivially
find that δZ(ap̂−η̄2 , p̂) = 0 and the inverse propagator has a scaling form from the onset
without being affected by the non-linearity of the theory. Moreover, note that if we do
not sit exactly on the RG fixed point but flow towards it, we will find that h flows with
k. The coupling defined in Eq. (2.87) is the asymptotic value that the RG flow reaches
when k → 0.
The scaling function, g(a), is determined by a supplementary condition on the large-k
limit of δZ(ω̂, p̂). In this limit the cut-off function Rk (p) is huge. We have an effective
theory with a very large mass term. This is very different from the physical theory (when
k → 0) and we can expect the scaling to be different from that of Eq. (2.79). Then the
asymptotic form of δZ(ap̂−η̄2 , p̂) must be
δZ(ap̂−η̄2 , p̂) = −g (a) +
f (ap̂−η̄2 , p̂)
,
p̂η̄1
p̂ ≪ 1,
(2.88)
in order for the physical (small-k) scaling to be undone when k is very large. When
this is inserted in Eq. (2.75) we find that g(a) is cancelled out ant that the pre-factor of
the second term on the right-hand side of Eq. (2.88), p̂−η̄1 , removes the η̄1 -scaling and
replaces it with something else given by f (ω̂, p̂),
(2)
lim Γk (ω, p) = kz̄1 f (ω̂, p̂).
(2.89)
k→∞
In Eq. (2.88) f (ap̂−η̄2 , p̂) p̂−η̄1 contains all the p̂-dependence of the asymptotic form of
δZ(ap̂−η̄2 , p̂). Simply defining f (ap̂−η̄2 , p̂) p̂−η̄1 as the asymptotic form of δZ(ap̂−η̄2 , p̂)
minus its constant (in p̂) part is ambiguous. One can however, see that f (ap̂−η̄2 , p̂) is
well defined by looking at Eq. (2.85). Indeed, using the boundary condition given by
Eq. (2.76) we can write Eq. (2.85) as7
δZ(ap̂−η̄2 , p̂) = h
Z
∞
dy
p̂
ˆ −η̄2 , y)
I(ay
.
y η̄1 +1
(2.90)
We see that the p̂−η̄1 pre-factor is already present in the integrand. In the limit y ≪ 1,
ˆ −η̄2 , y) will be simple because the cut-off function is dominating the flow. Let us
I(ay
assume that we have the form
ˆ −η̄2 , y) = y α (a0 + a1 y + ... ) ,
I(ay
y ≪ 1.
(2.91)
The exponent α and the coefficients ai depend on a, ηi , g(a) and h. Because in this
limit everything is dominated by the cut-off we expect this dependence to be relatively
simple. In the case of Burgers stochastic hydrodynamics we where able to compute α
and its relation to η̄i exactly. See Section (3.3) and Eqs. (3.94) and (3.101). Since we
are talking about the limit p̂ → 0 we can introduce p̂ < ϵ ≪ 1 and brake the integral
7
Here we assume that this integral is convergent. See Section (2.2.2.1) where the general case is
discussed.
34
2. The Functional Renormalisation Group
of Eq. (2.90) in two parts: one integral from p̂ to ϵ and one from ϵ to ∞. The we can
insert Eq. (2.91) into the first part,
"Z
#
ϵ
ˆ −η̄2 , y)
ˆ −η̄2 , y) Z ∞
I(ay
I(ay
dy
δZ(ap̂−η̄2 , p̂) = h
dy
+
y η̄1 +1
y η̄1 +1
ϵ
p̂
"Z
∞
ˆ −η̄2 , y)
a0
a1
I(ay
α−η̄1
+ϵ
+ϵ
+ ...
=h
dy
y η̄1 +1
α − η̄1
α − η̄1 + 1
ϵ
a0
a1
α−η̄1
− p̂
+ p̂
+ ... ,
(2.92)
α − η̄1
α − η̄1 + 1
with ϵ ≪ 1 small enough that Eq. (2.91) applies for p̂ < ϵ. We can now make the
identifications
"Z
#
∞
ˆ −η̄2 , y)
a
a
I(ay
0
1
+ϵ
g(a) = −h
dy
+ ϵα−η̄1
+ ... ,
(2.93)
y η̄1 +1
α − η̄1
α − η̄1 + 1
ϵ
a0
a1
−η̄2
α
+ p̂
+ ... .
f (ap̂ , p̂) = −h p̂
(2.94)
α − η̄1
α − η̄1 + 1
We define f (ap̂−η̄2 , p̂) p̂−η̄1 as the part of δZ(ap̂−η̄2 , p̂) from which a factor of p̂−η̄1 can
be extracted. What is left will not depend on p̂ by construction. The scaling function
is then constrained by enforcing Eq. (2.93). Note that in practice the sum on the righthand side of Eq. (2.91) will be truncated at a finite order in y. Then it is only an
approximation and ϵ should be taken as small as possible in order to minimise the error.
We should therefore take the limit ϵ → 0 in Eq. (2.93). This limit will only be finite if
the order at which we truncate Eq. (2.91) is large enough for all the terms that grow
as ϵ decreases to be taken into account. We need to consider all terms of order smaller
than y m with m > η̄1 − α.
With this definition of f (ω̂, p̂) we can now use Eq. (2.88) to constrain g(a). We are
then left with one of the exponents η̄i , one of the pre-factors zi0 and the coupling h
to determine. We use the normalisation constraints on g(a) given in Eq. (2.77) to do
this. Note that zi0 only enter Eqs. (2.85) and (2.88) through h. Eqs. (2.77) therefore
constrain the values of the second exponent and of the coupling. The last pre-factor
is left undetermined. This is a desirable property since the value of the pre-factor of
(2)
Γk (ω, p) depends on the particular system of units that we use and is fixed by comparing
the results with an experiment. It is not a property of an RG fixed point.
Let us now summarise the method to extract the RG fixed point properties. See Table
(2.3). First we define the different properties of the fixed point η̄i , zi0 , g(a) and δZ(ω̂, p̂)
in Eqs. (2.75), (2.76), (2.77) and (2.80). Next we use the RG flow equation Eq. (2.27)
and the initial condition given by Eq. (2.76) to constrain δZ(ω̂, p̂). This naturally leads
to a constraint on one of the two scaling exponents η̄i and to the definition of the fixed
point coupling h. This constraint is a necessary condition for the fixed point version of
the flow equation (2.85), to be truly independent of the cut-off scale. Then we constrain
g(a) by requiring that the solution of the RG flow equations be different in the two limits
35
2. The Functional Renormalisation Group
Steps
Equations
Free parameters
Fixed parameters
Define the different
parameters.
(2.75), (2.76),
(2.77), (2.80)
η̄i , zi0 ,
δZ(ω̂, p̂)
Use the RG flow
equation and the initial condition.
(2.27), (2.76)
η̄i , zi0 , g(a)
Remove
kdependence
from
the RG flow equation.
(2.85), (2.86)
η̄1 , z10 , h, g(a)
Enforce that the limits k → 0 and k → ∞
be different
(2.88), (2.93)
η̄1 , z10 , h
g(a)
Enforce the normalisation constraints of
g(a).
(2.77)
z10
η̄1 , h
g(a),
δZ(ω̂, p̂)
η̄2
Table 2.3.: The different steps that lead to extracting all of the fixed point properties
(2)
of Γk (ω, p). The steps are to be followed from top to bottom. They are
outlined in the first column and the relevant equations are given in the second
column. The third column lists the parameters that are undetermined at
the corresponding step. The fourth column gives the parameters that are
fixed by the corresponding step. They can be completely determined if the
parameters of the third column are given. We have made the choice of fixing
η̄2 and expressing z̄20 in terms of h at the third step. This is of course
arbitrary either one of the η̄i , z̄i0 could have been chosen. Note that z10 is
never fixed since it depends on the particular system of units that we use.
p̂ ≪ 1 and p̂ ≫ 1. Finally we use the normalisation constraints given by Eqs. (2.77) to
constrain the values of the remaining scaling exponent and the coupling.
It is surprising that the two most interesting properties of the fixed point, namely η̄i
and h are fixed by the arbitrary normalisation given in Eqs. (2.77). Indeed, it looks like
we could choose a different normalisation and get different values of η̄i and h. This is of
course not the case. A change of the normalisation (as compared to Eqs. (2.77)) of g(a)
can be reabsorbed into a redefinition of δZ(ω̂, p̂), ω̂ and p̂. Then we recover the same
set of equations as before. For example, if we choose g(1) = b, we can simply re-define
p̂′ = p̂ b1/η̄1 ,
ω̂ ′ = ω̂ bη̄2 /η̄1 ,
δZ ′ (ω̂ ′ , p̂′ ) = δZ(ω̂, p̂)/b,
g ′ (a) = g(a)/b.
Then we are back to where we started without having touched η̄i or h.
36
(2.95)
2. The Functional Renormalisation Group
Note that in the case of a Gaussian fixed point the theory is quadratic in v(t, x). Then
we simply have δZ(ω̂, p̂) = 0 and h = 0 and there is no need to impose the relation given
by Eq. (2.86) in between η̄i since Eq. (2.85) is trivially independent of k in this case.
If we have a Gaussian theory the RG flow of the effective average action is trivial. We
(2)
(n)
simply get k∂k Γk (ω, p) = 0 and Γk = 0 for n > 2. Then all one has to do to find
(2)
a fixed point is to choose a scaling form for Γk (ω, p). The exponents and the scaling
function are not constrained by the fixed point equations.
2.2.2.1. Ultraviolet divergent fixed points
In the previous section we have assumed that the integral on the right-hand side of
Eq. (2.90) is convergent. This is a self-consistent assumption since if it is not true we
can not have Eq. (2.76),
lim δZ(ap̂−η̄2 , p̂) = 0.
p̂→∞
(2.96)
We discuss here the fact that this is only a formal restriction on the form of Γk [v].
Indeed, it is physically perfectly acceptable that
lim δZ(ap̂−η̄2 , p̂) = ∞,
p̂→∞
(2.97)
since no fixed point is perfectly realised in nature. There is always a cut-off to make
everything finite. Such a divergence happens when the exact fixed point theory is UV
divergent and it not defined in the limit k → ∞. See Section (3.3.5.3) where this
is discussed in the context of stochastic hydrodynamics and cascades of energy. The
FRG program (start with Γk→∞ [v] = S[v] and progressively lower the cut-off scale to
Γk→0 [v] = Γ[v]) can still be carried out but the initial condition must be taken at an
arbitrary but not infinite cut-off scale Γk=Λ [v] = S[v]. All momenta larger than Λ are
not physical. In this case we can still find a theory with correlation functions that are
asymptotically scale invariant but there will always be a trace of the initial scale Λ which
restricts the scaling range. The actual value of Λ depends on the particular details of
the UV theory. It is not universal. Since we are investigating the scaling regime we will
take Λ as large as possible.
We refer to such fixed points as UV divergent and to fixed points where Eq. (2.96)
is realised as UV convergent. Note that the case limp̂→∞ δZ(ap̂−η̄2 , p̂) = const ̸= 0
¯
is UV convergent. Indeed, one can simply redefine δZ(ω̂,
p̂) = δZ(ω̂, p̂) − const and
ḡ(a) = g(a) + const and Eq. (2.96) is true for the new variables.
In the case of UV divergent fixed points Eq. (2.90) has to be supplemented with an
UV cut-off which modifies the fixed point conditions. This can be done in two different
ways. One can either modify the flow equation Eq. (2.85) in such a way that the RG
flow does not build up infinities in the UV. Or one can abandon the strict fixed point
conditions and only require that we are close to an RG fixed point. We will see that
these two options are actually equivalent. In both cases one must keep a finite (although
large) UV cut-off to compute the universal properties of the system, η̄i and g(a). Then
37
2. The Functional Renormalisation Group
one must check that the latter are independent of the former and increase the value of
the UV cut-off if they are not.
In the first case we replace Eq. (2.85) by
ˆ −η̄2 , p̂)
d
I(ap̂
δZ(ap̂−η̄2 , p̂) = −h
θ(B − p̂).
dp̂
p̂η̄1 +1
(2.98)
θ(x) is the usual step function. This means that the right-hand side of the flow equation
Eq. (2.27) is truncated for momenta larger than Λ ≡ kB. Then Eq. (2.96) is equivalent
to
lim δZ(ap̂−η̄2 , p̂) = 0,
(2.99)
p̂→B
since δZ(ap̂−η̄2 , p̂) is simply constant for p̂ > B. For any finite value of B Eq. (2.99)
can always be satisfied. The original RG flow equations are then recovered by taking
B ≫ 1. The fact that we have an UV divergent fixed point prevents us from taking the
limit B → ∞. We must instead choose it to be large enough for the momentum scale
that we probe to be correctly reproduced by the modified RG flow equation. Indeed,
properties of Γk [v] which depend on momentum p and satisfy k ≪ p ≪ kB = Λ will not
be sensitive to what is happening at the scale Λ. This is apparent from the diagrammatic
expression of the flow equation of the inverse propagator,
(2)
k∂k Γk (ω, p) =
3
3
−
1
2
.
(2.100)
4
(2)
The thick lines denote Gk = (Γk + Rk )−1 , the thin lines external momenta and frequencies, the black dots insertions of the derivative k∂k Rk of the regulator and the circled
numbers the corresponding derivatives of Γk [v]. The k∂k Rk insertions limit the loop
momentum to be close to the cut-off scale. Then all the momenta that enter Eq. (2.100)
are not far from k, p or p ± k. We see that when k ≪ p ≪ Λ, Λ does not enter the flow
equations.
If instead we do not wish to modify the flow equations but prefer to move slightly
away from the RG fixed point we write
Z B(a;k)
ˆ −η̄2 , y)
I(ay
δZ(ap̂−η̄2 , p̂) = h
dy
,
(2.101)
y η̄1 +1
p̂
instead of Eq. (2.90). B(a; k) is a function of the cut-off scale that marks the limit in
between close-to and far-from-the fixed point in the (ω̂, p̂, k) space. It will be defined
more precisely below. The idea here is to make the fixed point conditions weaker. We
only assume that the RG flow takes us close to the fixed point but does not actually
38
2. The Functional Renormalisation Group
stop on it. This is a more natural situation than simply truncating the flow equations
but leads an explicit k dependence in δZ(ω̂, p̂). This means that we are not at an RG
fixed point and the fixed point conditions do not strictly apply any more.
Let us assume that the RG flow passes close to one of its fixed points. We define
Λ1 and Λ2 as the scales in between which Γk [v] approximately behaves as a fixed point
theory. For Λ1 < k < Λ2 the re-scaled parameters of the flowing effective are almost
constant. Then a scaling range emerges. Indeed, any correlation function takes the form
O(ω, p) = k dO +ηO O(ω̂, p̂; g(k)),
for all k.
(2.102)
g(k) are the parameters that characterise Γ̂k [v̂] and dO and ηO are the canonical and
anomalous dimensions of O(ω, p) respectively. g(k) change with the cut-off scale precisely
in the right way for Eq. (2.102) to be true. Close to an RG fixed point we have
g(k) ∼
= g∗,
for Λ1 ≪ k ≪ Λ2 .
(2.103)
Then we are free to choose p̂ = p/k = 1 and extract a scaling form for O(ω, p) as long as
Λ1 ≪ p ≪ Λ2 . In our case Γ̂k [v̂] is characterised by an infinite set of parameters since
have flowing functions of momentum. We can write this as g(k) = g(ω̂, p̂; k). Then the
close-to-a-fixed-point condition can be written as
k∂k g(ω̂, p̂; k) ≪ g(ω̂, p̂; k).
(2.104)
This separates the 3d space spanned by (ω̂, p̂, k) into regions where Eq. (2.104) is satisfied
and regions where it’s not. B(a; k) is defined as the value of p̂ on the upper limit of the
region where Eq. (2.104) is true,
B(a; k) = max {p̂| Eq. (2.104) is satisfied} .
(2.105)
We can use the scaling range Λ1 ≪ p ≪ Λ2 to define η̄i , g(a), z̄i as before. Then
δZ(ω̂, p̂) is defined through Eq. (2.75). Note that since we are not exactly at a fixed
point anymore it contains an additional explicit k-dependence δZ(ω̂, p̂) → δZ(ω̂, p̂; k).
Outside of the scaling range δZ(ω̂, p̂; k) changes with k in such a way that the scaling
form breaks down. As long as k is restricted to Λ1 ≪ k ≪ Λ2 Eq. (2.85) is not modified.
Only its initial condition Eq. (2.96), is changed to
lim
p̂→B(a;k)
δZ(ap̂−η̄2 , p̂; k) = 0.
(2.106)
The picture that emerges is that as k is lowered from Λ2 to Λ1 p̂ increases. This in turn
decreases δZ(ap̂−η̄2 , p̂; k) and brings it to zero when the RG flow leaves the fixed point.
I.e. for k = Λ1 . Note that in this regime the system is close to the fixed point. The
explicit dependence of δZ(ap̂−η̄2 , p̂; k) on k is weak. For smaller values of the cut-off
scale, δZ(ap̂−η̄2 , p̂; k) becomes strongly dependent on k again. If p is out of the scaling
range δZ(ap̂−η̄2 , p̂; k) increases again and the scaling form is destroyed. On the other
hand if p is in the scaling range δZ(ap̂−η̄2 , p̂; k) stays roughly constant as p̂ is increases
while k is decreased.
39
2. The Functional Renormalisation Group
Note that the existence of the scaling range implies
lim δZ(ω̂, p̂; k) ∼
= 0,
for Λ1 ≪ p ≪ Λ2 ,
k∂k δZ(ω̂, p̂; k) ≪ δZ(ω̂, p̂; k),
for Λ1 ≪ k ≪ Λ2 .
k→0
(2.107)
The first equation ensures that we recover scaling in the physical limit k → 0. The
initial condition given by Eq. (2.106) and the fact that δZ(a(p/k)−η̄2 , p/k; k) does not
change very much for k < Λ1 and p ≫ k ensure that it is realised. The second equation
is related to the fact that we have an approximate fixed point for Λ1 ≪ k ≪ Λ2 . It
states that δZ(ω̂, p̂; k) only weakly depends on k close to the fixed point. This equation
relies on the fact that B(a; k) depends weakly on k. If this is not true then we need to
restrict the (ω̂, p̂, k) space a little more by replacing B(a; k) by
B(a) = mink [B(a; k)] .
(2.108)
Then Eq. (2.101) becomes
δZ(ap̂
−η̄2
Z
B(a)
, p̂) = h
dy
p̂
ˆ −η̄2 , y)
I(ay
.
y η̄1 +1
(2.109)
This picture provides approximate fixed point conditions. Using B(a) instead of B(a; k)
reduces the quality of the approximation but eliminate all the explicit k-dependence
from the fixed point equations. This reduction of the quality of the approximation is
in principle a practical limitation but does not make any difference in the end. Indeed,
we can assume that the RG flow come asymptotically close to the fixed point. Then
the closer we come to it the bigger B(a) will be. It then acts as a cut-off exactly as in
Eq. (2.98). We can then even use B = mina [B(a)]. The only strict requirement is that
we choose B ≫ 1.
We remark that for a fixed momentum p we can not lower k below p/B. We can only
recover physical results if B ≫ 1 since in this limit there is a large range of momenta
(2)
p ≫ k which are resolved. Then we recover a scaling form for Γk (ω, p) with a restricted
scaling range given by k ≪ p ≪ kB ≡ Λ. We see that the UV cut-off Λ = kB is
proportional to k. In the limit k → 0 the scaling range shrinks to a point and only
the largest scales are resolved. However since B is arbitrary it can always be chosen
large enough for the momentum scale that we are interested in to be within the range
k ≪ p ≪ Λ and the physical properties are that of an RG fixed point.
Note that a completely different solution to this problem of finding the properties of
UV divergent fixed points would be to reverse the direction of the RG flow. If one uses
a cut-off function that vanishes for p ≪ k and diverges for p ≫ k Eq. (2.27) still applies
but the boundaries of the flow are reversed Γk→0 [v] = S[v] and Γk→∞ [v] = Γ[v]. The
interpretation is a little strange since instead of coarse-graining fluctuations on scales
smaller than k we do the inverse. Large scale fluctuations are integrated out first. This
does not matter here since we look for fixed points. I.e. theories where there is no RG
40
2. The Functional Renormalisation Group
flow anyway. The fixed point equations then stay unchanged except for the Eq. (2.96)
that becomes,
lim δZ(ap̂−η̄2 , p̂) = 0.
p̂→0
(2.110)
We see that UV divergent fixed points will not bother us any more. We may however
encounter IR divergent fixed points where δZ(ap̂−η̄2 , p̂) diverges at p̂ = 0.
41
3. Burgers Turbulence
In this section we discuss the stationary states of the stochastic Burgers equation. We
start with a brief introduction to the physics that we wish to describe and set up its
mathematical formulation. In Section (3.1) we review the results of the literature with a
focus on the scaling properties of correlation functions. Next we establish a path integral
formulation of the steady state generating functional in Section (3.2). In particular
we compute its action (see Eqs. (3.42) or Eq. (3.43)). Section (3.3) is the heart of
the present work. We apply the Functional Renormalisation Group (FRG) and the
fixed point equations established in Section (2.2.2) to compute the two-point correlation
function of the stochastic Burgers equation. We establish an approximation scheme in
Section (3.3.1) and write Renormalisation Group (RG) flow equations for its parameters
in Section (3.3.2). Next we adapt the fixed point equations of Section (2.2.2) to our
truncation in Section (3.3.3). The different quantities that we have introduced in Sections
(3.3.1) and (3.3.3) are related to physical observables in Section (3.3.4). Finally the RG
fixed point equations are analysed in detail in Section (3.3.5) and fixed points as well as
their properties are discussed.
Let us start by discussing the classical turbulence described by Burgers’ equation [149],
E[v] ≡ ∂t v + [v · ∇] v − ν∆v = 0.
(3.1)
v(t, x) is a space and time dependent velocity field and ν is the kinematic viscosity
(see [151, 152] for reviews). Burgers’ equation is equivalent to the Navier–Stokes (NS)
equation if the equation of state is assumed to impose a constant pressure, P = const
[150]. It can be interpreted as a model for fully compressible hydrodynamics since there
is no pressure to stop two fluid elements from being squeezed together. In the limit
ν → 0, this leads to the appearance of spatial discontinuities in the velocity field after a
finite time even when smooth initial conditions are chosen [182, 183]. This is illustrated
in Figure (3.1) where two shocks are shown to appear from smooth initial conditions for
d = 1. Such shocks correspond to situations where the velocity field has a strong enough
gradient that fluid elements which are moving with a large velocity are advected onto
the slow ones too quickly for the latter to get out of the way. Then the shock propagates
through the system and "eats up" the slow particles on the way. Fluid accumulates at
singular points of space. Note that this picture looks a little different in the frame that
it moving with the shock. In this case the whole velocity profile is shifted vertically such
that the net velocity is zero. The strong gradient then implies that the velocity changes
sign. The picture is then that of two masses of fluid flowing towards each other and
colliding at the position of the shock. For d > 1 the shocks are not restricted to being
points and can have a rich topology [152]. Note that a small but non-vanishing value of
42
Velocity
Velocity
3. Burgers Turbulence
Position
Position
Figure 3.1.: A typical velocity profile for Burger’s equation in 1d at two successive times
(left, then right)
ν leads to shocks which are actually smooth at the dissipation scale, l ∼
= ν/vtypical . We
can see this in Figure (3.1). The shocks are actually rounded off because of the finite
value of ν.
Burgers’ equation is often considered as a toy model for classical hydrodynamics.
Given an initial velocity configuration it can be solved analytically in the limit ν → 0 with
the method of characteristics. Moreover, contrarily to the incompressible NS equation,
Burgers’ equation can be used to study 1d turbulence1 . This simplifies greatly numerical
computations and makes it possible to study stochastic versions of Eq. (3.1), [152, 182–
184].
Burgers’ equation has however many physical applications ranging from the modelling
of dust in the early universe to polymers in random media (see Ref. [151, 152] and
references therein). The irrotational Burgers equation can be mapped onto the KardarParisi-Zhang (KPZ) equation [5],
∂t θ +
λ
(∇θ)2 = ν∇2 θ,
2
(3.2)
with v = ∇θ. The coupling constant λ must be set to one if Kardar–Parisi–Zhang (KPZ)
equation is to be mapped on Burgers’ equation. Indeed, in hydrodynamic theories the
non-linearity is part of advective time derivative and can not be rescaled independently
of the partial time derivative. When KPZ equation is not used for hydrodynamics, λ
can be arbitrary and measures the strength of the non-linearity as compared to the
dissipative term. In particular when d ≥ 2 one finds a phase transition in the dynamics
of the stochastic stationary KPZ equation. For small values of λ the non-linearity is
irrelevant and the properties of the steady state are the same as for the linear theory.
On the other hand when λ is large enough the non linear term becomes relevant and the
steady state dynamics is more complicated. See e.g. [185] for an overview.
1
Note that a 1d incompressible fluid is trivial.
43
3. Burgers Turbulence
The KPZ equation is typically used to describe non-linear interface growth but can
also be applied to the dynamics of phase fluctuations in an ultracold Bose gas described
by the Driven-Dissipative Gross–Pitaevskii Equation (DDGPE) [186, 187], or to directed
polymers in random media [188–190]. Shocks can also appear in the Gross–Pitaevskii
equation (GPE) but, due to the definition of the phase on a compact circle, lead to the
creation of (quasi) topological defects, e.g. dissolve into soliton trains [191, 192].
In the present work we are mainly interested in out-of-equilibrium stationary states. In
order to study such systems we need to include a forcing mechanism. Then the dynamics
of the system adapts in order to transfer the energy from the forcing to the dissipation
scale and an asymptotic steady state can be reached. A natural set-up would be to add
a deterministic forcing that is periodic in time, for example, to the right-hand side of
Eq. (3.1). We would then need to choose a particular form of such a forcing in space and
time before we can extract a particular solution. We are however not really interested
in the detailed response of the fluid to a particular forcing because it is expected that
averages over space or time will provide smooth and symmetric correlation functions
independently of the details of the forcing mechanism. Instead we consider all possible
types of forcing and weigh them with a given probability distribution. We therefore
consider the stochastic Burgers equation,
E[v] = ∂t v + (v · ∇)v − ν∆v = f .
(3.3)
f is a force with zero average and Gaussian fluctuations
⟨fi (t, x)fj (t′ , x′ )⟩ = δij δ(t − t′ )F x − x′ .
⟨f ⟩ = 0,
It has the following probability distribution2
Z
1
1
′
−1 ′
P [f ] =
exp −
f (t, x) · f (t, x ) F
x−x
.
N
2 t,x,x′
(3.4)
(3.5)
N is a constant normalisation factor. We assume that the dynamics of the fluid is
ergodic in the sense that we can trade the spatio-temporal mean for the average over the
stochastic forcing. Such an averaging procedure makes it possible to define an out-ofequilibrium steady state which is fully invariant under space and time translations. There
is no need to take some kind of spatio-temporal averaging to recover universal properties
as in the case of a deterministic forcing. F (|x − x′ |) determines the fluctuations of the
forcing and is left undetermined at this point. The integral on the right-hand side of
Eq. (3.5) takes a simple form in Fourier space,
Z
1
1
f (ω, p) · f (−ω, −p)
P [f ] =
exp −
.
(3.6)
N
2 ω,p
F (p)
We see that the Fourier modes of f fluctuate independently from each other, follow
Gaussian distributions and that their variance is given by
Z
2
σ (ω, p) = F (p) ≡
eip·x F (x) .
(3.7)
x
2
We use the definition
R
y
F
−1
′
(|x − y|) F (|y − x |) = δ(x − x′ ) for F −1 (|x − y|).
44
3. Burgers Turbulence
To distinguish different types of forcing we define the exponent β as,
F (p) ∼ pβ .
(3.8)
Hence, β determines the degree of non-locality of the forcing. For β > 0 the energy is
mainly injected into the Ultraviolet (UV) modes while for β < 0 the forcing acts on large
Infrared (IR) scales. The case β = 0 corresponds to a forcing delta correlated in space.
Note that the forcing correlation function that we defined in Eq. (3.4) is not consistent
with a potential forcing field. Indeed, if there is a function u(t, x) such that f (t, x) =
∇u(t, x) the correlation function of the forcing must have the form
⟨fi (t, x)fj (t′ , x′ )⟩ =
∂ ∂
⟨u(t, x)u(t′ , x′ )⟩.
∂xi ∂x′j
(3.9)
If we require that the forcing be invariant under space and time translations and rotations
we can restrict the correlation function of the potential to satisfy
⟨u(ω, p)u(ω ′ , p′ )⟩ = δ(ω + ω ′ )δ(p + p′ ) U (p).
(3.10)
Then the correlation function of the forcing is
⟨fi (ω, p)fj (ω ′ , p′ )⟩ = δ(ω + ω ′ )δ(p + p′ ) pi pj U (p).
(3.11)
This is consistent with Eq. (3.4) only when d = 1. Then we can identify F (p) = p2 U (p).
For d > 1 there is a difference in the tensor structure of the correlation function of
Eq. (3.11) as compared to Eq. (3.4). This originates in the fact that the definition of
the forcing mechanism given in Eqs. (3.4) implies that we are forcing vorticity as well as
energy into the fluid. The correlation function of the vorticity injection is
⟨[∇ × f (t, x)] · [∇ × f (0, 0)]⟩ = δ(t) 2(1 − d)∇2 [F (x)] ̸= 0,
(3.12)
where F ′ (x) = dF/dx|x .
3.1. Scaling and correlation functions
Within such a set-up observables that one can compute are the moments of the velocity
field averaged over the different realisations of the forcing. Typically one looks at the
velocity increments [11],
Dh
r iq E
Sq (τ, r) = (v(t + τ, x + r) − v(t, x)) ·
,
(3.13)
r
because of their symmetry properties. Indeed, the out-of-equilibrium steady state of the
stochastic Burgers equation is expected to be invariant under time and space translations
and spatial rotation. Moreover Sq (τ, r) contain a velocity difference so that they are
symmetric under Galilei boosts as well. Additionally, the turbulent state is expected
45
3. Burgers Turbulence
to be invariant under scale transformations for a large class of forcing mechanism. In
particular we define the exponents ζq through the equal time structure function,
Sq (0, λr) = λζq Sq (0, r),
for λ > 0,
and the exponents χ and z through the the second order structure function,
τ S2 (τ, r) = r2(χ−1) g z .
r
(3.14)
(3.15)
Note that we have ζ2 = 2(χ − 1). g(a) is the scaling function and is analogous to g(a)
defined in Section (2.2.2). It is well known from analytical [5, 58, 90, 92, 93, 112, 152]
as well as numerical [184, 193–200] calculations that χ and z are related by
χ + z = 2.
(3.16)
This can be attributed to Galilean invariance which prohibits an anomalous scaling of
the velocity field. Since the non-linearity of E[v] is part of the advective derivative of
the fluid velocity, it must scale in the same way as the partialptime derivative. This is
only possible if the velocity scales as position divided by time ( S2 (0, r) ∼ r/rz ), which
implies, the relation (3.16) when the exponents are matched.
Note that this relation holds for stationary stochastic KPZ dynamics as well. In
this case the relevant symmetry is not Galilei invariance but infinitesimal tilting of the
interface, θ(t, x) → θ(t, x + tλv) + x · v − λv 2 /2. On the other hand the relation (3.16)
does not hold any more if the forcing correlation function is not white in time. Indeed,
if we make the replacement
(3.17)
δ(t − t′ ) F (x − x′ ) → F t − t′ , x − x′ ,
in Eq. (3.4) the forcing correlation function looses its invariance under Galilei transformations3 . See [112] where this case is examined.
The dependence of ζq on the order of the structure function q, and more precisely
their deviations from a linear behaviour is called intermittency because it has its origin
in rare and large events within the hydrodynamic flow [11]. For Gaussian statistics we
get ζq = ζ2 q/2. Intermittency is an active topic of research within the incompressible
NS turbulence set-up [78, 85, 86, 95, 97, 98, 201, 202] and is still not fully understood
within Burgers’ equation, especially for d > 1.
In the case d = 1 Burgers’ and KPZ equations are equivalent and many results are
available. For a forcing correlation function of the type given in Eq. (3.8) and for β > 0
the forcing fluctuates mainly on small spatial scales and the shocks are overwhelmed.
Then there is no intermittency [183, 184]. One finds that ζq = (χ − 1)q. In this regime
perturbative RG techniques apply [112, 162–164] and it was found that
1 β
χ = max
,− + 1 .
(3.18)
2
3
3
Under Galilei transformations x − x′ goes to x − x′ − (t − t′ )v 0 . This quantity is only invariant if
t = t′ .
46
3. Burgers Turbulence
When β < 0 non-linearities become relevant and perturbative RG techniques fail. The
case β < 3 is however relatively simple. When β is small enough the forcing acts on
large spatial scales and its realisations are differentiable and smooth enough for shocks to
form naturally (see Figure (3.1)) and propagate through the system. Then the stationary
state contains a finite density of shocks and the scaling of the velocity increments in the
regime where |r| is much smaller than the average distance in between shocks but still
much larger than the shock size can be estimated by a simple argument [151]. Indeed,
we can write
Sq (0, r) = P (r passes a shock) ∆vshock + P (r does not pass a shock) [(∂x v) |r|]q .
(3.19)
∆vshock is the typical jump in velocity across a shock and P (event) is the probability of
’event’. By ’r passes a shock’ we mean that there is a shock in between x and x + r4 .
Since the shocks are discontinuous ∆vshock does not depend on r. On the other hand if
there is no shock the velocity profile is differentiable and the velocity increment is simply
proportional to |r|q . We compute the average velocity difference by summing the product
of the probability that r passes over a shock with the corresponding velocity difference
and the complementary probability multiplied by the corresponding linearised velocity
difference. Note that the velocity difference in the second term of Eq. (3.19) is smooth
enough to be linear in |r| by definition. The discontinuous velocity differences are in the
first term. In order to compute ζq we estimate the two probabilities as behaving as
P (r touches a shock)
∼ |r| ,
P (r does not touch a shock) = 1 − P (r touches a shock)
∼ 1 − |r| .
(3.20)
Indeed, when |r| is much bigger than the shock size the latter appear point-like and the
probability of finding one with a vector of length |r| is simply proportional to |r|. We
finally keep only the leading (for small |r|) behaviour in the second term and write
Sq (0, r) ∼ ∆vshock |r| + [(∂x v)]q |r|q ,
(3.21)
and see that ζq = min (1, q). The regime −3 < β < 0 is not as well understood.
Dimensional analysis and matching with the known scaling for β = 0 and β = −3
suggests ζq = min (−qβ/3, 1) [152]. This assumption seems to be supported by [184].
Note that for a forcing correlation function of the type given in Eq. (3.8) and for β = 2
an analytic expression for the probability distribution of the single point velocity field
was obtained. See [203, 204] and references therein for detailed discussions.
Much less is known about the cases d > 1 and apart from [153, 154], the literature
seems to focus exclusively on the case of a potential forcing (as in Eq. (3.11)) and
very often directly assumes v = ∇θ in order to simply switch to KPZ equation. Little is
known about ζq and intermittency. The two-point correlation function and its exponents
4
Remember that we average over many realisations of the forcing such that x actually drops out of the
final results.
47
3. Burgers Turbulence
χ and z are studied in more detail. Much interesting work has been done in this context
using a variety of non-perturbative methods. Most of these concentrate on the case β = 2,
corresponding to white-noise forcing in space. In this context, predictions for the scaling
exponents, scaling functions and upper critical dimension have been made, using, e.g. ,
the mode coupling approximation [205–208], the self-consistent expansion [209, 210], or
the weak-noise scheme [211–213]. The case β < 0 of a forcing that is concentrated on
large scales was tackled in Ref. [190] by means of a replica-trick approach being exact
in the limit d → ∞, and bi-fractal scaling of the velocity increments was obtained. The
tails of the probability distribution of velocity differences were addressed in [214] using
an operator product expansion in d = 1, and in [182, 215] within an instanton approach.
Decaying Burgers turbulence was studied in [216].
We close this section by mentioning the work of [55, 57, 58, 61] since it is closely
related to ours. In these papers the stationary state of the stochastic KPZ equation is
studied for β = 2 [55, 57, 58] with the FRG. A non-perturbative approximation scheme
that respects Galilei invariance is devised and RG fixed points are found for d = 1, 2, 3.
Corresponding scaling exponents as well as scaling functions are estimated. For d = 1
the exact scaling exponents of Eq. (3.18) are recovered and the obtained scaling function
compares very well with the exact result [203, 204]. For d ≥ 2 the Gaussian fixed point
becomes attractive as well and a phase transition occurs. For general β the authors of
[61] assume that the noise correlation function takes the form5 ,
(3.22)
U (p) = D 1 + w pβ−2 ,
and look into the cases of β ≤ 2. It was found that for β < βc = 4 − d there is no stable
Gaussian fixed point. When β < βtrans (d) ∼
= 2−d/2 the non-local noise is relevant and we
get χ = (4 − d − β)/3. On the other hand for βc < β < βtrans (d) the RG flow is attracted
to a fixed point with a local forcing (w = 0) and χ = (4 − d − βtrans (d))/3 ∼
= (4 − d)/6
does not depend on β any more. Eq. (3.18) is generalised to

 1/4 for d = 1
4 − d − βtrans (d) 4 − d − β
∼
0.863 for d = 2 . (3.23)
χ = max
,
, βtrans (d) =

3
3
0.100 for d = 3
See Eq. (3.108) with χ = η1 − d for the corresponding values of χ when β > βtrans (d).
Note that βtrans (d) ∼
= 2 − d/2 is a simple estimation based on a linear interpolation in
between known results at d = 1 and d = 4 [162, 164]). The values of βtrans (d) given
in Eq. (3.23) are average values from the KPZ literature (see [193, 194, 196–200] and
Table I. of Ref. [58]. For β > βc the Gaussian fixed point becomes attractive as well as
the local one. We see again a phase transition. We remark that [162, 164] tackles this
problem perturbatively and postulates most of these results.
5
Note that this is work about the KPZ equation. The forcing is a potential field. See Eq. (3.11) where
U (p) is defined.
48
3. Burgers Turbulence
3.2. Functional treatment
In this section we apply the Martin–Siggia–Rose/Janssen–de Dominicis (MSR/JD) formalism [165, 217–220] and express the generating functional of velocity correlation functions as a path integral. We extract the corresponding action functional which is given
in Eqs. (3.42) and (3.43). We follow closely the derivation of [165], chapter 4, where
classical Langevin equations are analysed.
We define the generating functional of velocity field correlation functions as
Z[J ] = ⟨e
R
J ·v
⟩.
(3.24)
All the velocity field correlation functions can be extracted from Z[J ] by taking derivatives with respect to J (t, x) before evaluating at J (t, x) = 0. Eq. (3.24) contains an
averaging over the stochastic forcing f (t, x). We write it as
Z
R
Z[J ] = D[f ] e J ·v[f ] P [f ].
(3.25)
v[f ] is the solution of Eq. (3.3) with the realisation f inserted and P [f ] is the probability
density functional of the forcing which is given in Eqs. (3.5) and (3.6). Not that we do
not worry about initial conditions here because we are interested in the steady state
where the initial time is far away in the past, t0 → −∞. Next we reformulate Eq. (3.25)
by including an integration over E(t, x) and enforcing the equations of motion with a
delta functional,
Z
R
Z[J ] = D[f ]D[E] e J ·v[E] P [f ] δ[E − f ].
(3.26)
We can then change variables from E to v under the integral,
Z
R
δE[v] J ·v
.
Z[J ] = D[f ]D[v] e
P [f ] δ[E[v] − f ] δv The Jacobian of the coordinate change is given by
δE[v] = det ∂Ei [v](t, x) .
δv ∂vj (t′ , x′ )
(3.27)
(3.28)
Note that this is a functional determinant. It is the determinant of an operator which
acts on the space of functions of spatio-temporal variables (t, x). It can be included
in the weight of the path integral by virtue of the identity log (det [A]) = Tr (log [A]).
Finally we evaluate the delta function and insert Eq. (3.5) to write
Z
R
Z[J ] = D[v] e−S[v]+ J ·v ,
(3.29)
with
1
S[v] =
2
Z
′
E[v](t, x) · E[v](t, x ) F
t,x,x′
−1
(x − x′ ) −
49
∂Ei [v](t, x)
log
(t, x, t, x).
∂vj (t′ , x′ )
t,x
(3.30)
Z
3. Burgers Turbulence
As for the determinant of Eq. (3.28) we have here a functional logarithm. It can be
interpreted as a power series in its argument. We have now arrived at an action for the
steady state of the stochastic Burgers equation. We can however simplify it. Indeed,
we show in the following that the second term of Eq. (3.30) does not contribute to the
steady-state dynamics.
Let us start by giving an expression for the operator that is inside the logarithm in
coordinate space,
∂v (t, x) ∂Ei [v](t, x)
i
′
′
2
′
′
. (3.31)
= δ(t − t )δ(x − x ) δij ∂t − ν∇x′ + vl (t, x)∂xl +
∂vj (t′ , x′ )
∂xj
Defining the time derivative operator and its inverse,
Dt (t, x; t′ , x′ ) = δ(t − t′ )δ(x − x′ ) ∂t′ ,
It (t, x; t′ , x′ ) = θ(t − t′ ) δ(x − x′ ),
we can extract Dt from Eq. (3.31). We write ∂E/∂v = Dt · (1 + X) with6
∂v (t, x) l
′
′
′
′
2
.
X(t, x; t , x ) = θ(t − t )δ(x − x ) δij −ν∇x′ + vl (t, x)∂x′l +
∂xl
(3.32)
(3.33)
This form for δE[v]/δv makes the trace of its logarithm particularly simple. First note
that the multiplicative factor that we extracted does not depend on the velocity field
and can be absorbed into the normalisation of Z[J ]. Furthermore the step function in
front of X implies that X n will be proportional to (t − t′ )n−1 in the limit t → t′ . Let us
illustrate this with a simpler example,
A(t, t′ ) = θ(t − t′ ) a(t′ ).
(3.34)
a(t′ ) is a function of t′ as in Eq. (3.33). The general case is completely analogous but
contains heavy expressions. It is easy to check that
Z
Z t−t′
A2 (t, t′ ) ≡ θ(t − τ )a(τ ) θ(τ − t′ )a(t′ ) = θ(t − t′ )
dτ a(t − τ )a(t′ )
τ
0
∼
= θ(t − t′ )(t − t′ )a(t)a(t′ ).
(3.35)
The last equality was obtained by inserting a(t − τ ) ∼
= a(t) which is asymptotically exact
in the limit t → t′ . Using Eq. (3.35) it can be shown by recurrence that
An (t, t′ ) ∼
= θ(t − t′ )
(t − t′ )n−1
a(t)n ,
(n − 1)!
as t → t′ .
(3.36)
We can now insert this into Eq. (3.30) and use the fact that the second term is a trace
and is evaluated at (t, x) = (t′ , x′ ). Then only the term linear in X contributes. Once
the constant contribution arising from Dt is subtracted we get
Z
Z
1
S[v] =
E[v](t, x) · E[v](t, x′ ) F −1 (x − x′ ) −
X(t, x, t, x).
(3.37)
2 t,x,x′
t,x
6
∂E/∂v denotes the operator with matrix elements given by Eq. (3.31) in real space. 1 is the identity
operator. θ(x) is the step function. It is zero if x < 0 and 1 if x > 0.
50
3. Burgers Turbulence
Note that X(t, x, t, x) is not well defined since it contains a delta function which is
evaluated at zero and since it is proportional to θ(0). The value of the step function
evaluated at zero is related to the precise definition of the continuum limit of the theory
and has to be chosen consistently. See [130, 221] or chapter 4 of [165] for a detailed
discussion. A natural choice (that we do not make at this point and) which corresponds
to a forward discrete time propagation is the Itō prescription θ(0) = 0.
The δ(0) pre-factor looks however more troubling at first but can be shifted into the
normalisation of the path integral in the same way as the Tr [log (Dt )] factor. Performing
the trace over the spatial indexes and assuming that the delta distribution has been
regularised in some way, we formally write the second term of Eq. (3.37) as
Z Z
∂ 2 δ (0)
∂vl (t, x)
∂δ (0)
d −ν
X(t, x, t, x) = θ(0)
+ δ (0)
. (3.38)
+ vl (t, x)
∂xl ∂xl
∂xl
∂xl
t,x
t,x
We already see that the first term does not depend on v and only contributes as a
shift of S[v] and that the third term only depends on the velocity field evaluated at the
boundary of space x → ∞. We discard them both. Finally we can see that the middle
term actually behaves as the third. Indeed, if we write it as
Z
Z
∂δ (x − x′ )
dθ(0)
∂ [δ (x − x′ )]2
′
dθ(0)
δ(x − x ) vl (t, x)
=
vl (t, x), (3.39)
∂xl
2
∂xl
t,x,x′
t,x,x′
we can integrate it by parts and perform the x′ integration. It becomes
Z
Z
∂δ (x − x′ )
dθ(0)δ(0)
∂vl (t, x)
′
dθ(0)
δ(x − x ) vl (t, x)
=−
.
∂xl
2
∂ xl
t,x,x′
t,x
(3.40)
Note that we can avoid the manipulation of the square of a deltaR function by defining
the delta function evaluated at zero as a volume factor, δ(0) = y 1/(2π)d . Then we
have
Z
Z
2
θ(0)
∇
·
v(t,
x)
+
d
νy
+
i
y
·
v(t,
x)
.
(3.41)
X(t, x, t, x) =
(2π)d t,x,y
t,x
The first two terms can be neglected as before. The third term however
R now vanishes
because it contains an integration over all space of an odd function, y y = 0. If one
is not comfortable with such manipulations one can always choose to work with Itō’s
prescription and choose θ(0) = 0. It is however interesting to note that the way in which
we enforce causality at the microscopic level, i.e. choice of the value of θ(0), does not
seem to play a role here. In fact we will never have to choose a value of θ(0) in this work
because we work with a steady state and do not look into response functions.
We finally write the action for the stochastic Burgers equation as
Z
1
S[v] =
E[v](t, x) · E[v](t, x′ ) F −1 (|x − y|),
2 t,x,y
E[v](t, x) = ∂t v(t, x) + [v(t, x) · ∇] v(t, x) − ν∆v(t, x),
51
(3.42)
3. Burgers Turbulence
or
E[v](ω, p) · E[v](−ω, −p)
,
F (p)
ω,p
Z
2
v(ω ′ , p′ ) v(ω − ω ′ , p − p′ ) · p′ ,
E[v](ω, p) = iω − νp v(ω, p) − i
S[v] =
1
2
Z
(3.43)
ω ′ ,p′
in Fourier space.
Eq. (3.29) together with Eqs. (3.42) or (3.43) contains the full information on the
steady state of Burgers turbulence. It will however be more practical to work with
different variables. As in thermal equilibrium one can define the Schwinger functional,
δnW
W [J ] = log (Z[J ]) , ⟨vi1 (t1 , x1 )..vin (tn , xn )⟩c =
, (3.44)
δJi1 (t1 , x1 )..δJin (tn , xn ) J =0
which generates connected correlation functions of the velocity field. Then we switch
from the representation in terms of J (t, x) to the one in terms of ⟨v(t, x)⟩ by taking the
Legendre transform of W [J ],
Z
Γ[v] = sup −W [J ] + J · v .
(3.45)
J
Γ[v] is the 1 Particle Irreducible (1PI) effective action. It takes into account the fluctuations of the velocity field through an infinite series of 1PI diagrams. The physical
velocity field expectation value is the extremum of Γ[v]
δΓ = 0,
(3.46)
δvi (t, x) v=⟨v⟩
and the higher order correlation functions are computed through the higher order derivatives of Γ[v] evaluated at v = ⟨v⟩. See Section (2.1.4) where this is explained in detail.
Note that the action (3.43) is rather complicated. Its inverse propagator,
(2)
Sij [0](ω, p) = δij F −1 (p) ω 2 + ν 2 p4 ,
(3.47)
contains a non-trivial dependence on momentum through F −1 (p) and S[v] contains
two different vertexes which depend on momentum, frequency and spatial indexes in
(3)
a complicated way. We do not give explicit expressions for Sijl [0](ω, p; ω ′ , p′ ) and
(4)
Sijlm [0](ω, p; ω ′ , p′ ; ω ′′ , p′′ ) because Eq. (3.43) is formally very similar to the ansatz that
we will choose for the flowing effective action Γk [v] in Section (3.3.1). The vertexes of
S[v] can be obtained by making the replacements
Γk [v] → S[v],
Fk−1 (p) → F −1 (p),
in Eqs. (B.6) and (B.8) of Appendix B.
52
νk (p) → ν,
(3.48)
3. Burgers Turbulence
Let us close this section by noting that the MSR/JD formalism is usually used with
an additional response field. See e.g. [96, 139, 222] where this is done. Instead of directly
performing the integration over the forcing in Eq. (3.27) we could have taken the Fourier
transform of the delta function,
Z
R
(3.49)
δ [E[v] − f ] = D[ṽ] ei ṽ·[E[v]−f ] ,
and performed the f -integration with ṽ(t, x) left free since the argument of the exponential stays quadratic in the forcing. This has three advantages. First the resulting
action
Z
1
S [v, ṽ] =
F (p) ṽ(ω, p) · ṽ(−ω, −p) − 2i ṽ(ω, p) · E[v](−ω, −p),
(3.50)
2 ω,p
does not contain the inverse of the forcing correlation function any more. Indeed, when
one integrates over the forcing the F −1 (|x − x′ |) term in the probability distribution (see
Eq. (3.5)) is inverted. Then we can choose F (p) to vanish for certain values of p without
having to worry about divergences in the action. Secondly the action that one obtains
in terms of v(t, x) and ṽ(t, x) has a simpler structure than the one of Eq. (3.42) since it
only contains one vertex. Finally even though it was introduced as a ghost field ṽ(t, x)
can be interpreted as a response field. If one adds a deterministic part to the stochastic
forcing of Eq. (3.3) f (t, x) → f (t, x) + Φ(t, x) on can write the response function as
δ
′
′ = −i ⟨vj (t′ , x′ )ṽi (t, x)⟩.
(3.51)
⟨vj (t , x )⟩
δΦi (t, x)
Φ(t,x)=0
Note however that the in theory of both fields which is described by S [v, ṽ], we must
deal with twice as much degrees of freedom which makes it necessary to consider all the
objects of the theory as tensors in space as well as field indexes indexes.
3.3. Functional renormalisation group calculation
We have seen in Section (3.1) that scale invariance is an essential property of Burgers
turbulence. We therefore turn to the RG and look for fixed point of the field theory
defined by Eq. (3.29) together with Eq. (3.42) or Eq. (3.43). Perturbative RG methods
yield a continuum of non-Gaussian IR attractive fixed points with exponents given by
Eqs. (3.18) and (3.16) for d = 1 and for β > 0 [112, 162–164]. Outside of this range nonperturbative effects become important and a more sophisticated approach is necessary.
We turn to the non-perturbative RG which is outlined in Section (2.1.3). We take as
starting point the exact flow equation Eq. (2.27),
−1
1
(2)
k∂k Γk [v] = Tr Γk [v] + Rk
k∂k Rk ,
(3.52)
2
53
3. Burgers Turbulence
for the effective average action, Γk [v]. In order to find self-similar turbulent configurations we look for IR fixed points of the flow, i.e. for solutions of (3.52) which are scaling
in ω and p in the limit k → 0.
Our main goal is to get a handle on the second order velocity increment S2 (τ, r) of
the stochastic Burgers equation and on the exponents χ and z. For this we start by
making a non-perturbative approximation for the flowing effective action Γk [v]. See
Section (3.3.1). This approximation contains two arbitrary functions of momentum and
is formally similar to S[v]. Next we write RG flow equations for these two functions
in Section (3.3.2). The RG fixed point equations of Section (2.2.2) are adapted to our
approximation in Section (3.3.3). The two unknown functions of momentum are related
to the physically observable quantities S2 (τ, r), χ and z in Section (3.3.4). Finally we
solve the RG fixed point equations in both asymptotic limits of momentum much larger
and much smaller than the cut-off scale and discuss the properties of the fixed points
that we find in Section (3.3.5).
3.3.1. Approximation scheme
Eq. (3.52) relates the flowing effective action with its second moment and therefore creates an infinite hierarchy of integro-differential equations for all the field derivatives of
Γk [v]. To allow their solution in practice, truncations are in order. We have discussed
different approach to such truncations in Section (2.2) and we choose to truncate the
high order correlation functions of the theory in favour of a more precise treatment of
the frequency and momentum dependence of the two-point correlation function. Indeed,
both vertexes of S[v] contain a non-trivial momentum dependence because of the advective non-linearity of fluid dynamics and through the forcing correlation function F (p).
Moreover, the three-point vertex depends on frequency as well as momentum. Momentum dependence is already an essential property of S[v]. To not take this into account
in the flowing effective action will not give good results. See [181] where this was tried
with the stochastic KPZ equation.
Note that the Blaizot–Mendez–Wschebor (BMW) approach [173] can not be used
here because it relies on evaluating Eq. (3.52) at a constant field while including the
momentum (and frequency) dependence of low order correlation functions only. The
strength of the BMW approximation is that it takes into account the full dependence
of the derivatives of Γk [v] on this constant field as in the case of the local potential
approximation. Here Galilei invariance implies that the derivatives of the effective action
behaves as7
(n)
(n)
Γi1 i2 ..in [u](ω1 , p1 ; ..; ωn−1 , pn−1 ) = Γi1 i2 ..in [0](ω1 + p1 · u, p1 ; ..; ωn−1 + pn−1 · u, pn−1 ),
(3.53)
when u(t, x) = u is a constant velocity field. We see that the frequency, momentum and
field dependence are all related by Galilei invariance in such a way the BMW approximation can not be applied. Indeed, Galilei invariance implies that truncating the frequency
7
This can be seen by applying a Galilei boost, v → v − u.
54
3. Burgers Turbulence
and momentum dependence of Γk [v] is equivalent to truncating its dependence on the
constant field u.
The solutions resulting from our truncation should preserve the symmetries of the
underlying theory. For this we make the ansatz
Z
1
Γk [v] =
E k [v](ω, p) · E k [v](−ω, −p) Fk−1 (p),
2 ω,p
Z
v(ω − ω ′ , p − q) · q v(ω ′ , q), (3.54)
E k [v](ω, p) = iω + νk (p)p2 v(ω, p) − i
ω ′ ,q
for the effective average action, in terms of the k-dependent inverse force correlator
Fk−1 (p) and kinematic viscosity νk (p). Γk [v] has the same form as the action S[v]
of the underlying Burgers equation (3.43), but with the inverse force correlator and
the kinematic viscosity allowed to be k-dependent. This ensures that no symmetry is
broken by the ansatz. In particular this truncated action functional is manifestly Galilei
invariant. We anticipate that νk (p) will become p-dependent as a result of the RG flow
because the advective derivative renders the cubic and quartic couplings of the velocity
field momentum dependent.
With the above ansatz, we keep a general dependence of the inverse propagator on p
while taking into account the ω-dependence in an expansion to first order in the frequency
squared,
(2)
(2)
Γk,ij (ω, p) ≡ δij Γk (ω, p) = δij νk (p)2 p4 + ω 2 Fk−1 (p).
(3.55)
The sole dependence on the norm p reflects the assumed rotational invariance. Note that
isotropy of the problem implies as well that δΓk [v = 0]/δv = 0. I.e. ⟨v(t, x)⟩ = 0. The
truncation of the frequency dependence ensures that the integrand on the right hand side
of the flow equation (3.52) is a rational function of ω. We will see that this enables the
analytic integration over the frequency variable in the flow equation so that there is no
need to explicitly cut off the frequency degrees of freedom. Rk (ω, p) = Rk (p) is sufficient
(see Appendix B for details). Finally, note that Γk [v] is chosen such that the inverse
force correlator Fk−1 (p) and thus the inverse propagator are diagonal in momentum p
and in the field indices i, j. As a consequence, this truncation is only able to capture
solutions of the RG flow equations where vorticity is injected in the steady state. A
correlation function of the form (3.11) can not be accounted for in the classical limit
k → ∞ and is not allowed to develop as the cut-off scale is lowered.
νk (p) and Fk−1 (p) can be interpreted as an effective viscosity and forcing correlation
function that emerges on the large scales when the small scales are integrated out.
−1
Indeed, one can choose νk→∞ (p) = ν and Fk→∞
(p) = 1/F (p) as initial conditions and
compute the whole k-dependence of both quantities from the flow equation. For each
value of k along the way to k = 0, Γk [v] can be used as an effective bare action as long as
the modes with momentum larger than k are cut off. Such a procedure is not guaranteed
to lead to a scale invariant solution when k → 0 since any value can be chosen for ν
and the function F (p) must be specified. Each different choice can lead to a different
55
3. Burgers Turbulence
solution which will most likely still depend on the scales that where chosen at k → ∞.
In contrast we will write in Section (3.3.3), directly a set of RG fixed point equations
without ever having to specify νk (p) and Fk−1 (p). The equations provide solutions which
are scale invariant by construction.
3.3.2. Flow equations
Before we go on to the RG fixed point equations let us write flow equations for νk (p)
and Fk−1 (p). These are the main ingredient of the fixed point equations but can be used
as well to compute the full RG flow. The main difference in between these and the fixed
point equations resides in the initial conditions. We insert the truncated effective action
(3.54) into the flow equation (3.52) and define
−1
1
(2)
Ik [v] ≡ Tr Γk [v] + Rk
k∂k Rk ,
(3.56)
2
as the term on the right-hand side of Eq. (3.52) with the truncation (3.54) inserted. See
Section (2.1.3) and Eqs. (2.28) to (2.31) for precise definitions. The k-dependence of
(2)
νk (p) and Fk−1 (p) can be extracted from the flow equation of Γk (ω, p) by expanding
everything in powers of ω 2 . From Eq. (3.52) we get8
(2)
(2)
k∂k Γk,ij (ω, p) = Ik,ij (ω, p).
(3.57)
The truncation (3.54) leads to an inverse propagator (3.55), that is diagonal in the spatial
indexes ij. We therefore further project Eq. (3.57) onto its diagonal part by taking its
trace,
(2)
(2)
(2)
d k∂k Γk (ω, p) = Ik,ii (ω, p) ≡ d Ik (ω, p).
(3.58)
(2)
Ik (ω, p) is computed by taking two field derivatives of Ik [v], evaluating at v = 0, taking
the trace over the field indexes and dividing by d. It has the following diagrammatic
representation
(2)
Ik (ω, p) =
3
3
−
1
2
.
(3.59)
4
(2)
The thick lines denote Gk = (Γk + Rk )−1 , the thin lines external momenta and frequencies, and the black dots insertions of the derivative k∂k Rk of the regulator. The 3- and
8
(2)
(2)
We use a definition analogous to the definition of Γk,ij (ω, p) in Eq. (2.74) for Ik,ij (ω, p). See Eqs. (F.15)
and (F.16).
56
3. Burgers Turbulence
4-vertices are given in Appendix B, in Eqs. (B.6) and (B.8), respectively. The equations
for νk (p) and Fk−1 (p) are then
(2)
(2)
k∂k Fk−1 (p)νk (p)2 p4 = Ik (0, p),
(2)
k∂k Fk−1 (p) =
∂Ik (0, p)
.
∂ω 2
(3.60)
(2)
Ik (0, p) and ∂Ik /∂ω 2 (0, p) are computed in a straightforward although lengthy way.
We refer to Appendix B for details on their calculation and simply state the result here
Z
−1
−1
kd
(2)
Fk (k)Fk−1 (q)ν̃k ν̃q (ν̃k + ν̃q )
Ik (0, p) = −
2d
h Ω
× Fk−1 (q)2 ν̃q (ν̃q + ν̃k ) d(p2 + k 2 ) − 2ker · p
− θ q 2 − k 2 Fk−1 (q)2 ν̃q (ν̃k + ν̃q ) d(p2 + k 2 ) − 2k p · er
+ Fk−1 (k)2 ν̃k (ν̃k + ν̃q ) d(p2 + q 2 ) − 2p · q
+ Fk−1 (p)2 ν̃p 2 d(k 2 + q 2 ) + 2k q · er
− Fk−1 (p)Fk−1 (q)ν̃p ν̃q 2d p · q
− Fk−1 (p)Fk−1 (k)ν̃p ν̃k 2d k p · er
i
,
Z
(2)
−1
∂Ik
kd
(0, p) =
θ q 2 − k 2 Fk−1 (p) Fk−1 (k)Fk−1 (q)ν̃k ν̃q (ν̃k + ν̃q )3
2
∂ω
2d Ω
× Fk−1 (p) (ν̃k + ν̃q )2 − ν̃p 2 d(q 2 + k 2 ) + 2k q · er
+ Fk−1 (q) ν̃q ν̃k + ν̃q + ν̃p 2d p · q
+ Fk−1 (k) ν̃k ν̃k + ν̃q + ν̃p 2d k p · er .
We have used the sharp cut-off function Rk (p) = k d z1 R̃k (p), with
=0
if p ≥ k
,
R̃k (p)
→ ∞ if p < k
(3.61)
(3.62)
(3.63)
which makes it possible to analytically perform the radial part of the momentum inteR
(2)
gration
Ik (ω,R p). We are left with its angular part, Ω which is defined through9
R
R ∞in d−1
dp Ω . The vector er which appears in Eqs. (3.61) and (3.62) is of unit
p
⃗ = 0 p
length and contains the remaining angular integration. It points in the direction defined
by the angle Ω. We define as well q = p − ker and the short-hand notation ν̃p = νk (p)p2 .
Finally note that the dependence on the full vector p is only apparent. Indeed, the trace
over the spatial indexes makes everything isotropic. It can be check in Eq. (B.13) that
(2)
Ik (ω, p) only depends on p and the square of the frequency ω 2 .
9
For d = 1, this reduces to
R
Ω
f (p) = [f (p) + f (−p)]/(2π).
57
3. Burgers Turbulence
3.3.3. Fixed point equations
We now apply the fixed point conditions of Section (2.2.2) to the truncation (3.54). We
follow precisely the reasoning of Section (2.2.2) and relate the equations that we write
here to the corresponding equations of Section (2.2.2). We give short explanations of
the most important features the fixed point equations here and reffer to Section (2.2.2)
where everything is explained in great detail.
(2)
Since we have truncated the frequency dependence of Γk,ij (ω, p) the equations of
Section (2.2.2) can be simplified as well. The form (3.55) for the inverse propagator
implies that the scaling function is
g(a) = 1 + a2 .
(3.64)
Note that it is no longer possible to impose Eqs. (2.77) on the derivative of g(a). The
normalisation of the scaling function is however a matter of convention. We choose
instead
dg g(0) = 1,
= 1.
(3.65)
d(a2 ) a=0
The parametrisation made in Eq. (2.75) applied to Eq. (3.55) can be written as
(2)
= k d z1 p̂η1 [1 + δZ1 (p̂)] ,
Γk (0, p)
(2)
∂Γk ∂ω 2 2
= k d−4 z2 p̂η2 [1 + δZ2 (p̂)] .
(3.66)
(ω =0,p)
and
p
p̂ = ,
k
1
ω̂ = 2
k
r
z2
ω,
z1
v̂(ω̂, p̂) = k d+1 v(ω, p).
(3.67)
The parameters νk (p) and Fk−1 (p) are easily related to the ones we just introduced.
Comparing Eqs. (3.55) and (3.66) we can directly write
s
p
1 + δZ1 (p̂)
νk (p) = z1 /z2 p̂(η1 −η2 −4)/2
,
1 + δZ2 (p̂)
Fk−1 (p) = k d−4 z2 p̂η2 [1 + δZ2 (p̂)] .
(3.68)
As for the normalisation of the scaling function we have chosen slightly different definitions for the scaling pre-factors zi and exponents ηi as compared to Section (2.2.2).
We have the following correspondences
η1 = η̄1 ,
z1 = z̄1 ,
η2 = η̄1 + η̄2 ,
z2 = z̄1 z̄2 .
58
(3.69)
3. Burgers Turbulence
Finally we get an additional truncation on the ω̂ behaviour of δZ(ω̂, p̂) (which is defined
in Eq. (2.75)) because of the frequency truncation. It is constrained to the form
δZ(ω̂, p̂) = δZ1 (p̂) + ω̂ 2 p̂η1 −η1 δZ2 (p̂),
(3.70)
with Eq. (2.76) becoming δZi (p̂ → ∞) = 0.
The parametrisation (3.66) of the inverse propagator is made in such a way that the
latter has the two following fixed point properties. First when it is rescales with z1 k d and
expressed in terms of the rescaled variables of Eqs. (3.67) is loses completely its explicit
dependence on the cut-off scale k. Secondly the property δZi (p̂ → ∞) = 0 ensures that
in the physical limit k → 0 the inverse propagator assumes a scaling form,
(2)
Γk→0 (ω, p) = k d z1 p̂η1 1 + p̂η2 −η1 ω̂ = k d−η1 z1 pη1 + k d−η2 −4 z2 pη2 ω 2 .
(3.71)
As in Eqs. (2.79) and (2.80) this expression is independent of k only if
z1 = z10 k η1 −d ,
z2 = z20 k η2 +4−d .
(3.72)
(2)
We now express Ik (ω, p) in terms of the re-scaled variables (3.67) and the non-scaling
part of the inverse propagator δZi (p̂),
r h
i
z2 ˆ(2)
(2)
(2)
d
Ik [0] (ω, p) = k
(3.73)
I1 (p̂) + ω̂ 2 Iˆ2 (p̂) + O ω̂ 4 .
z1
(2)
The flow integrals Iˆi (p̂), are obtained by inserting Eqs. (3.68) in Eqs. (3.61) and (3.62),
dividing out all the dimensionless pre-factors zi and the dimensionfull powers of k and
expanding in powers of ω̂ 2 .
(2)
Note that Iˆi (p̂) contain the scaling exponents ηi and the functions δZi (p̂) but the
cut-off scale as well as the re-scalling pre-factors zi have been completely extracted and
are part of the pre-factors of Eq. (3.73). Explicit expressions are given in Appendix C
in Eqs. (C.2) and (C.3) with p
(C.4) to (C.10) inserted. We can now compare this with
Eq. (2.84) and identify z̄3 = z2 /z1 . Since there are only two free parameters in the
truncation (3.54), no additional parameter enters the flow equations and z̄3 only depends
on zi . Eqs. (2.86) and (2.87) become
3η1 = η2 + 4 + 2d,
(3.74)
and
r
h≡
z2 1
.
z 1 z1
(3.75)
These equations arrise form the requirement that there be no explicit cut-off dependence
at the fixed point. Indeed, taking the cut-off derivative of Eqs. (3.66) and taking into
acount Eqs. (3.72) we find that the flow equation of the inverse propagator (3.57) can
be written as
(2)
Iˆ (p̂)
dδZi (p̂)
= −h i η +1 ,
dp̂
p̂ i
59
(3.76)
3. Burgers Turbulence
for i = 1, 2. The h factor that arrises is defined in terms of zi in Eq. (3.75). Eq. (3.74)
has to be satisfied in order for h not to have any explicit cut-off dependence. We see
that we have only one free scaling exponent.
h is the fixed point coupling of the theory. We can see this by inserting the dimensionless re-scaled variables into the truncated effective action we find Γk [v] = Γ̂k [v̂]/h. This
shows that when h → 0 all the fluctuations are damped and only the extremum of Γ̂k [v̂]
contributes to the path integals. The theory is classical in the sense that its fluctuations
are Gaussian. On the other hand when h is large all the fluctuations are important and
we have a strongly interacting theory. Note that in the case of a Gaussian fixed point
h = 0 and there is no need to impose the relation (3.74) in between ηi . Instead we
find δZi (p̂) = 0 and the inverse propagator assumes a scaling form from the onset. The
scaling exponents are then completely free.
Finally the constraint on the k → ∞ limit of the fixed point equations (2.88) are now
fi (p̂)
.
p̂ηi
δZi (p̂ → 0) = −1 +
(3.77)
Eqs. (3.77) are the consequence of the requirement that the fixed point effective action
be qualititatively different in the two limits k → 0 and k → ∞. Indeed, we can see by
inserteing them in Eqs. (3.66) that the form (3.77) for δZi (p̂) is such that the scaling in
terms of ηi is removed and replaced by the functions fi (p̂).
3.3.4. Computing observable quantities
Before we move on to the solutions of the RG flow equations we relate ηi , h νk (p) and
Fk−1 (p) to the physically observable quantities S2 (τ, r), χ and z. The physical quantities
are only related to the flowing quantities in the limit k → 0. We can however compute a
scale dependent velocity increment from the truncated effective action (3.54). This one
reduces to the physical quantity when the cut-off is removed k → 0.
First we express the second order velocity increment in terms of νk (p) and Fk−1 (p).
For this we write it in terms of the Fourier modes of the velocity field. Using Eqs. (2.32)
and (F.22) we can write
⟨vi (ω, p)vj (ω ′ , p′ )⟩ = δ(ω + ω ′ )δ(p + p′ )δij
(2π)d+1
.
+ νk (p)2 p4 ]
Fk−1 (p) [ω 2
(3.78)
Then we take the double Fourier transform of Eq. (3.78) and write
Z
1
⟨vi (t + τ, x + r)vj (t, x)⟩k = δij
ei(ωτ −p·r) −1 2
Fk (ω + νk (p)2 p4 )
ω,p
=
δij
2
Z
p
60
2
e−ip·r
e−νk (p)p |τ |
.
−1
Fk (p)νk (p)p2
(3.79)
3. Burgers Turbulence
We have used the residue theorem in order to perform the frequency integration and
obtain the second equality. We finally get
Z
1 − exp −ip · r − νk (p)p2 |τ |
S2,k (τ, r) =
Fk−1 (p)νk (p)p2
p
.
Inserting Eqs. (3.68) and taking the limit k → 0 we can write
i
h
q
Z
z1 (η1 −η2 )/2 2
r
p̂
k
|τ
|
1
−
exp
−ip̂
·
kr−
2
z2
z2 k
.
S2 (τ, r) =
(η
+η
)/2
1
2
z1 z2
p̂
(3.80)
(3.81)
p̂
The apparent k-dependence can be removed by changing the variable integration to p
and inserting Eqs. (3.72). Since k is a free parameter we can chose it to be k = 1/r and
get
h
i
q
Z
z10
(η1 −η2 )/2 |τ |
r
1
−
exp
−ip̂
−
(r
p̂)
z
z20
z20 1
S2 (τ, r) = r(η1 +η2 −2d)/2
. (3.82)
z10 z20
p̂(η1 +η2 )/2
p̂
p̂z is the z-component of p̂. From this we can identify
χ=
η1 + η2 − 2d
+ 1,
4
z=
η1 − η2
.
2
(3.83)
Inserting Eq. (3.74) this can be written as
χ = η1 − d,
z = −η1 + 2 + d.
(3.84)
Note that we recover Eq. (3.16), χ + z = 2, from the fixed point equations. This is the
first non-trivial result that emerges form our approach.
Let us finally note that Eq. (3.82) can be slightly simplified by using Eqs. (3.75) and
(3.74) to express z2 and η2 in terms of z1 , h and η1 ,
r2(η1 −d−1)
S2 (τ, r) =
ĝ
2
z10
τ
rd+2−η1
1
z10
,
d
ĝ(x) =
h
Z
p̂
1 − exp −p̂d+2−η1 x/h−ip̂z
.
p̂2η1 −2−d
(3.85)
We see that even if all the fixed point parameters are known z10 stays undetermined and
can be fixed by comparing S2 (τ, r) to an experiment.
Eq. (3.78) can be used to compute the kinetic energy of the system
Z
Z
Z
V
1
V
1
1
2
=
.
Ek,kin ≡
⟨v (t, x)⟩k =
−1
−1
2
2
4
2 x
2 ω,p Fk (p) [ω + νk (p) p ]
4 p Fk (p)νk (p)p2
(3.86)
61
3. Burgers Turbulence
R
V = x is the volume of the system. As in Eq. (3.79) we used the residue theorem to
perform the frequency integration. We can as well identify the kinetic energy spectrum
from Eq. (3.86),
Z
1
1
1
ϵk,kin (p) ≡
⟨v(ω, p) · v(−ω, −p)⟩k =
.
(3.87)
−1
2 ω
2 Fk (p)νk (p)p2
which in the limit k → 0 scales as
ϵ0,kin (p) = ϵkin (p) ∼ p−(η1 +η2 )/2 = pd+2−2η1 ,
(3.88)
at the RG fixed point. See Sections (3.3.5.3) or (4.1) for a discussion of its physical
interpretation.
3.3.5. Asymptotic properties of the flow integrals
In this section we investigate the solutions of the fixed point equations. In particular
(2)
we look into the properties of Iˆi (p̂) for very large and very small arguments and solve
the fixed point euqations in the assymptotic regimes. We consider the consequences of
Eqs. (3.76) with ηi related through Eq. (3.74) without imposing the conditions given by
Eqs. (3.77) yet. η1 and h are to be considered as free parameters first. The idea is to
solve the differential equations for arbitrary values of ηi and h and to choose the values
for which Eqs.(3.77) are true later. Note that the following discussion only applies to
non Gaussian fixed points where h ̸= 0 and Eq. (3.74) is mandatory. We start by writing
our flow integrals as10
(2)
Iˆ1 (p̂) = p̂ F1,1 (p̂)
+ p̂(η1 +η2 )/2+2
p
1 + δZ1 (p̂)
p
1 + δZ2 (p̂) F1,2 (p̂)
+ p̂η1 +η2 +2 δd1 [1 + δZ1 (p̂)] [1 + δZ2 (p̂)] F1,3 (p̂),
(2)
Iˆ2 (p̂) = p̂(η1 +η2 )/2+2
(3.89)
p
p
1 + δZ1 (p̂) 1 + δZ2 (p̂) F2,1 (p̂)
+ p̂η2 +2 [1 + δZ2 (p̂)] F2,2 (p̂)
+ p̂η1 +η2 +2 δd1 [1 + δZ1 (p̂)] [1 + δZ2 (p̂)] F2,3 (p̂)
+ p̂2η2 +2 δd1 [1 + δZ2 (p̂)]2 F2,4 (p̂).
(3.90)
Explicit expressions for the functions Fi,j (p̂) are given in Eqs. (C.4) to (C.10) of Appendix
C. These functions where defined in such a way that they are analytic and non vanishing
′ (0) p̂ + O(p̂2 ).
at p̂ = 0. They can be Taylor expanded Fi,j (p̂ → 0) = Fi,j (0) + Fi,j
Expressions (in terms of η1 and δZi (1)) for the first term of their Taylor expansions are
given in Eqs. (C.11) and (C.12) of Appendix C.1. Similarly, their large p̂ asymptotic
behaviour can be computed and turn out to be power laws. Details on how to compute
their exponents and pre-factors are given Appendix (C.2) and explicit expressions are
given in Eqs. (C.16) to (C.18) and (C.31) to (C.33).
10
Note that the exponents of the following expression contain the Kronecker delta δd1 . Indeed, cancellations in the leading pre-factors of F1,3 (p̂), F2,3 (p̂) and F2,4 (p̂) occur for d = 1 and the sub-leading
terms must be taken into account.
62
3. Burgers Turbulence
3.3.5.1. Scaling limit (p ≪ k)
We start by looking at the limit p̂ → 0. Since all but the the constant terms of Fi,j (p̂)
(2)
are sub-dominant in the limit p̂ → 0 the asymptotic form of Iˆi (p̂ ≪ 1) only depends
on Fi,j (0). This makes apparent the asymptotic behaviour of our flow integrals as a
sum of power laws multiplied by combinations of [1 + δZi (p̂ ≪ 1)] and Fi,j (0). We see
that different asymptotic behaviours of δZi (p̂ ≪ 1) and values of ηi lead to different
terms dominating. The true dominating terms must be determined self-consistently.
The asymptotic form of Eqs. (3.76) is
p
δZ1
h n
(p̂) ∼
= − η1 F1,1 + F1,2 p̂(η1 +η2 )/2+1 [1 + δZ1 (p̂)] [1 + δZ2 (p̂)]
dp̂
p̂
o
η1 +η2 +2δd1 −1
+F1,3 p̂
[1 + δZ1 (p̂)] [1 + δZ2 (p̂)] ,
n
p
δZ2
h
(p̂) ∼
= − η2 F2,1 p̂(η1 +η2 )/2+1 [1 + δZ1 (p̂)] [1 + δZ2 (p̂)]
dp̂
p̂
+F2,2 p̂η2 +1 [1 + δZ2 (p̂)] + F2,3 p̂η1 +η2 +2δd1 −1 [1 + δZ1 (p̂)] [1 + δZ2 (p̂)]
o
+F2,4 p̂2η2 +2δd1 −1 [1 + δZ2 (p̂)]2 ,
(3.91)
with the short-hand notation Fi,j = Fi,j (0). The asymptotic behaviour of δZi (p̂) can
be extracted from these equations with the method of dominant balance [223]. This
is however a lengthy process which requires the solution of 12 different sets of coupled
differential equations since each combinations of terms must be considered separately.
We instead make the further assumption that both functions behave as power laws at
small momenta. More precisely, we will use the form
δZi (p̂ → 0) ∼
= ci +
Ai
p̂αi −ηi ,
αi − ηi
(3.92)
which is consistent with Eq. (3.77) only when ci = −1 and fi (p̂) ∼
= Ai p̂αi /(αi − ηi ). Note
that this is a self consistent assumption. Indeed, it gives δZi (p̂ → 0) ∼ p̂min(0,αi −ηi )
which implies that we have power laws on both sides of Eq. (3.91). Moreover since this
expression is motivated by the differential equation (3.76), αi is really defined through11
δZi′ (p̂ → 0) ∼ p̂αi −ηi −1 and ci is an integration constant. Then the case αi − ηi = 0 is to
be understood as δZi (p̂ → 0) ∼ ci + Ai log(p̂).
We can now insert this ansatz into Eq. (3.91) and match the exponents on both sides.
The asymptotic form δZi (p̂ → 0) ∼ p̂min(0,αi −ηi ) implies that we need to consider the
four cases given by αi − ηi < 0 and αi − ηi > 0 separately if we want to extract values
of αi that provide solutions to Eqs. (3.76). Note that the cases αi − ηi = 0 correspond
to logarithmic divergences of δZi (p̂) and should be treated separately as well.
• We first consider the case of both αi − ηi < 0. Then the constants ci are subdominant and do not need to be taken into account on the right-hand-side of
11
We use the notation δZi′ (p̂) = dδZi (p̂)/dp̂.
63
3. Burgers Turbulence
Eqs. (3.91). Note that this case also arises when both ci = −1, i.e. at solutions of
the full RG fixed-point equations. We see that, ηi do not appear any more in the
exponents. αi must satisfy
α
α2
1
α1 = min 1,
+
+ 2, α1 + α2 + 2δd1 ,
α 2 α 2
2
1
+
+ 2, α2 + 2, α1 + α2 + 2δd1 , 2α2 + 2δd1 .
(3.93)
α2 = min
2
2
This implies that αi are restricted to
(α1 , α2 ) ∈ ([−2δd1 , 1] , −2δd1 ) ,
(α1 , α2 ) = (1, 5) .
(3.94)
See Figure (3.2) where these solution are plotted in red and black (red and blue
for d = 1) vertical lines and big dots. This asymptotic solutions can only arise if
αi − ηi < 0 or ci = −1.
• Next we consider the case α1 − η1 < 0 and α2 − η2 > 0. In this case, matching the
exponents on both sides of Eqs. (3.91) leads to
α
η2
1
+
+ 2, α1 + η2 + 2δd1 ,
α1 = min 1,
α 2 η 2
1
2
(3.95)
α2 = min
+
+ 2, η2 + 2, α1 + η2 + 2δd1 , 2η2 + 2δd1 .
2
2
Then we get
α1 ∈


{1}
if η2 > −2δd1
]−∞, 1] if η2 = −2δd1 ,

∅
if η2 < −2δd1
(3.96)
and α2 is a function of α1 and η2 given by the second line of Eqs. (3.95). Inserting
Eq. (3.74) we can express η2 in terms of η1 in Eq. (3.96) and check that we always
get solutions that satisfy α1 − η1 < 0. However second condition for this case to
be realised α2 − η2 > 0 restricst the solution further. Indeed, we have α2 − η2 > 0
only for η2 ∈ ]−2δd1 , 5[. This in turn implies that the only consistent solutions are
η2 ∈ ]−2δd1 , 5[ ,
α1 = 1,
α2 = min
5 η2
+ , η2 + 2, η2 + 1 + 2δd1 , 2η2 + 2δd1 .
2
2
(3.97)
These solutions are plotted on Figure (3.2) as a horizontal thin green line. Note
that they seem to connect the different solutions of Eqs. (3.93).
• The same analysis can be done for all the other cases: α1 − η1 > 0, α2 − η2 < 0,
both αi − ηi > 0 and logarithmic divergences. None of these cases has a consistent
solution.
64
3. Burgers Turbulence
4
2
α1
0
−2
−2
0
2
4
6
α2
Figure 3.2.: A graphical representation of the values of the bare exponents α1 and α2 ,
defined in Eqs. (3.92) in blue and red for d = 1 and black and red for d ̸= 1,
which characterise the non-Gaussian fixed points. These different solutions
are connected by a thin horizontal green line which marks the solutions of
Eqs. (3.95) where Eqs. (3.77) can not be satisfied. Note that it stops at the
upper black dot when d ̸= 1 and at the upper blue for d = 1. It connects
the solutions of Eq. (3.93) in all dimensions. The scaling exponents of the
original stochastic Burgers equation, as described by S[v], Eqs. (3.42) or
(3.43), are related by α1 = α2 + 4 and are shown as a black line. The
blue dot at (α1 , α2 ) = (1, −2) corresponds to the fixed point investigated
in [58, 112], while the top half of the blue line (for α1 > 0) represents the
set of points found in [112] for different types of forcing, corresponding to a
bare action with exponents given by the dotted black line. In all dimensions
we find a continuum of fixed points, and a further, new fixed point that
may arise if the fluid is forced on large scales (red dot). Along the blue
line we find that the scaling exponent satisfies η1 = 2 − α1 /2 while it is
η1 = 2 − α1 /2 + d along the black line. η2 is related to η1 through Eq. (3.74).
The second case α1 − η1 < 0 and α2 − η2 > 0 provides an asymptotic solutions to the
differential equation (3.76). However, it requires that
−1 + δZ2 (p̂ → 0) ∼ const ̸= 0.
(3.98)
If const = 0 then the power law, −1 + δZ2 (p̂ → 0) ∼ p̂α2 −η2 becomes dominant again
and we are back to the first case. In other words we have found two sets of solutions of
the differential equations (3.76) but only one of them can contain solutions of Eqs. (3.77)
as well. We focus on these solutions in the remainder of this Section.
The above results suggest that the allowed combinations (α1 , α2 ) correspond to the
different possible non-Gaussian fixed points of the theory. For the different dimensions,
we find, for the scaling exponents relevant in the limit p ≪ k, a connected interval for
α1 and an additional point at (α1 , α2 ) = (1, 5).
65
3. Burgers Turbulence
Having identified the different fixed points of the RG flow we can now study their
properties. In principle this can be done by numerically solving Eqs. (3.76) for different
values of η1 and h and tuning both parameters to satisfy Eqs. (3.77). Note that Eq. (3.74)
makes η2 a simple function of η1 . We instead look a little deeper into the analytic
properties of the fixed point equations. Indeed, we show now that the value of the
scaling exponents ηi at the different fixed points can be inferred by looking at the p̂ ≪ 1
limit of Eqs. (3.76).
Along the lines of fixed point given by (α1 , α2 ) = (] − 2δd1 , 1[, −2δd1 ) and shown in
red and black in Figure (3.2) we can relate the physical scaling exponents ηi to the UV
exponent α1 in a simple way. Inserting Eqs. (3.92) and keeping only the dominating
terms we have seen that the right hand side of Eqs. (3.91) is a linear combination of
monomials. Since we assume that αi − ηi < 0 the exponents of these monomials only
depend on αi . In fact the different exponents are given by the different arguments of the
minimum functions on the right-hand side of Eqs. (3.93) minus (1 + ηi ) (see Eqs. (D.2)
in Appendix D). We see that once we have chosen values of αi from the solutions of
Eqs. (3.93) we can compare the different exponents and identify the leading monomial
in the limit p̂ → 0. It then becomes possible to match the pre-factors on both sides of
Eqs. (3.91) and extract equations containing both Ai . These are given in Appendix D
in Eqs. (D.7) to Eqs. (D.10) and Eqs. (D.3) to Eqs. (D.6) for d = 1 with the short-hand
notation, ai = Ai (αi − ηi ). Contrarily to Eqs. (3.93) these are not closed and can not
be solved independently of Eqs. (3.76). However, for the interval sets of αi marked by
the black and blue lines in Figure (3.2), the ratio of the equation for Ai , Eqs. (D.4) and
(D.8), only contains α1 and ηi and makes it possible to relate both quantities. Taking
the ratio of Eqs. (D.4) and using the fact that F1,3 = F2,4 gives
a1 η1 − α1
a1
= ,
a2 η2 + 2
a2
(3.99)
for d = 1 and the ratio of Eqs. (D.8) gives
a1 η1 − α1
a1
= ,
a2 η2
a2
(3.100)
for d ̸= 1. We see that ai drop out of these equations and that, inserting Eq. (3.74), η1
can be related to α1 ,
2 − α1 /2,
d=1
.
(3.101)
η1 =
2 − α1 /2 + d, d ̸= 1
Specifically, for d = 1, we get 3/2 < η1 < 3 and, for d ̸= 1, 3/2 + d < η1 < 2 + d. These
intervals are marked by dark grey shading in Figure (3.4). In view of Refs. [61, 162], we
note that we find an upper bound to the regime of allowed η1 . Specifically, Eq. (3.84)
implies that above this bound, the dynamical critical exponent would become negative.
It is not possible to relate η1 to αi at the other solutions of Eq. (3.93) in a similar way.
At these points, the full solution of Eq. (3.76) is necessary to verify the existence of the
RG fixed points and extract the values of the ηi .
66
3. Burgers Turbulence
Note that there is a qualitative difference in between the intervals of fixed points,
(α1 , α2 ) = (]−2δd1 , 1[, −2δd1 ) and their end points (α1 , α2 ) = ({−2δd1 , 1}, −2δd1 ). Along
the blue and black lines of Figure (3.2) only one of the monomials of the right-hand side
of Eq. (3.91) dominates while at the end points two terms contribute to the asymptotic
behaviour (see Appendix D and Eqs. (D.2)). As a result Eq. (3.101) does not apply at
the end points since the corresponding equations do not simplify. The values of ηi is
still undetermined. However, under the assumption that η1 is a continuous function of
α1 , one obtains η1 = 2 − α1 /2 + d − δd1 also at the end points. We remark that the
calculation reported in Ref. [112] provides a continuum of fixed points, with η1 = 2−α1 /2
and 0 < α1 < 1, and an additional fixed point with η1 = 3/2, the end-point value.
Finally let us remark that none of these resulting small-p̂ scalings corresponds to that
of the action S[v] for Burgers’ equation. In S[v], ν is p-independent, which implies that
(2)
(2)
the ratio of Γk→∞ (0, p) and ∂Γk→∞ /∂ω 2 (0, p) scales as p4 , see Eq. (3.55). In the bare
limit k → ∞ we have
(2)
∂Γ
A
A2
(2)
1
k→∞ ∼
Γk→∞ (0, p) ∼
(3.102)
pα1 ,
pα2 .
=
=
2
α1 − η1
∂ω α2 − η2
(0,p)
Hence, taking the ratio of Eqs. (3.102) and requiring it to scale as in the bare case gives
(2)
α1 = α2 +4. Analogously, Eq. (3.8) implies that ∂Γk→∞ /∂ω 2 (0, p) scales as p−β and thus
that α2 = −β. The resulting possible combinations (α1 , α2 ) = (4 − β, −β) are marked
by the black (dashed/solid) line in Figure (3.2). As expected, there is no choice of the
forcing exponent β that makes the stochastic Burgers equation sit at a non-Gaussian
RG fixed point for all values of k.
Finally note that the relatively large value of α2 = 5 at the isolated fixed point suggests
that it is realised for a highly non local forcing, β = −5.
3.3.5.2. Scaling limit (p ≫ k)
In the opposite limit of vanishing cut-off all fluctuations are integrated out and the full
effective theory emerges. We have seen in Section (2.2.2.1) that two possibilities arise.
Either δZi (p̂ → ∞) = 0 or δZi (p̂ → ∞) = ∞. In the first case, once the cut-off scale
is sent to zero, see Eq. (3.66), one finds fixed points with correlations given by scaling
functions across all momentum scales. If, on the other hand, limp̂→∞ δZi (p̂) = ∞, an
RG fixed point can only exist if the scaling range is restricted to momenta smaller than
some upper cut-off Λ. Then scaling only arises within the range of physical momenta
and δZi (1 ≪ p̂ < Λ/k) ∼
= 0. In this situation of an UV divergent fixed point, the theory
is not well defined exactly at the fixed point but the latter can be approached arbitrarily
by choosing Λ accordingly large. See Section (2.2.2.1) for a detailed discussion.
Here, we consider UV finite fixed points and take the limit p̂ → ∞ in the flow integrals.
The boundary condition δZi (p̂ → ∞) = 0 then allows us to write Eqs. (3.76) in the
67
3. Burgers Turbulence
9
12
8
9
7
6
6
β1
3
5
β2 0
4
−3
3
2
−6
1
−9
0
1
2
3
4
5
−12
6
−6 −3
0
η1
3
6
9
η2
Figure 3.3.: Plots of β1 (left) and β2 (right) with respect to η1 , η2 respectively for d = 1
(blue solid lines) and d = 3 (black dotted lines). ηi is plotted on both plots
as a thin red line. We see that we can only have βi − ηi < 0 for a finite range
of η1 . As always η2 is related to η1 through Eq. (3.74).
integral form
Z
δZi (p̂) = h
∞
dy
p̂
(2)
Iˆi (y)
.
y ηi +1
(3.103)
For p̂ ≫ 1, also the integration variable y exceeds 1 by far such that we can approximate
(2)
δZi (y) = 0 in the integrals Iˆi (y). The flow integrals can be further approximated by
keeping only their leading term as p̂ → ∞,
(2)
Iˆi (p̂ → ∞) ∼ p̂βi .
The exponents βi are determined in


 4 − 2δd1 − 2η1 + d
3η1 − 2d − 2
β1 =


2η1 − d

3(η1 − d − 2)



 5η − 4d − 6
1
β2 =

3η1 − 2d − 2



4η1 − 3d − 4 − 2δd1
(3.104)
Appendix C.2. We give the final result here,
if (6 + 3d − 2δd1 )/5 ≥ η1
if (6 + 3d − 2δd1 )/5 < η1 ≤ d + 2 ,
(3.105)
if η1 > d + 2
if η1 ≤ d/2
if d/2 < η1 ≤ d + 2
if d + 2 < η1 ≤ d + 2 + 2δd1
if d + 2 + 2δd1 < η1
68
.
(3.106)
3. Burgers Turbulence
These are piece-wise affine functions of η1 . They are plotted for d = 1 (blue solid lines)
and d = 3 (black dashed lines) in Figure (3.3) as an example. Note that these equations
are strictly valid only when βi − ηi < 0 since they where computed with the assumption
δZi (p̂ → ∞) = 0. In order to obtain finite integrals on the right hand side of Eqs. (3.103),
it is necessary that βi < ηi . We see from Figure (3.3) that this requirement restricts η1
(and η2 ) to a finite range. A careful analysis of Eqs. (3.105) and (3.106) shows that this
range is given by
d=1:
d
< η1 < d + 1,
(3.107)
d ̸= 1 : (d + 4)/3
which is shown as the white area in Figure (3.4).
As we discussed in the beginning of this section Eq. (3.107) limits the range of UV
convergent fixed points. Outside of the white area of Figure (3.4) the flow integrals need
to be regularised in the UV before they can be used to extract fixed point properties.
The introduction of such an UV cut-off introduces a scaling range where the fixed point
properties are realised. In such cases the fixed point theory is not well defined since it
contains an infinity in the UV. It can however be approached arbitrarily close by taking
a large enough value of the UV cut-off.
Figure (3.4) contains the range of UV convergent fixed points as well as the values of
η1 that can be computed form the literature on KPZ equation. The blue dots correspond
to a potential12 delta correlated in space, i.e. β = 2. These values of η1 are average values
from the KPZ literature (see [193, 194, 196–200] and Table I. of Ref. [58]). They are
given by,
η1 = 3/2
for d = 1,
η1 = 2.379(15)
for d = 2,
η1 = 3.300(12)
for d = 3.
(3.108)
The vertical blue lines are the fixed points found in [61] for values of β > βtrans (d).
See the discussion at the end of Section (3.1) and Eq. (3.23) for more details. The
question of weather these lines continue for arbitrarily small (and negative) values of β
(i.e. arbitrarily large values of η1 ) is not addressed in [61] and only mentioned briefly
in [162]. It is however clear that something must happen because as β is decreased the
relation13 η1 = (4 + 2d − β)/3 implies that the dynamical critical exponent z decreases
and even reaches zero for β = −d − 2. The black area of Figure (3.4) corresponds to
negative values of z.
We see that the local KPZ fixed points (blue dots) are UV convergent. For d = 1
the analysis of Section (3.3.5.1) is in good agreement with the results of [61]. We find a
continuum of fixed point with η1 > 3/2 and an additional fixed point at the edge of this
12
Note that this is work about the KPZ equation. The forcing is a potential field. See Eq. (3.11) where
U (p) is defined.
13
Again, see the discussion at the end of Section (3.1) and Eq. (3.23) where the results of [61] are
summarised.
69
3. Burgers Turbulence
3
η1 2
1
1
2
d
3
Figure 3.4.: The range of values of η1 (defined in Eqs. (3.66) and related to χ and z in
Eqs. (3.84)), which correspond to UV convergent non-Gaussian fixed-points
for different spatial dimensions d (white area). The white vertical stripe at
d = 1 only applies to this single dimension. In the regions shaded in light
and dark grey any potential fixed point is UV divergent such that the RG
flow integrals must be regularised in the UV. The blue dots correspond to
the average literature values for the exponent η1 , [193, 194, 196–200]. Their
values, as given in Table I. of Ref. [58], are given in Eq. (3.108). They
correspond to a delta correlated forcing in the KPZ framework, i.e. the
exponent β, defined in Eq. (3.8), is equal to two. The vertical blue lines
marks the range of possible exponents found in [61] for different forcings,
with the exponent β < 2 It is not known where this set of fixed points
ceases to exist. The dark-grey shaded area marks the range of values of η1
that we find at the sets (α1 , α2 ) shown as the blue and black solid lines at
α2 = −2 and 0 in Figure (3.2), respectively. It continues accordingly also for
η1 > 7/2. In between the two dotted lines there is a direct cascade of kinetic
energy. In the black area at the left top, the dynamical critical exponent z
is negative.
continuum. Strictly speaking we can not infer the value of η1 at the blue dots, but it is
reasonable to assume that it is a continuous function of α1 . Then we obtain η1 = 3/2 on
the blue dot at d = 1. The same structure was found in [61]. There the value η1 = 3/2
corresponds to the singular short-range fixed point which is realised for β ≥ 3/2. And
the values η1 > 3/2 to the set of non-local fixed points with β < 3/2.
For d ̸= 1 however Eq. (3.101) provides fixed points which are UV divergent. Moreover, based to the different values of η1 that they find, the authors of [61] seem to find
more fixed points than we do. Our calculation has however the advantage of explicitly
70
3. Burgers Turbulence
excluding any fixed point with z < 0. We explain these discrepancies by the fact that
our truncation is not consistent with KPZ equation for d ̸= 1 since it can not describe
an irrotational fluid. Even though they have overlapping values of η1 , the fixed points
that we find are most likely different from the ones of [61] which are purely irrotational.
3.3.5.3. Implications for driving and turbulent cascades
The bounds (3.107) on η1 have distinct physical interpretations: While the lower bound
can be expressed as a regularity condition on the type of forcing that is sampled by the
stochastic process, the upper bound marks the onset of a direct energy cascade.
Locality of the forcing
Let us discuss first the lower bound at η1 = (d + 4)/3 and η1 = 1 for d = 1. As the
forcing is a Gaussian random variable, it follows the probability distribution
Z
1
1
2 −1
Pk [f ] =
|f (ω, p)| Fk (p) .
(3.109)
exp −
N
2 ω,p
N is a normalisation factor. This distribution implies
the probability of a spatially
R that
−1
local force field f (t, x) ∼ δ(x − xf ) can be finite if p Fk (p) is finite and is necessarily
zero otherwise. Furthermore, for the latter integral to be finite, the critical exponent
at an UV finite fixed point needs to fulfil η1 < (d + 4)/3, as one finds by inserting the
parametrisation Eq. (3.68) for Fk−1 . We conclude that in the lower grey shaded area
of Figure (3.4), η1 < (d + 4)/3, local Gaussian forcing of the type f (t, x) ∼ δ(x − xf )
is included while it is suppressed for η1 > (d + 4)/3. We have shown above that UV
finite non-Gaussian fixed points require η1 > (d + 4)/3. Hence, for an RG fixed point
to be UV finite, the forcing needs to be sufficiently regular in space-time, specifically
limp→∞ |f (ω, p)| = limω→∞ ∂ω |f (ω, p)| = 0.
Note that, in the case d = 1, the regularity condition is modified. The fixed points
are UV finite above the relatively lower limit η1 > 1. This is a consequence of the fact
that in d = 1 dimension, point-like shocks are stable solutions. Applying a force such as
f (t, x) ∼ δ(x − xs ) is comparable to inserting a shock at the position xs .
Energy cascades
The upper bound, η1 = d + 1, can be related to the appearance of a direct cascade
of energy. We briefly sketch the argument leading to this result in the following. We
look at the kinetic energy spectrum, ϵkin (q), which is defined through the kinetic energy
density of the system
Z
Z
1
1
2
Ekin ≡
⟨v (t, x)⟩ = V ϵkin (q),
ϵkin (q) = ⟨v(t, q) · v(t, −q)⟩.
(3.110)
2 x
2
q
71
3. Burgers Turbulence
R
V = x is the volume of the system. ϵkin (q) characterises the distribution of kinetic
energy over the different Fourier modes of the stationary state. Note that we consider a
uniform system and abuse the notation by defining,
⟨v(t, q) · v(t, q ′ )⟩ ≡ δ(q + q ′ )(2π)d ⟨v(t, q) · v(t, −q)⟩.
(3.111)
By definition the kinetic energy density as well as the kinetic energy spectrum are timeindependent in a stationary state. Then one can compute the time derivative of ϵkin (p),
insert the equation of motion (3.1) and equate this to zero. We find
∂t ϵkin (q) = −ϵν (q) + ϵf (q) + ϵadv (q) = 0,
(3.112)
with14
ϵν (q) = νp2 ⟨v(t, q) · v(t, −q)⟩,
1
ϵf (q) = [⟨f (t, q) · v(t, −q)⟩ + ⟨v(t, q) · f (t, −q)⟩] ,
2
(3.113)
and
ϵadv (q) =
i
2
Z
p · {⟨v(t, q − p) [v(t, p) · v(t, −q)]⟩ + ⟨v(t, −q − p) [v(t, p) · v(t, q)]⟩} .
p
(3.114)
Note that we use the equivalent of the notation of Eq. (3.111) for the three-point function
here (see Eq. (F.11)). This makes possible the study of fluxes of energy at the steady
state. ϵν (q) and ϵf (q) represent the energy and dissipation rates respectively while
ϵadv (q) is the amount of kinetic energy that arrives at the Fourier mode q from the
other modes. ϵf (q) is hard to estimate but ϵν (q) and ϵadv (q) are both expressed in
terms of velocity field correlation functions and can be computed from the effective
average action. See Section (2.1.4) and Eqs. (2.32) and (2.35). ϵf (q) is then determined
by Eq. (3.112). In terms of Fk−1 (p) and νk (p) we have,
ϵk,ν (q) =
dν
1
,
−1
2 Fk (q)νk (q)
(3.115)
and
1
ϵk,adv (q) =
8
Z
p
1
ν̃p + ν̃q + ν̃|p−q|
"
dp2 + p · (q − p)
Fk−1 (|p − q|)Fk−1 (p) ν̃|p−q| ν̃p
#
p2 − p · q
p · (q − p) − dp · q
+ −1
+
Fk (q)Fk−1 (p) ν̃q ν̃p Fk−1 (|p − q|)Fk−1 (q) ν̃|p−q| ν̃q
"
Z
1
1
dp2 − p · (p + q)
+
8 p ν̃p + ν̃q + ν̃|p+q| Fk−1 (|p + q|)Fk−1 (p) ν̃|p+q| ν̃p
#
p2 + p · q
dp · q − p · (p + q)
+ −1
+
.
Fk (q)Fk−1 (p) ν̃q ν̃p Fk−1 (|p + q|)Fk−1 (q) ν̃|p+q| ν̃q
14
The notation of Eq. (3.111) is used here as well for ⟨f (t, q) · v(t, −q)⟩.
72
(3.116)
3. Burgers Turbulence
The physical correlation functions are recovered in the limit k → 0. As in the case of
Eqs. (3.61) and (3.62) we use the short-hand notation ν̃p = νk (p)p2 . We see that we are
able to identify the kinetic energy transport kernel. Since our system is isotropic all the
quantities of this section only depend on the norm of momenta. We define the energy
transport kernel Ek,trans (p, q) through,
Z
Z ∞
d−1
ϵk,adv (q) =
dq Ek,trans (p, q),
(3.117)
Ek,adv (q) ≡ q
Ω
0
which invariant under rotations by construction. We have simply averaged ϵk,adv (q)
over all the directions in which q may point and multiplied by the surface factor q d−1 .
Ek,adv (q) represents the total kinetic energy transfer to the momentum shell of radius q.
Ek,trans (p, q) is then the double angular average of the integrand of Eq. (3.116) multiplied
by two surface factors (qp)d−1 . It measures the transfer of kinetic energy from the
momentum shell p to q. It characterises the flux of energy across the different scales of
the system.
In the limit k → 0, E0,trans (p, q) ≡ Etrans (p, q) is a function of h and η1 only. Evaluating
Etrans (p, q) in this way, a direct energy cascade, i.e. , transport which is local in momentum space on a logarithmic scale, can be identified numerically for d+1 < η1 < d+3/2. In
this regime, Etrans (p, q) is non-vanishing only for p ∼
= q (locality), positive for p < q and
negative for p > q (positive directionality), and Etrans (p, q) ∼
= −Etrans (p, −q) (balance of
driving and dissipation, i.e. , inertial turbulent transport).
Note that it is natural to have a direct cascade requiring a UV regulator: Physically,
a cascade is realised only in a given inertial range. For example, in 3d Kolmogorov
turbulence, energy is injected on the largest scales and transported to smaller scales
by the non-linear dynamics which leads to larger eddies feeding into smaller ones. The
kinetic energy is dissipated into heat once it reaches the end of the inertial range set
by the viscosity. At an RG fixed point, the inertial range by definition extends over all
momenta. Hence, the UV cut-off of the dissipation scale is absent. As a result, energy in
a direct cascade is transported to infinitely large momenta, leading to a UV divergence
of the fixed-point theory.
73
4. Ultra-cold Bose Gases
In this section we consider the dynamics of dilute gases of bosons with a contact interaction. We start by summarising relevant results concerning out-of-equilibrium steady
states in closed systems. We introduce the concept of Non-Thermal Fixed Points (NTFP)
and their scaling properties in Section (4.1). In Section (4.2) we introduce the DrivenDissipative Gross–Pitaevskii Equation (DDGPE) as a model for a dilute gas of Bosons
in contact with external reservoirs of particles and energy. The inclusion of driving
and dissipation into the closed system enables the mapping of such a system onto the
stochastic Burgers equation which was introduced in Part 3. Finally we assume that
both approaches, the closed NTFP and the open DDGPE, describe the same out-ofequilibrium steady state in Section (4.3) and apply the results of Section (3.3) in order
to extract non-trivial scaling relations in the context of out-of-equilibrium Bose gases at
a NTFP. In particular we show that Eq. (3.16) which originates in the Galilei invariance
of the hydrodynamic theory has a dual expression in terms of the scaling exponents of
the Bose gas in Section (4.3.1) and Eq. (4.22), and we use the values of χ and z known
form the Kardar–Parisi–Zhang (KPZ) literature for d = 1, 2 and 3 to compute anomalous correlation to the scaling of the compressible kinetic energy spectrum of the Bose
gas, see Section (4.3.2), Eqs. (4.27) and Figure (4.2).
In order to describe the ultra-cold Bose gas we consider a field theory defined by the
action
Z i ∗
1 ⃗
g
∗
∗
∗
∗ 2
⃗
S=
(ψ ∂t ψ − ψ∂t ψ ) −
∇ψ · ∇ψ + µ ψψ − (ψψ ) ,
(4.1)
2m
2
t,x 2
and bosonic commutation relations. Here and in the following we set ~ = 1. g is
the interaction constant which is proportional to the boson-boson s-wave scattering
length, m is the mass of the particles and µ is the chemical potential. Outside of
thermal equilibrium there is no thermodynamic reservoir and µ is not precisely defined.
There is however a fixed average density of particles which is fixed by a the Lagrange
multiplier µ, ⟨n⟩ = f (µ). Then calling µ "chemical potential" is an abuse of language.
What we actually mean is the inverse of f , µ ≡ f −1 (⟨n⟩). The particular form of the
interaction potential that we consider g (ψψ ∗ )2 applies in the dilute limit where the
inter-particle distance is much larger than their scattering length. Only the contact
interaction remains.
The full quantum dynamics of the theory can be expressed as a path integral as in
Section (3.2) by means of the Schwinger-Keldysh formalism [224–228]. At the mean field
level the dynamics follow directly from the Euler-Lagrange equations and are described
74
4. Ultra-cold Bose Gases
by the Gross–Pitaevskii equation (GPE) [229, 230] (see [26] for an overview),
1 2
2
i∂t ψ = −
∇ − µ + g |ψ| ψ.
2m
(4.2)
Because of the Bose condensation the low energy modes are strongly occupied and
classical fluctuations are much stronger than quantum fluctuations. Then it becomes
meaningful to make a semi-classical approximation where the quantum fluctuations are
handled approximately (see e.g. [25, 34, 44, 231, 232]). In particular the c-field methods
achieve this by introducing an Ultraviolet (UV) cut-off which separates the phase space
into a classical (low energy, high occupation number) and quantum (high energy, small
occupation number) regions.
Within this set-up one can write a Fokker–Planck equation for the time evolution of
the Wigner quasi-probability distribution (see e.g. [233, 234]) and project it on the classical region. When the terms containing three field derivatives of the Wigner distribution
are neglected, the time evolution of the different correlation functions can be expressed
in terms of stochastic differential equations. Then the field expectation value in the
classical region is described in terms of the mean field equation (GPE) with additional
collision integrals to model interactions with the quantum region which is assumed to be
at thermal equilibrium. See [34, 44] and references therein for an overview. When the
classical region is highly occupied one can neglect its coupling with the quantum region.
Quantum fluctuations then are included in the mean field description by sampling initial
conditions from the initial Wigner distribution and integrating the otherwise deterministic GPE. Averaging over the initial conditions enables the calculation of fluctuating
observables. This is the truncated-Wigner approximation [235, 236]. It gives good results as long as the low energy modes of the theory are strongly occupied while the high
energy are not. See e.g. [237, 238] where the domain of validity of this approximation is
discussed.
Note that requiring strong occupation numbers for the low energy modes is somewhat
incompatible with the diluteness assumption made on the interaction of the bosons.
The physics that we describe here is in the middle region where both approximations
are valid. We have enough particles for the semi-classical approximation to hold but not
to much for the contact interaction to loose its applicability.
4.1. Super-fluid turbulence and non-thermal fixed points
As in Section (3) we are mainly interested here in non-equilibrium steady states. We focus, in particular, on NTFPs. See [16–18] or [30, 47] and references therein for overviews
about quantum turbulence and NTFPs. Such fixed points arise when an initial inhomogeneous distribution of conserved charges prevents the rapid thermalisation of the
system. Indeed, if the initial state of the system contains spatially separated regions of
positive and negative charge both parts need to mix before the uniform thermal state
can be reached. The differently charged particles need to cross the whole volume of the
system before they can meet and annihilate. We will see later in this Section that the
75
4. Ultra-cold Bose Gases
α
Figure 4.1.: Two very different paths to thermal equilibrium [43]. The initial states are
taken from the region on the left-hand side. The latter is seperated in two
regions. If the system is initiated on the right of the grey separation it relaxes
to thermal equilibrium. On the other hand if we give it initial conditions
from the left it is attracted to the NTFP.
conserved quantity that prevents the thermalisation in the case of the ultra-cold Bose gas
is the angular momentum and that the entities that need to meet in order to annihilate
are opposite sign vortices.
Another way of thinking of NTFPs is in terms of dynamical systems. Let us assume
that the state of the system during the out-of-equilibrium dynamics can be characterised
by a vector in Rn . Then the only stable and attractive fixed point of the time evolution
map corresponds to thermal equilibrium. However, if there is another fixed point with
only a few repulsive directions and if the initial state of the system lies very close to
the basin of attraction of such a fixed point then the time evolution will lead towards it
and stay there for as long as it takes for the relevant directions to take over and bring
the system to thermal equilibrium. We see that two scenarios emerge. Either the initial
state of the system is far enough from the basin of attraction of the NTFP and it relaxes
directly to its thermal state or it is attracted to the NTFP in the initial stage of its time
evolution and stays there for a very long time before its thermalisation (see Figure (4.1)).
A striking property that arises in both cases is that the state of the system during the
(quasi-)stationary state is not sensitive to the particular choice of the initial conditions.
We expect the properties of the system to be universal. In particular we will see that
physical quantities exhibit scale invariance at NTFPs as in the case of driven-dissipative
classical turbulence (see Section (3.1)).
Such NTFPs were identified by means of a strong-wave-turbulence analysis on the basis
76
4. Ultra-cold Bose Gases
of 2 Particle Irreducible (2PI) dynamic field equations [16–18] as well as semi-classical
field simulations [36, 38–40, 42, 43, 46–48, 68, 71, 72, 239] within the truncated-Wigner
approximation. Before we discuss the outcome of these calculations let us introduce the
observables that we use to probe the system.
As in the case of Burgers turbulence we focus on two-point correlation functions. The
time ordered Greens function1 ,
⟨ψ(t + τ, x + r)ψ(t, x)⟩c ⟨ψ(t + τ, x + r)ψ † (t, x)⟩c
G(τ, r) = T
,
(4.3)
⟨ψ † (t + τ, x + r)ψ(t, x)⟩c ⟨ψ † (t + τ, x + r)ψ † (t, x)⟩c
contain all the two body information. It is usually decomposed,
i
sign(τ )ρ(τ, r),
2
in term of the statistical and spectral functions F (τ, r) and ρ(τ, r) which have direct
physical interpretations (see e.g. [240] for more details). Here and in the following we
describe the steady-state properties of the system. The correlation functions therefore
only depend on the relative coordinates (τ, r). In terms of field expectation values F (τ, r)
and ρ(τ, r) can be written as,
G(τ, r) = F (τ, r) −
1
Fab (τ, r) = ⟨{ψa (t + τ, x + r), ψb (t, x)}⟩c , ρab (τ, r) = i⟨[ψa (t + τ, x + r), ψb (t, x)]⟩,
2
(4.4)
with ψ1 = ψ, ψ2 = ψ † and [·, ·] and {·, ·} the commutator and anti-commutator of two
fields respectively. Note that the statistical function F (τ, r) which we just introduced
has nothing to do with the forcing correlation function of Eq. (3.4). We will not talk
about the forcing correlation function in this section. F (τ, r) is always the statistical
function.
The kinetic energy of the system can be computed from F (τ, r),
Z
1
V
Ekin ≡ −
⟨ψ † (t, x)∇2 ψ(t, x)⟩ = −
Tr ∇2 F (0, 0) .
(4.5)
2m x
2m
R
V = x is the volume of the system. Ekin can be decomposed into its Fourier modes
in order to define the kinetic energy spectrum. Taking the Fourier transform of the
complex field ψ we can write2
Z
Z
V
Ekin ≡
p2 ⟨ψ † (t, p)ψ(t, −p)⟩ ≡ V ϵkin (p).
(4.6)
2m p
p
We have defined
p2 †
1
ϵkin (p) =
⟨ψ (t, p)ψ(t, −p)⟩ =
2m
2m
Z
1
=−
eip·x ∇2 F (0, x).
2m x
1
2
Z
eip·x ⟨∇ψ(t, x) · ∇ψ † (t, 0)⟩
x
(4.7)
T is the time ordering operator. It moves the operator with the largest time argument to the left.
We have inserted ⟨ψ † (t, p)ψ(t, p′ )⟩ ∼ δ(p + p′ ) which is a consequence of the invariance of the system
under space translations. See Section (F), Eq. (F.11) for a precise definition.
77
4. Ultra-cold Bose Gases
Physically ϵkin (p) is the amount of kinetic energy stored in the Fourier mode p. It can be
used as a measure of the energy content of the different scales of the system. We define
the particle density spectrum in a completely analogous way. The number of particles is
Z
Z
Z
†
†
(4.8)
N = ⟨ψ (t, x)ψ(t, x)⟩ ≡ V ⟨ψ (t, p)ψ(t, −p)⟩ ≡ V n(p).
p
x
p
We see that both spectra are related by ϵkin (p) = p2 n(p)/(2m).
We now briefly summarise the results of Ref. [18]. We will build upon these in Section
(4.3). Stationary scaling solutions for the statistical and spectral two-point correlators,
F (sz ω, sp) = s−2−κ F (ω, p) ,
ρ (sz ω, sp) = s−2+η ρ (ω, p) ,
(4.9)
respectively, were predicted by means of a non-perturbative wave-turbulence analysis of
the 2PI dynamic equations for these correlation functions.
The critical behaviour is characterised by the exponents κ and η, as well as the dynamical exponent z. η is an anomalous critical exponent which determines the deviation
of the spectral scaling from the free behaviour. In Ref. [18] two possible solutions were
found, corresponding to different strong-wave-turbulence cascades, with scaling exponents
κP = d + 2z − ηP ,
κQ = d + z − ηQ ,
(4.10)
between κ, η, z, and d. (κP , ηP ) correspond to an energy cascade while (κQ , ηQ ) reflect a
quasi-particle cascade in the wave turbulent system. Both represent NTFPs of the nonequilibrium Bose gas. The scaling of the statistical correlation
function F implies scaling
R
of particle density and kinetic energy spectra n(p) = dωF (ω, p) and ϵkin (p) = p2 n(p):
ϵkin (p) ∼ p−ξ ,
n(p) ∼ p−ξ−2 ,
with ξ = κ − z.
(4.11)
Assuming ηQ = 0, the distribution nQ (p) ∼ p−d−2 corresponds, for d = 2, 3 to the
scaling of the flow field v ∼ r−1 with the distance r from a vortex core [241, 242] and,
equivalently, of a random distribution of vortices [36, 39], as we will discuss in more detail
in Section (4.3.2). Such an Infrared (IR) divergence can be interpreted as an inverse
cascade of particles. This plays an important role in the equilibration and condensation
process [243–245] after a strong cooling quench in a Bose gas [43, 63] since the cascade
builds up the Bose condensate by accumulating particles on the largest scales.
In Refs. [36, 39, 40, 42, 47], the above non-thermal fixed points of the dilute superfluid gas were discussed in the context of topological defect formation and super-fluid
turbulence. A key result is that nearly degenerate Bose gases in d = 2, 3 dimensions,
quenched parametrically close to the Bose-Einstein condensation (in d = 2 Berezinsky–
Kosterlitz–Thouless) transition, can evolve quickly to a quasi-stationary state exhibiting
critical scaling [39] and slowing-down behaviour [42]. The critical scaling exponents ξ
of the kinetic energy spectra ϵkin (p) ∼ p−ξ corroborated the predictions ξQ = d − ηQ
of the strong-wave-turbulence analysis of Ref. [18] for a quasi-particle cascade, with a
78
4. Ultra-cold Bose Gases
very small value of ηQ . Within the numerical precision it was found that ξQ = 2 in
d = 2 and ξQ = 3 in d = 3 [39]. These exponents turned out to be related to randomly
distributed vortices and (large) vortex rings occurring during the approach of the critical
state [36, 39].
4.2. Driven-dissipative Gross–Pitaevskii equation
In order to make contact with classical turbulence and the results of Section (3) we
include driving and dissipation into the description of the Bose gas given by Eqs. (4.1)
and (4.2). Physically this amounts to coupling the Bose gas to an external reservoir which
can absorb particles and energy. A well known example of such system are Exciton–
Polariton Condensates (EPC) in solid-state systems as well as in ultra cold atomic gases
(see e.g. [186] for a review). It was found experimentally [246–254] as well as theoretically
[255–260] that such systems undergoes Bose condensation and exhibit vortices and superfluid turbulent dynamics similar to the isolated case even though the steady state is
maintained by the competition of driving and dissipation.
The driven-dissipative super-fluid dilute Bose gas can be described in an a similar way
as the isolated gas. See e.g. [261, 262] for applications of the c-field methods to EPCs.
In the semi-classical approximation we can simply replace the GPE by the DDGPE,
1
2
2
i∂t ψ = −
− iν ∇ − µ + g |ψ| ψ + ζ.
(4.12)
2m
We allow for the necessary dissipation, loss, and gain of energy and particles by allowing
µ = µ1 + iµ2 and g = g1 − ig2 to become complex, including an effective particle gain or
loss µ2 , as well as two-body interaction and loss parameters g1,2 . The diffusion term ∝ ν
is generated through the coarse graining of high-frequency modes. It was first proposed
empirically and shown to provide a good model of the EPC dynamics [263, 264]. It
was derived from first principles in [59, 60, 222] using Renormalisation Group (RG)
techniques. ζ is a Gaussian, delta-correlated white noise that satisfies
⟨ζ(t, x)⟩ = 0,
⟨ζ ∗ (t, x)ζ(t′ , x′ )⟩ = γ δ(t − t′ )δ(x − x′ ).
(4.13)
It accounts for the noise induced by the driving mechanism. The overall driving intensity
in turn is given by µ2 . When µ2 > 0 the steady state is maintained by the competing
of the pumping of individual particles into the system and the two body losses that are
taken into account by g2 .
We have seen in the previous section that scaling at NTFPs is the outcome of a
cascade of particles or energy. In a closed system this can not go on forever. The
system eventually thermalises. During the cascade process the system is however quasistationary. The cascade is maintained because the non-thermal distribution of particles
and energy allows for depletions and over-occupations of the different Fourier modes of
the system which act as sinks and sources. During the particles cascade the condensate
is not fully formed yet and there is still a lot of room in the zero mode. We have a
79
4. Ultra-cold Bose Gases
sink in the IR. In order to understand the source in the UV we must look a little more
into the early the time evolution of the system, i.e. before it reaches the NTFP. Indeed,
when the far-from-equilibrium initial conditions contain a few very highly occupied IR
modes, the field expectation value initially oscillates in time and space. This oscillating
background then drives the occupation of the higher energy modes just like parametric
resonances in classical physics [265, 266]. See [240] for a detailed discussion of the
parametric resonance mechanism and [42, 47] for details on early time dynamics of the
Bose gas. Then when the system reaches the NTFP there are a lot of particles in the UV
which act as a source for the cascade towards the IR. Note that this source of particles
is in the UV as compared to the zero mode. Actually it is at intermediate scales, at
momentum of about one half of the inverse healing length3 [39, 43]. There is still a lot
of depleted modes in the (far) UV for energy to accumulate there. What is observed in
numerical simulations ([47] and references therein) is that the large number of particles
that cascade to the IR looses most of its energy to a small number of particles which
constitute the UV part of the spectrum. In this way while particles are cascading to the
IR, energy is simultaneously cascading to the UV.
On the other hand, an EPC contains built-in driving and dissipation such that cascades
can be sustained forever. The driving mechanism is incoherent pumping of particles. It is
simply proportional to the local (in momentum space) particle spectrum. The dissipation
happens through two-body losses which is non-linear. It is proportional to the product
of two spectra at different momenta. Then if there is a Bose condensate most of the
dissipated particles will be lost to the condensate since it is macroscopically occupied as
compared to the rest of the spectrum. We see that once more we find a sink in the IR.
It then becomes reasonable to assume that the cascades in both open an closed systems
are a properties of the underlying dynamics instead of the detailed driving mechanism.
In the rest of this section we assume that this is the case. We will study the scaling
properties of the non-equilibrium steady state of the EPC and compare them to results
known in the context of NTFPs in the closed system.
Super-fluid turbulence [28, 30, 32, 35, 82–84, 87–89, 99, 100] manifests itself in selfsimilar field configurations in the domain of long-wavelength hydrodynamic excitations.
The hydrodynamic formulation of the DDGPE results by introducing the parametrisa√
tion ψ = n exp[iθ] in terms of the fluid density n and velocity fields v = m−1 ∇θ.
The phase angle θ then obeys a Langevin equation of the KPZ type which is equivalent
to Burger’s equation (3.3) for the curl-free velocity field v, under the condition that
f = m−1 ∇U , with a random potential field U . We therefore relate the DDGPE to
the stochastic Burgers equation (3.3). Indeed, the hydrodynamic decomposition of the
3
The healing length is the scale which one can use to make Eq. (4.2) dimensionless. If the distance is
measured in units of ξ = (2mµ)−1/2 then the pre-factor of the kinetic term is equal to the chemical
potential and can be scaled into a dimensionless interaction parameter ĝ = gmξ 2−d .
80
4. Ultra-cold Bose Gases
complex field, ψ =
√
n exp{iθ}, makes it possible to write Eq. (4.12) as
1
(∇θ)2 − ν∇2 θ = U,
2m
1
= S.
∂t n + ∇ · (n∇θ)
m
∂t θ +
(4.14)
This is formally similar to the equations that arise from the conservative GPE, with the
addition that the continuity equation is in-homogeneous and that the KPZ equation has
a non-zero dissipative term,
ν
1
∇n
Re(ζe−iθ )
√
√ ∇· √
+ ∇n · ∇θ + µ1 − g1 n −
U=
,
n
4m n
n
n
√
√
∇n
− 2νn (∇θ)2 − 2µ2 n − 2g2 n2 + 2 n Im(ζe−iθ ).
(4.15)
S = ν n∇ · √
n
These equations are coupled non-linear Langevin equations. If the fluctuations of the
field amplitude are sub-dominant the former can be decoupled by assuming that U plays
the role of the potential of the stochastic forcing f = m−1 ∇U , with noise correlator
⟨U (ω, p)U (ω ′ , p′ )⟩ = δ(ω + ω ′ ) δ p + p′ u(ω, p).
(4.16)
This describes particles being injected and removed as amplitude fluctuations, such that
the system reaches a state where they can be described by a (not necessarily thermal)
distribution and feed energy to the phase fluctuations. Burgers’ equation is obtained by
setting v = m−1 ∇θ. Note that a very similar mapping was first introduced in [187].
The kinetic energy spectrum can be written in therms of θ and n by means of the
density and phase representation. It is decomposed into three parts,
Z
Z
Z
ρ
1
(∇n)2
1
⟨(∇θ)2 ⟩ +
⟨
⟩+
⟨δn (∇θ)2 ⟩
2m x
2m x 4n
2m x
= Ehydro
+ Equantum
+ Eexchange .
Ekin =
(4.17)
The amplitude of ψ is separated into n = ⟨n⟩ + δn ≡ ρ + δn. At sufficiently low energies,
the average value of the amplitude is much larger than its fluctuations and the major
contribution to the kinetic energy is Ehydro . Then,
Z
Z
ρ
2
∼
Ekin =
⟨(∇θ) ⟩ ≡ V ϵkin (p),
(4.18)
2m x
p
where V is the volume of the system. Hence ϵkin (p) can be related to the two-point
correlation function of the Burgers fluid,
Z
mρ
ϵkin (p) =
⟨v(ω, p) · v(−ω, −p)⟩.
(4.19)
2 ω
81
4. Ultra-cold Bose Gases
4.3. Strong wave and quantum turbulence
In the following we discuss results of the present work. We exploit the mapping of the
DDGPE onto the stochastic Burgers equation which was introduced in Section (4.2)
to extract information on quantum turbulence in dilute Bose gases as described by the
GPE.
4.3.1. Galilei invariance and Kolmogorov scaling
The kinetic energy spectrum of the ultra-cold Bose gas is expressed in terms of the
velocity field of Burgers’ equation in Eq. (4.19). One can relate the scaling exponents
of the ultra-cold Bose gas κ, η and z to the exponents of Burgers turbulence χ and z
by computing the scaling exponent of the kinetic energy spectra in both set-ups. In the
case of the Bose gas we insert Eqs. (4.9) into Eq. (4.7) and get,
ϵkin (p) ∼ pz−κ .
(4.20)
For the Burgers fluid we insert Eqs. (3.66) into Eq. (4.19), take the limit k → 0 and get
Eq. (3.88), ϵkin (p) ∼ p2+d−2η1 . By matching this scaling with Eq. (4.20) we conclude
that,
z − κ = d + 2 − 2η1 .
(4.21)
In the case of the Burgers fluid, inserting Eq. (3.74) into Eqs. (3.83) gives Eqs. (3.84).
In particular z = 2 + d − η1 . If we assume that the dynamical critical exponent z is the
same for the Bose gas and classical turbulence Eq. (4.21) turns into κ = η1 . Equivalently,
Eqs. (3.84) provide,
κ + z = d + 2.
(4.22)
This is a non trivial relation in between the scaling exponents of the ultra-cold Bose
gas. It has its origin in Galilei invariance through Eq. (3.16) which reduces by one the
number of free scaling exponents.
This can thus be used to eliminate z from Eqs. (4.10) and write
κP = d + 4/3 − ηP /3,
κQ = d + 1 − ηQ /2.
(4.23)
In turbulenceR theory, one considers the scaling of the radial kinetic energy distribution
E(p) = pd−1 Ω ϵkin (p) ∼ pd−1 ϵkin (p). Indeed, this observable is isotropic by construction. It measures the energy content of the momentum shell of radius p. Combining the
above results, one finds that the direct energy and inverse particle cascades have radial
single-particle kinetic energy distributions
EP (p) ∼ p−5/3+2ηP /3 ,
EQ (p) ∼ p−1+ηQ ,
(4.24)
respectively. We find that, for the direct energy cascade, the strong-wave-turbulence
scaling [18] of EP (p) is equivalent to the classical Kolmogorov law [10, 11], with an
82
4. Ultra-cold Bose Gases
intermittency correction 2ηP /3. Kolmogorov-5/3 scaling has been reported to be possible
in a super-fluid both experimentally [82–84], and in simulations [87, 88] of the GPE.
Given the relation (4.24) of the scaling laws with hydrodynamics and topological and
geometric properties of the superfluid gas, we call the exponents −5/3 and −1 canonical
while the effects of fluctuations are captured by the anomalous corrections 2ηP /3 and
ηQ , respectively.
4.3.2. Acoustic turbulence in a super-fluid
Let us return to the KPZ dynamics. In order to make contact with scaling in acoustic
turbulence in a super-fluid, we insert the average literature values for η1 computed
within the KPZ framework into Eq. (3.88) or equivalently into Eq. (4.20) with Eq. (4.21)
inserted. These are given in Table I. of Ref. [58] and Figure (3.4). They correspond to
a forcing potential field delta-correlated in space. Cf. Eq. (3.10) with U (p) = 1. Note
that modulo the difference in the tensor structure of ⟨fi (t, x)fj (t′ , x′ )⟩ this corresponds
to β = 2 in Eq. (3.8). We obtain ϵkin (p) ∼ p−ξ , with
ξ = 0,
for d = 1,
ξ = 0.758(30),
for d = 2,
ξ = 1.600(24),
for d = 3.
(4.25)
These results can be compared with scaling behaviour observed in acoustic turbulence
in ultra-cold Bose gases, as summarised in the following.
Results related to the quantum turbulence discussed in the previous section were
obtained for a 1-dimensional Bose gas in Ref. [40]. There, the relation between critical
scaling of the single-particle momentum spectrum and the appearance of solitary wave
excitations was pointed out. It was found that this spectrum, as for a thermal quasicondensate, has a Lorentzian shape if the solitons are distributed randomly in the system,
with the width of the Lorentzian being related to the mean density of solitons. The
latter is in general different from and independent of the thermal coherence length of a
gas with the same density and energy. The kinetic energy spectrum, in the regime of
momenta larger than the Lorentzian width, correspondingly shows a momentum scaling
ϵkin (p) ∼ p2 n(p) ∼ p0 . This, in turn, is in full agreement with the above result quoted in
Eq. (4.25), corresponding to a white-noise forcing, i.e. β = 2. The power law is consistent
with that occurring in the single-particle spectrum of a random distribution of grey and
black solitons in a one-dimensional Bose gas.
The fixed points found in Ref. [58], at which the exponents (4.25) apply, describe
critical dynamics according to the KPZ equation describing, e.g. the unbounded propagation of an interface moving with coordinates (θ, x) in a two-component statistical
system. On the contrary, the KPZ equation derived for the phase angle θ of the complex
field ψ evolving according to the GPE, see Section (4.2), is subject to the additional
constraint that the range of angles 0 < θ ≤ 2π is compact. This constraint plays an
important role if the phase excitations are large enough to allow for (quasi) topologi-
83
4. Ultra-cold Bose Gases
cal defects. Hence, one can not expect the predictions (4.25) to necessarily match the
scalings occurring when defects such as vortices are present.
We now show that the scalings (4.25) are present at the NTFP. Note that, while the
strong-wave-turbulence prediction Eqs. (4.11) and (4.10), ξQ = d − ηQ ∼
= d, is consistent
with vortices dominating the infrared behaviour of the single-particle spectrum [39], it
does not apply to the 1d case where there are no vortex defects. Indeed, ξQ = d = 1
is by 1 larger than the exponent ξ = 0 appearing in the Lorentzian distribution at
large momenta. However, also in d = 2 and d = 3, a scaling ξc ∼
= d − 1 appears as
a result of kink-like structures and longitudinal, compressible sound excitations. See
Section (E) for definitions of the different components of the kinetic energy spectrum.
In Ref. [39], it was demonstrated that the single-particle spectrum of the compressible
component of the super-fluid turbulence can show power-law behaviour, with an exponent ϵcomp (p) ∼ p−d+1 . The exponent ξc ∼
= d − 1 is present at the NTFP but it appears
in the compressible part of the kinetic energy spectrum which is sub-dominant as compared to its incompressible component4 , ϵinc (p) ∼ p−d . This power-law with exponent
ξc = d − 1, was ascribed to sound wave turbulence on the background of the vortex gas,
in particular to the density depressions remaining for some time in the gas after a vortex
and an anti-vortex have mutually annihilated [267], cf. Fig. 15 of Ref. [39]. mutually
annihilated [267], cf. Fig. 15 of Ref. [39].
If we assume that the scaling of Eq. (4.25) only applies to the incompressible spectra
instead of the full ϵkin (p) we can modify Eq. (4.11) to
ϵcomp (p) ∼ p−ξc ∼ pz−κ+1 .
(4.26)
The exponent is greater by one than in Eq. (4.11) because these are sub-dominant
excitations. Then comparing this with Eqs. (4.25) and assuming that the particle cascade
is realised (i.e. inserting the second of Eqs. (4.10)) we get
η = 0,
for d = 1,
η = 0.242(30),
for d = 2,
η = 0.400(14),
for d = 3.
(4.27)
For d = 1, the scaling (4.25) corresponds to that of the Lorentzian of a random soliton
gas as discussed above. Furthermore, within the numerical precision, the power laws seen
in Fig. 15 of Ref. [39] are found to be consistent with the values (4.25). We reproduce
the data in Figure (4.2), comparing it with the IR scaling exponents (4.25) for d = 2, 3
(blue solid lines). The figures show the occupation number spectrum n(p) ∼ p−2 ϵkin (p)
obtained from a numerical simulation of the GPE. The particular scalings occur shortly
after the decay of the last topological excitations, i.e. the last vortex-antivortex pair
for d = 2 or vortex ring for d = 3. At the time the picture is taken, the compressible
excitations dominate and their scaling exponent can be measured. The relatively large
4
Remember that we are dealing with a particle cascade towards the IR. We consider the case of p being
very small.
84
4. Ultra-cold Bose Gases
anomalous predictions of Eq. (4.27) fit the data very well. It was found numerically in
[39] that the particle cascade is indeed, realised in the IR.
We remark that the grey solitary-wave excitations as well as the density depressions
remaining after vortex-anti-vortex annihilation are consistent with the absence of the
compactness constraint on θ in the KPZ equation. The soliton gas can be dominated
by grey solitons [40] which imply only weak density depressions at the position of the
phase jump. The weaker the depression, the smaller the phase kink and the less relevant
the compactness of the range of possible θ. Similarly the density depressions leading to
ξc ∼
= d − 1 do not require the phase to vary over the full circle. Hence, we expect in these
cases that KPZ predictions for critical exponents apply also to the GPE, as defects do
not play a role.
Figure 4.2. (following page): Acoustic turbulence in a d = 2 (left) and d = 3 (right
panel) dilute super-fluid Bose gas. Shown are occupation
number spectra of the late-stage evolution of a closed system after an initial quench, briefly after the last vortexantivortex pair (d = 2) or vortex ring (d = 3) has disappeared, cf. Fig. 15 of Ref. [39]. The blue solid lines correspond to the predictions obtained in the present work, given
in Eq. (4.25). Note that n(p) ∼ p−ξ−2 . The black solid
lines mark the canonical scaling, n(p) ∼ p−d−1 . The latter scaling is expected on geometrical grounds, e.g. results
from randomly distributed plane-wave density depression
waves or solitons [39]. It is emphasised that the comparatively large deviations (∼ p−2.758 for d = 2 and ∼ p−3.600 for
d = 3) are found independently and contrast the small deviation from the canonical scalings (4.24) of quantum turbulent spectra in the presence of vortices.
P The radial momentum is given in lattice units, k = [2 di=1 sin2 (2πni a/L)]1/2 ,
with ni ∈ Z, −L/(2a) ≤ ni ≤ L/(2a), a being the grid
constant and L its side length. kξ = 2 sin(π/(2ξ)) is the
momentum corresponding to the inverse healing length.
85
4. Ultra-cold Bose Gases
Occupation number n(k)
106
n(k)
nq (k)
nc (k)
ni (k)
k −3
k −2
k −2.758
104
102
1
0.02
0.1
kξ
1
3
Radial momentum k
106
Occupation number n(k)
0.3
n(k)
nq (k)
nc (k)
ni (k)
k −4
k −2
−3.6
k
104
102
1
0.1
0.3
Radial momentum k
86
1
kξ
3
5. Conclusions and Outlook
In this thesis we have studied steady states of driven-dissipative dynamics. We investigated the classical turbulence described by the stochastic Burgers equation and the
stationary quantum turbulence described by the Gross–Pitaevskii equation (GPE).
The major focus of this work is on the stochastic Burgers equation. There we have applied the Functional Renormalisation Group (FRG) in order to look for non-perturbative
scaling solutions. We established an approximation scheme which takes into account the
necessary momentum dependence and is expressed in terms of effective parameters. We
then proceeded to write a set of Renormalisation Group (RG) fixed point equations in
order to look directly for scaling solutions. The fixed point equations where thoroughly
studied analytically and solved in both asymptotic regimes of infinite and vanishing
rescaled momentum.
In all spatial dimension a continuum of fixed points was found and the corresponding
scaling exponents where extracted. These are constrained within a given range that
depends on the number of spatial dimensions. Moreover, we find an additional isolated
fixed point as well.
For d = 1 our results compare well with the literature and even suggest the existence
of another yet unknown fixed point. For d ̸= 1 however, the range of vales that we find
for the scaling exponents is smaller than the one that appears in related work about
Kardar–Parisi–Zhang (KPZ) equation [61]. This case is however only equivalent to the
dynamics of Burgers’ equation when the velocity field is constrained to be the gradient
of a potential. There is not much literature about Burgers’ equation with vorticity.
Authors almost exclusively assume that the velocity field is the gradient of a potential
and switch to the related KPZ problem (see e.g. [151, 152]). On the other hand, our
approximation is devised to mimic 3d incompressible Navier–Stokes (NS) turbulence
where no such potential is introduced. It contains a forcing which is not the gradient of
a potential. I.e. vorticity is being pumped into the system.
Our results indicate that there is a qualitative difference in between the potential
and rotational systems. This can be further investigated by using a more sophisticated
approximation scheme which is able to capture both situations. However it might be
simpler and more straightforward to simply simulate the stochastic Burgers equation
and numerically measure the scaling exponents. For a forcing correlation function with
exponent β = 2 the two-point correlation function of the velocity field scales with and
exponent η1 ∼
= 2.379 (d = 2) and η1 ∼
= 3.300 (d = 3) when the flow is constrained to be
potential. On the other hand if the forcing and therefore the velocity field contains a
vortical component our study predicts 3.5 ≤ η1 ≤ 4 (d = 2) and 4.5 ≤ η1 ≤ 5 (d = 3)
(see Table (5.1)).
The full numerical solution of the fixed point equations remains as a goal beyond the
87
5. Conclusions and Outlook
d
2
3
Potential forcing
η1
η1
∼
= 2.379
∼
= 3.300
Vortical forcing
3.5 ≤ η1 ≤ 4
4.5 ≤ η1 ≤ 5
Table 5.1.: Predictions to be checked numerically for β = 2. The first column is the
dimension of space, the second shows the approximate value that η1 assumes
when the forcing as well as the velocity field are constrained to be potential
and the third gives the predictions of this work which apply when the forcing
is pumping vorticity into the system.
scope of this thesis. It is emphasised that the RG fixed point equations introduced here
go far beyond previous approaches. Experience based on [161] suggested that the fixed
point equations could be solved iteratively. First δZi (p̂) = 0 and arbitrary values of h
and η1 would be inserted in the right-hand side of the fixed point equations Eq. (3.76).
This would then provide a first estimation for its left-hand side as well as values for η1
and h. This would then have been inserted back into the right-hand side of Eq. (3.76).
Such a procedure was successfully implemented in the case of thermal equilibrium YangMills theory [161] but converged immediately to the Gaussian fixed point in our case.
Indeed, inserting δZi (p̂) = 0 in Eq. (3.76) returned h = 0 and δZi (p̂) = 0. In the
case of stochastic hydrodynamics the vertexes of the action are locked to its quadratic
part because of Galilei invariance. This makes it impossible to truncate the propagator
(assume δZi (p̂) = 0) at the intermediate steps of the calculation. Since it is analytic
our analysis of the asymptotic fixed point equations provide insight into their solutions
without having to approach them numerically. In particular we have learned that they
have multiple solutions from which the correct ones need to be selected.
Our RG study of the Burgers and KPZ problems opened a new interesting view on
aspects of quantum turbulence. Out-of-equilibrium steady states of super-fluid Bose
gases where related to the classical problem of the stochastic Burgers equation by considering an ultra-cold Bose gas coupled to external reservoirs of particles and energy.
Viscosity is an essential property of stochastic hydrodynamics even in the limit ν → 0.
In order to include this in the description of the Bose gas the GPE was upgraded to
the Driven-Dissipative Gross–Pitaevskii Equation (DDGPE) which contains driving and
dissipation encoded in its complex parameters. Then making the density and phase
√
decomposition of the average wave function of the Bose gas ψ = n eiθ one can write
a KPZ-like equation for the phase variable. Once more we could not apply our results
on Burgers’ equation because the dynamics of the phase do not contain vorticity. We
could however use results from the KPZ literature to make predictions about the scaling
properties of Non-Thermal Fixed Points (NTFP) in ultra-cold Bose gases.
The first step towards these predictions was to assume that the cascade dynamics and
its scaling exponents are a property of the underlying GPE dynamics and do not result
from the particular driving and dissipation mechanism. Then the predictions that we
make apply to quasi stationary non-equilibrium states of closed systems (NTFP) while
88
5. Conclusions and Outlook
our set-up is truly stationary by virtue of the balance of driving and dissipation.
From there we were able to make two predictions. First we related a well known
relation in between the scaling exponents of the stochastic KPZ equation to its dual
in terms of the exponents of the Bose gas. This provides an additional constraint in
between the different scaling exponents of the Bose gas. In the case of a direct cascade
of energy this can be used to identify a Kolmogorov −5/3 scaling of the kinetic energy
spectrum and its anomalous correction at the NTFP. Note that a similar scaling form
was pointed out in [18] in the case of an inverse cascade of particle cascade where the
dynamical critical exponent z does not play a role. Both cascades are expected to differ
qualitatively through their vortex distribution.
Secondly we could use precise estimations of the scaling exponents of the stochastic
KPZ equation found in the literature to compute the anomalous corrections to the values
of the scaling exponents of the compressible part of the kinetic energy spectrum of the
Bose gas. On the way this provided an insight on the nature of the KPZ dynamics.
When we use the density and phase decomposition to translate the GPE to the KPZ
equation we gets an additional constraint on θ because we are dealing with an angle.
I.e. θ is a compact field. This seemingly innocent constraint is actually crucial since
quantum turbulence is basically made out of vortices which are the loci of phase jumps.
In such a situation there is no chance for the predictions of the KPZ literature to be
any good. However the decomposition of the kinetic energy spectrum of the Bose gas
makes it possible to separate the different contributions to the kinetic energy spectrum.
Then the dominant contribution of the topological excitations can be discarded and we
can look at the sub-dominant ones. For this reason we have compared the predictions of
the KPZ literature to the scaling of the compressible kinetic energy spectrum computed
from the far-from-equilibrium closed GPE. We find an excellent agreement (see Figure
(4.2)). This seems to indicate that we where right to assume that both steady states (in
the quasi-stationary closed system or in the stationary open system) can be described
in terms of the same NTFP.
As a last remark we note that this suggests that there is a qualitative difference in
the scaling exponents of the traditional KPZ equation and of its compactified version.
Indeed, if θ is interpreted as an angle instead of a height then vortices become available.
Our experience with GPE hints towards the fact that if the traditional KPZ two-point
correlation function scales as1 ,
Σ2 (λz τ, λ r) = λ2χ Σ2 (τ, r),
(5.1)
then its compact version scales as
Σ2 (λz τ, λ r) = λ2χ+1 Σ2 (τ, r).
(5.2)
In the GPE however the density vanishes at vortex cores. Hence even if the phase is
non-analytic at a vortex the wave function stays smooth because of the vanishing density.
It is not clear if vortex solutions can exist without such a regularisation.
1
We introduce the KPZ increment Σ2 (τ, r) = ⟨[θ(t + τ, x + r) − θ(t, x)]2 ⟩.
89
Appendices
90
A. Local Potential Approximation, Fixed
Point Coefficients
In this section we give explicit expressions for the fixed point coefficients Cn , An and Bn
defined in Section (2.2.1.2), Eq. (2.57),
λn = Cn (ρ0 , λ2 , .., λn ) + An (ρ0 , λ2 , λ3 ) λn+1 + B(ρ0 , λ2 ) λn+2 ,
n ≥ 3.
(A.1)
The coupling λn , defined in Eq. (2.53) is the n-th derivative of u(ρ) evaluated at ρ =
ρ0 . In order to establish Eq. (A.1) we start with the fixed point equation Eq. (2.52),
differentiate it n times and evaluate it at ρ = ρ0 . We get

d2 u
d3 u
n−1
ρ
+
3
2
3
2
d−2
Ωd d

d ρ
d ρ
λn =
(λn+1 ρ0 + nλn ) − 2 n−1  h
.
(A.2)
i
2 d
d d
ρ
d2 u
du
2 d2 ρ ρ + dρ + 1 ρ=ρ0
In the following we will manipulate Eq. (A.2) in order to extract all the terms that
contain λn+1 and λn+2 . Then Cn , An and Bn can be extracted in a straightforward way.
We start by using the fact that1
n
dn (f g) X
n!
ds f dn−s g
=
,
dn x
s!(n − s)! ds x dn−s x
s=0
(A.3)
in order to expand the derivative on the right-hand side of Eq. (A.2). We get
λn =
d−2
(λn+1 ρ0 + nλn )
d
n−1
Ωd X
(n − 1)!
(2λn+2−s ρ0 + [3 + 2(n − 1 − s)] λn+1−s ) X(s, 2).
− 2
d
s!(n − 1 − s)!
(A.4)
s=0
We have defined
n
X(n, m) ≡

d 
1
h 2
n
d ρ
2 d u ρ + du
d2 ρ
dρ
im +1 ρ=ρ0
+
d

= −m n−1  h
im+1 2
d
ρ
2 dd2uρ ρ + du
dρ + 1

3
2 dd3uρ ρ
n−1
2
3 dd2uρ
.
(A.5)
ρ=ρ0
1
This identity is true for any couple of differentiable functions and can be proven by recurrence on n.
91
A. Local Potential Approximation, Fixed Point Coefficients
We see straight away that the first two terms of the sum on the right-hand side of (A.4),
for s = 1 and 2, contain λn+2 and λn+1 . Looking at Eq. (A.5), we see that the coupling
with the largest n that appears in X(n, m) is λn+2 . Therefore, neither λn+2 nor λn+1
appear in the summands of the right-hand side of (A.4) if 2 ≤ s ≤ n − 2. We extract
the terms that contain λn or λn+1 and write
λn =
d−2
(λn+1 ρ0 + nλn )
d
Ωd
− 2 {2λn+2 ρ0 + [3 + 2(n − 1)] λn+1 } X(0, 2)
d
Ωd
− 2 (n − 1) {2λn+1 ρ0 + [3 + 2(n − 2)] λn } X(1, 2)
d
n−2
Ωd X
(n − 1)!
− 2
{2λn+2−s ρ0 + [3 + 2(n − 1 − s)] λn+1−s } X(s, 2)
d
s!(n − 1 − s)!
s=2
Ωd
− 2 (2λ3 ρ0 + 3λ2 ) X(n − 1, 2).
d
(A.6)
We see that a structure similar to Eq. (A.1) is starting to emerge. We still have to extract
the factors of λn+1 and λn+2 from the last term of Eq. (A.6) before the expression can
be rearranged in to the desired form. We apply once more Eq. (A.3) on X(n − 1, 2) and
write
X(n − 1, 2) = −2
n−2
X
s=0
(n − 2)! X(s, 3)
(2λn+1−s ρ0 + [2(n − s) − 1] λn−s ) .
s!(n − s − 2)!
(A.7)
It is now clear that λn+2 does not appear in X(n − 1, 2) while λn+1 only appear in the
first term (s = 0) of the sum of Eq. (A.7). We can now put everything together and
write
Cn =
d−2
Ωd 2λ3 ρ0 + 3λ2
n λn + 2 2
{n [3 + 2(n − 2)] λn }
d
d (2λ2 ρ0 + 1)3
−2
+
Ωd (2λ3 ρ0 + 3λ2 )2
[3(n − 2) {2λn ρ0 + [3 + 2(n − 3)] λn−1 }]
d2 (2λ2 ρ0 + 1)4
n−2
Ωd X (n − 2)! (4λ3 ρ0 + 6λ2 )
{2λn+1−s ρ0 + [3 + 2(n − 2 − s)] λn−s } X(s, 3)
d2
s!(n − s − 2)!
s=2
−
n−2
(n − 1)!
Ωd X
{2λn+2−s ρ0 + [3 + 2(n − 1 − s)] λn+1−s } X(s, 2),
2
d
s!(n − 1 − s)!
s=2
(A.8)
d−2
Ωd
2λ3 ρ0 + 3λ2
3 + 2(n − 1)
An =
ρ0 + 2 4n ρ0
−
,
d
d
(2λ2 ρ0 + 1)3 (2λ2 ρ0 + 1)2
92
(A.9)
A. Local Potential Approximation, Fixed Point Coefficients
and
B = −2
Ωd
1
ρ0
.
2
d
(2λ2 ρ0 + 1)2
(A.10)
Finally, X(n, m) can be computed recursively. We have:
1
,
(2λ2 ρ0 + 1)m
n−1
X
(n − 1)!
X(n, m) = −m
[2λ3+s + (2 + 3s) λ2+s ] X(n − 1 − s, m + 1). (A.11)
s!(n − 1 − s)!
X(0, m) =
s=0
93
B. Flow Integrals
In this appendix we give details on the computation of the flow integrals of Eqs. (3.61)
and (3.62). We use the sharp cut-off Rk (p) = k d z1 (k)R̃k (p), with
=0
if p ≥ k
R̃k (p)
.
(B.1)
→ ∞ if p < k
See Section (2.1.3). As a result we can use the identity
1
(2)
Γk
+ Rk
k
1
∂Rk
1
= 2k 2 δ(p2 − k 2 ) (2) ,
(2)
∂k Γ + Rk
Γ
k
(B.2)
k
to evaluate the flow integrals. To proceed, we read off, from the ansatz (3.54), the
two-point function
(2)
δij
δ 2 Γk [v = 0]
=
δ(ω + ω ′ )δ p + p′ Γk (ω, p),
′
′
d+1
δvi (ω , p⃗ )δvj (ω, p⃗)
(2π)
(B.3)
(2)
with Γk (ω, p) defined in (3.55) and given by
(2)
Γk (ω, p) = νk (p)2 p4 + ω 2 Fk−1 (p),
(B.4)
the 3-vertex
δ 3 Γk [v = 0]
δ(ω + ω ′ + ω ′′ ) δ (p + p′ + q) (3)
=
Γk;ijl (ω ′ , p′ ; ω, p), (B.5)
δvi (ω ′ , p′ )δvj (ω, p)δvl (ω ′′ , q)
(2π)2(d+1)
with
2 ′
(3)
δil pj + δjl pi
Γk;ijl (ω ′ , p′ ; ω, p) = − Fk−1 (p + p′ ) ω + ω ′ + iνk (p + p′ ) p + p′ − Fk−1 (p′ ) −ω ′ + iνk (p′ ) p′2 δij pl − δil (p′j + pj )
(B.6)
− Fk−1 (p) −ω + iνk (p) p2 δij p′l − δjl (p′i + pi ) ,
and the 4-vertex
δ 4 Γk [v = 0]
=
δvi (ω, p)δvj (ω ′ , p′ )δvl (ω ′′ , q)δvm (ω ′′′ , q ′ )
δ(ω + ω ′ + ω ′′ + ω ′′′ )δ (p + p′ + q + q ′ ) (4)
Γk,ijlm (p, p′ , q),
(2π)3(d+1)
(B.7)
94
B. Flow Integrals
with
(4)
Γk,ijlm (p, p′ , q) = Fk−1 (p + p′ ) δmi pj + δmj p′i pl + p′l + ql − δli pj + δlj p′i qm
+ Fk−1 (|p + q|) (δim pl + δlm qi ) pj + p′j + qj − (δij pl + δjl qi ) p′m
+ Fk−1 (p′ + q ) δjm p′l + δlm qj pi + p′i + qi − δij p′l + δil qj pm .
(B.8)
Using these, the flow integral (3.59) can be written as
Z
k2
δ(p′2 − k 2 ) h (4)
(2)
Ik [0](ω, p) = −
Γ
(p, −p, p′ )
d ω′ ,p′ Γ(2) (ω ′ , p⃗′ ) k;iijj
k
′
i
2
2
θ (⃗
p − p⃗) − k
(3)
(3)
− 2 (2)
Γk;ijl (−ω ′ , −⃗
p′ ; ω ′ − ω, p′ − p⃗) Γk;jil (ω − ω ′ , p⃗ − p⃗′ ; ω ′ , p⃗′ ) .
Γk (ω ′ − ω, p⃗′ − p⃗)
(B.9)
The theta function arises because of the sharp cut-off. Since the cut-off diverges for p < k,
(2)
the propagator (Γk + Rk )−1 vanishes in this regime. This is irrelevant in the diagram
depending on the 4-point vertex, cf. (3.59), where the single appearing propagator carries
a ∂k Rk insertion and is thus evaluated at p2 = k 2 . Eq. (B.9) is divided by d in order to
be consistent with Eq. (3.58).
The index contractions in Eq. (B.9) are lengthy but straightforward. We find,
(3)
(3)
Γk;ijl (−ω ′ , −p′ ; ω ′ − ω, p′ − p) Γk;jil (ω − ω ′ , p − p′ ; ω ′ , p′ ) =
(2)
Γk (p, ω)Fk−1 (p) 2(d − 1)p′2 + dp2 − 2(d − 1)p′ · p
(2)
+ Γk (p′ , ω ′ )Fk−1 (p′ ) 2(d − 1)p2 + dp′2 − 2(d − 1)p′ · p
(2)
+ Γk (p′ − p, ω ′ − ω)Fk−1 (p′ − p) dp′2 + dp2 − 2p′ · p
+ 2dFk−1 (p′ )Fk−1 (p) ω ′ ω − νk (p′ )p′2 νk (p)p2 p′ · p
+ 2dFk−1 (p)Fk−1 (p − p′ )
h
2 i
× ω(ω − ω ′ ) − νk (p)p2 νk (p − p′ ) p − p′ p · p − p′
+ 2dFk−1 (p′ )Fk−1 (p − p′ )
h
2 i
× ω ′ (ω ′ − ω) − νk (p′ )p′2 νk (p − p′ ) p − p′ p′ · p′ − p ,
(B.10)
and
(4)
Γk;iijj (p, −p, p′ ) = Fk−1 p − p′ −2 p · p′ + d(p2 + p′2 )
+ Fk−1 p + p′ 2 p · p′ + d(p2 + p′2 ) .
95
(B.11)
B. Flow Integrals
Once Eqs. (B.10) and (B.11) are inserted into Eq. (B.9), it becomes apparent that as
a result of the truncation of the frequency dependence of the inverse propagator (3.55),
the integrand is a rational function of ω ′ . The ω integration can be done analytically.
Introducing the short-hand notation ν̃p = νk (p)p2 , etc., a straightforward application of
the residue theorem gives
Z
1
1
=
,
2
2
4
2νk (p)p2
ω ω + νk (p) p
ν̃q
Z
A
+
1
+ B f + C ν̃q ν̃q + ν̃p + f 2
2
ν̃p
A + Bω + Cω
=
h
,
2 i
2
2 (ω − f )2 + (ν̃ )2
ω ω + (ν̃p )
q
2ν̃ f 2 + ν̃ + ν̃
q
q
p
(B.12)
for A, B and C ∈ R and ν̃q , ν̃p > 0. The right-hand side of Eq. (B.9) can be cast into
a linear combination of such integrals for appropriate values of A, B and C. We finally
obtain
Z
k2
(2π)d δ(r2 − k 2 ) δ(p − q − r)
(2)
Ik (ω, p) = −
−1
d q,r Fk (q)Fk−1 (r)ν̃q ν̃r [ω 2 + (ν̃q + ν̃r )2 ]
h
× Fk−1 (q)2 ν̃q ω 2 + (ν̃q + ν̃r )2 d(p2 + r2 ) − 2p · r
− θ q 2 − k 2 Fk−1 (q)2 ν̃q ω 2 + (ν̃q + ν̃r )2 d(p2 + r2 ) − 2p · r
+ Fk−1 (r)2 ν̃r ω 2 + (ν̃q + ν̃r )2 d(p2 + q 2 ) − 2p · q
+ Fk−1 (p)2 ν̃q + ν̃r ω 2 + ν̃p2 d(q 2 + r2 ) + 2q · r
+ Fk−1 (p)Fk−1 (q)ν̃q ω 2 − ν̃p ν̃q + ν̃r 2d p · q
i
(B.13)
+ Fk−1 (p)Fk−1 (r)ν̃r ω 2 − ν̃p ν̃q + ν̃r 2d p · r .
Note that the terms ∝ d and independent of d originate from contractions
of the types
R
R
R∞
d−1 dr
δ ij δ ji pk q k and δ ij δ jl pi q l , respectively.
We
can
integrate
radially,
=
⃗
r
0 r
Ω,
R
which, for d = 1, reduces to Ω f (r) = [f (r) + f (−r)]/(2π). The delta distributions
allow to set p′ = ker and q = p − ker , with |r̂| = |er | = 1. Expansion in powers of ω 2
96
B. Flow Integrals
gives Eqs. (3.61) and (3.62).
(2)
Ik (0, p) = −
kd
2d
1
Z
−1
−1
Ω Fk (k)Fk (q)ν̃k ν̃q (ν̃k
+ ν̃q )
× Fk−1 (q)2 ν̃q (ν̃q + ν̃k ) d(p2 + k 2 ) − 2ker · p
− θ q 2 − k 2 Fk−1 (q)2 ν̃q (ν̃k + ν̃q ) d(p2 + k 2 ) − 2k p · er
+ Fk−1 (k)2 ν̃k (ν̃k + ν̃q ) d(p2 + q 2 ) − 2p · q
+ Fk−1 (p)2 ν̃p 2 d(k 2 + q 2 ) + 2k q · er
h
− Fk−1 (p)Fk−1 (q)ν̃p ν̃q 2d p · q
− Fk−1 (p)Fk−1 (k)ν̃p ν̃k 2d k p · er
i
.
(B.14)
and
Z
(2)
θ q 2 − k 2 Fk−1 (p)
∂Ik kd
=
∂ω 2 2d Ω Fk−1 (k)Fk−1 (q)ν̃k ν̃q (ν̃k + ν̃q )3
(0,p)
× Fk−1 (p) (ν̃k + ν̃q )2 − ν̃p 2 d(q 2 + k 2 ) + 2k q · er
+ Fk−1 (q) ν̃q ν̃k + ν̃q + ν̃p 2d p · q
+ Fk−1 (k) ν̃k ν̃k + ν̃q + ν̃p 2d k p · er .
97
(B.15)
C. Re-scaled Flow Integrals
In this section we write explicit expression for the re-scaled flow integrals which are
defined in Eq. (3.73) and Eqs. (3.89) and (3.90). In Sections (C.1) and (C.2) we study
their asymptotic behaviour for very small and very large arguments respectively.
The re-scaled flow integrals are obtained from Eqs. (3.61) and (3.62) by inserting
the fixed point parametrisation (3.66) and the re-scaled variables (3.67). We make the
replacements given by
r
z1
2
p = k p̂,
ω=k
ω̂,
z2
s
p
1 + δZ1 (p̂)
νk (p) = z1 /z2 p̂(η1 −η2 −4)/2
, Fk−1 (p) = k d−4 z2 p̂η2 [1 + δZ2 (p̂)] . (C.1)
1 + δZ2 (p̂)
The re-scaled flow integrals take the form
(2)
Iˆ1 (p̂) = p̂ F1,1 (p̂) + p̂(η1 +η2 )/2+2 S1 (p̂) S2 (p̂) F1,2 (p̂)
+ p̂η1 +η2 +2 δd1 [S1 (p̂) S2 (p̂)]2 F1,3 (p̂),
(C.2)
and
(2)
Iˆ2 (p̂) = p̂(η1 +η2 )/2+2 S1 (p̂) S2 (p̂) F2,1 (p̂) + p̂η2 +2 S2 (p̂)2 F2,2 (p̂)
+ p̂η1 +η2 +2 δd1 [S1 (p̂) S2 (p̂)]2 F2,3 (p̂) + p̂2η2 +2 δd1 S2 (p̂)4 F2,4 (p̂).
(C.3)
p
We have introduced the short-hand notation Si (p̂) = 1 + δZi (p̂). The functions Fi,j (p̂)
are given by1 ,
Z n
1
S2 (1)2 2
F1,1 (p̂) =
θ q̂ 2 − 1 q̂ −(η1 +η2 )/2
d(p̂ + q̂ 2 ) − 2p̂ · q̂
2d p̂ Ω
S1 (q̂)S2 (q̂)
2
o
S2 (q̂)
− θ 1 − q̂ 2 q̂ η2
d(p̂2 + 1) − 2 p̂ · er ,
(C.4)
S1 (1)S2 (1)
F1,2 (p̂) = − p̂
−2
Z
θ q̂ − 1 T (q̂)
Ω
1
2
p̂ · q̂
p̂ · er
+ (η +η )/2
1
2
S1 (1)S2 (1) q̂
S1 (q̂)S2 (q̂)
,
(C.5)
Note that the exponents of the following expression contain the Kronecker delta δd1 . Indeed, cancellations in the leading pre-factors of F1,3 (p̂), F2,3 (p̂) and F2,4 (p̂) occur for d = 1 and the sub-leading
terms must be taken into account.
98
C. Re-scaled Flow Integrals
p̂−2δd1
F1,3 (p̂) =
2d
F2,1 (p̂) = p̂
−2
Z
Z
θ q̂ 2 − 1 T (q̂)
Ω
θ q̂ − 1 T (q̂) 3
θ q̂ − 1 T (q̂) 2
2
Ω
F2,2 (p̂) = p̂
−2
Z
2
Ω
p̂−2δd1
F2,3 (p̂) = −
2d
Z
p̂ · q̂
p̂ · er
+ (η +η )/2
1
2
S1 (1)S2 (1)
q̂
S1 (q̂)S2 (q̂)
p̂ · q̂
p̂ · er
+ (η +η )/2
1
2
S1 (1)S2 (1)
q̂
S1 (q̂)S2 (q̂)
θ q̂ 2 − 1 T (q̂) 3
Ω
d(q̂ 2 + 1) + 2q̂ · er
,
q̂ (η1 +η2 )/2 S1 (q̂)S2 (q̂)S1 (1)S2 (1)
d(q̂ 2 + 1) + 2q̂ · er
,
q̂ (η1 +η2 )/2 S1 (q̂)S2 (q̂)S1 (1)S2 (1)
F2,4 (p̂) =F1,3 (p̂),
(C.6)
,
(C.7)
,
(C.8)
(C.9)
(C.10)
With er the unit vector pointing in the direction of Ω, q̂ = p̂ − er and the short-hand
notation (ν̃k + ν̃q )−1 = k −2 (z2 /z1 )1/2 T (q̂). These functions where defined in such a
way that they are analytic and non vanishing at p̂ = 0. They can be Taylor expanded
′ (0) p̂ + O(p̂2 ). Their asymptotic behaviour determines the
Fi,j (p̂ → 0) = Fi,j (0) + Fi,j
asymptotic form of the flow integrals and is studied in Sections (C.1) and (C.2).
C.1. Scaling limit (p ≪ k)
Here we give expressions for Fi,j (0). They are computed in a straightforward way by
Taylor expanding the integrands of Eqs. (C.4) to (C.10) up to leading order and performing the angular integration. In the following, we use the notation δZi′ (1) = dδZi /dp̂|p̂=1 .
Such terms arise in F1,1 (0) and F2,1 (0) because the respective integrands vanishes at
p̂ = 0 and the Taylor expansions of δZi (q̂) around q̂ = 1 enter the leading term. For
d = 1 we get
1
2(η1 − 1)S12 S22 − S22 δZ1′ (1) + S12 δZ2′ (1) ,
3
8πS1 S2
1
2 2
2
′
2
′
F1,2 (0) = −
4(η
−
1)S
S
+
S
δZ
(1)
+
S
δZ
(1)
,
1
1
2
2
1
1
2
8πS14 S22
1
F1,3 (0) = F2,4 (0) =
,
8πS13 S2
1 F2,1 (0) =
4(η1 − 1)S12 S22 + S22 δZ1′ (1) + S12 δZ2′ (1) ,
6
32πS1
1
2 2
2
′
2
′
F2,2 (0) =
4(η
−
1)S
S
+
S
δZ
(1)
+
S
δZ
(1)
,
1
1
2
2
1
1
2
16πS15 S2
S2
F2,3 (0) = −
.
32πS15
F1,1 (0) =
(C.11)
99
C. Re-scaled Flow Integrals
And for d ̸= 1 we have
2d Ωd Γ(d/2)2 2 2
2
′
2
′
2(η
−
1)S
S
−
S
δZ
(1)+S
δZ
(1)
,
1
1
2
2
1
1
2
16π(d − 1)! S13 S2
Ωd 4(η1 − 1)S12 S22 + S22 δZ1′ (1) + S12 δZ2′ (1) ,
F1,2 (0) = −
2
4
8dS1 S2
Ωd (d − 1)
F1,3 (0) = F2,4 (0) =
,
4dS13 S2
Ωd F2,1 (0) =
4(η1 − 1)S12 S22 + S22 δZ1′ (1) + S12 δZ2′ (1) ,
6
32dS1
Ωd
2 2
2
′
2
′
4(η
−
1)S
S
+
S
δZ
(1)
+
S
δZ
(1)
,
F2,2 (0) =
1
1
2
2
1
1
2
16dS15 S2
Ωd (d − 1)S2
F2,3 (0) = −
.
16dS15
F1,1 (0) =
(C.12)
We have introduced the short-hand notation Si = Si (1). This leads to the following
asymptotic form for the re-scaled flow integrals,
p
(2)
Iˆ1 (p̂) ∼
= F1,1 + F1,2 p̂(η1 +η2 )/2+1 [1 + δZ1 (p̂)] [1 + δZ2 (p̂)]
+ F1,3 p̂η1 +η2 +2δd1 −1 [1 + δZ1 (p̂)] [1 + δZ2 (p̂)] ,
p
(2)
Iˆ2 (p̂) ∼
= F2,1 p̂(η1 +η2 )/2+1 [1 + δZ1 (p̂)] [1 + δZ2 (p̂)]
+ F2,2 p̂η2 +1 [1 + δZ2 (p̂)] + F2,3 p̂η1 +η2 +2δd1 −1 [1 + δZ1 (p̂)] [1 + δZ2 (p̂)]
+ F2,4 p̂2η2 +2δd1 −1 [1 + δZ2 (p̂)]2 ,
(C.13)
with the short-hand notation Fi,j = Fi,j (0). Eqs. (3.91) follow by inserting this in
Eqs. (3.76).
C.2. Scaling limit (p ≫ k)
(2)
In this subsection, the asymptotic behaviour of the integrals Iˆ1,2 (p̂) for p̂ ≫ 1 is derived
from the respective dependence of the integrals Fi,j (p̂) given in Eqs. (C.4) to (C.10). We
discuss this for each Fi,j (p̂) separately, taking into account spherical symmetry.
We start by making a simplification which is valid only in the asymptotic limit p̂ ≫ 1
and for Ultraviolet (UV) convergent fixed points δZi (p̂ → ∞) = 0. For p̂2 ≫ 1, then
also q̂ ∼ p̂, i.e. , q̂ 2 − 1 > 0, and we can approximately set the theta functions and,
since δZi (q̂ → ∞) = 0, also the Si (q̂) to one. Separating out the leading UV scaling,
Fi,j (p̂ → ∞) ∼ p̂γi,j , we write the Fi,j (p̂) in the form
Z
γi,j
Fi,j (p̂) = p̂
fi,j (1/p̂, p̂ · er /p̂).
(C.14)
Ω
100
C. Re-scaled Flow Integrals
The fi,j are finite and non-vanishing at 1/p̂ = 0.
Note that, in Eqs. (C.5), (C.7) and (C.8), different terms can be leading in the UV such
that the above definition of the fi,j and γi,j depends on the values of the η1,2 . Moreover,
the denominator of T (q̂) in Eqs. (C.5)–(C.10) contains a divergence if η1 − η2 > 0 in
which case an additional factor p̂(η2 −η1 )/2 appears. This can be seen by recalling the
definition T (q̂) = (z1 /z2 )1/2 k 2 (ν̃k + ν̃q )−1 , which gives (recall q̂ = |p̂ − er |) the large-p̂
asymptotic behaviour
T (|p̂ − er |) ∼
=
∼
=
"
p
p̂ + 1 − 2p̂ · er
p
2
(η1 −η2 )/4
S1 (1)
+
S2 (1)
#−1
p̂−(η1 −η2 )/2 if η1 − η2 > 0
.
S2 (1)/S1 (1) if η1 − η2 < 0
(C.15)
Having identified the leading scaling behaviour, the integrals can be computed in the
limit p̂ → ∞ by neglecting sub-leading contributions to the integrands. We can approximate fi,j (1/p̂, p̂ · er /p̂) ∼
= fi,j (0, p̂ · er /p̂) in the integrands and perform the angular integration which gives, for those
R integrals where fi,j (0, y) does not depend on
y = p̂ · er /p̂, a surface factor Ωd = Ω = dπ d/2 [(2π)d Γ(d/2 + 1)]−1 . The asymptotic
behaviour of the integrals F1,1 (p̂), F1,3 (p̂), and F2,3 (p̂) can be derived in this way. The
result is (Si ≡ Si (1))
F1,1 (p̂ → ∞) ∼
= Ωd S22 [δd1 /2 + (d − 1)/d] p̂1−2δd1 p̂−(η1 +η2 )/2 ,
(C.16)
 −η
1
if η1 > η2

 p̂
Ω
d
2−2δ
−η
−1
d1
∼
p̂ 1 (1 + S1 /S2 )
if η1 = η2 ,
F1,3 (p̂ → ∞) =
p̂

2S1 S2
 −(η1 +η2 )/2
p̂
(S2 /S1 ) if η1 < η2
(C.17)
 −2η +η
1
2
if η1 > η2

 p̂
Ω
d
−3
p̂−η1 (1 + S1 /S2 )
if η1 = η2 .
p̂2−2δd1
F2,3 (p̂ → ∞) ∼
=−

2S1 S2
 −(η1 +η2 )/2
3
p̂
(S2 /S1 ) if η1 < η2
(C.18)
The calculation of the asymptotic behaviour of the integrals (C.5), (C.7) and (C.8) can
become more involved. Two possibilities arise. If η1 +η2 ≥ −2, the asymptotic behaviour
is determined in the same way as for F1,1 , F1,3 and F2,3 . However, for η1 + η2 < −2 the
leading term of fi,j (ϵ → 0, p̂·er /p̂) is proportional to p̂·er /p̂, and thus vanishes under the
angular integral. In this case, the asymptotically leading term is obtained by expanding
yT (q̂) ≡ yT (p̂, y) to order y 2 before the limit p̂ → ∞ is taken and the term that is linear
in y is neglected. This ensures that we only consider terms that contribute to the angular
integration. One can check that truncating at order y 2 does not affect the asymptotic
behaviour. Indeed y enters through the combination p̂2 − 2p̂ · er = (1 − 2y/p̂)p̂2 . We see
that the term of order y n is multiplied by 1/p̂n and can only dominate in the asymptotic
regime if all the lower order terms are irrelevant.
101
C. Re-scaled Flow Integrals
We discuss the procedure for F1,2 (p̂) and state the results for the two remaining
integrals F2,1 (p̂) and F2,2 (p̂). To simplify the derivation we use that (η1 + η2 )/2 =
2η1 − 2 − d from Eq. (3.74). We start by approximating δZi (q̂) ∼
= 0, θ q̂ 2 − 1 = 1 in
Eq. (C.5), which gives, defining ϵ = 1/p̂ such that p̂ · er = y/ϵ,
F1,2 (p̂) ∼
=
Z
Ω
with q̂ =
ϵy − 1
−2η1 +2+d
ϵy T (q̂),
− q̂
S1 S2
(C.19)
p
1 + ϵ2 − 2ϵy/ϵ. We factor out ϵ−2η1 +2+d from q̂ 2η1 +2+d in the second term:
F1,2 (p̂) ∼
=
Z
ϵy − 1
−d−1+2η1
2
(−2η1 +2+d)/2
.
−ϵ
y(1 + ϵ − 2ϵy)
T (q̂)
S1 S2
Ω
(C.20)
The asymptotic behaviour of T (q̂) is determined by the sign of −(η1 − η2 )/2 = η1 − 2 − d,
see Eq. (C.15). For both signs, different η1 will render either of the terms in Eq. (C.20)
dominating for large p̂ (ϵ → 0).
1. η1 < d + 2, T (q̂ → ∞) ∼ p̂η1 −d−2 : We write T (q̂) = ϵ−η1 +2+d T̃ (ϵ) such that
T̃ (ϵ → 0) = 1 and
F1,2 (p̂) ∼
=
Z
Ω
ϵ−η1 +d+2
ϵy − 1
− ϵη1 +1 (1 + ϵ2 − 2ϵy)(−2η1 +2+d)/2 y T̃ (ϵ).
S1 S2
(C.21)
There are three sub-cases to be distinguished: (a) For 2η1 < d+1, the second term, which
provides an extra scaling factor ϵη1 +1 , is dominant. Then the leading-power exponent
defined in Eq. (C.14) reads γ1,2 = −η1 − 1, and the integrand is
−2η1 +d+1 ϵy − 1
2
(−2η1 +2+d)/2
f1,2 (ϵ, y) = ϵ
− (1 + ϵ − 2ϵy)
y T̃ (ϵ).
(C.22)
S1 S2
The leading term f1,2 (0, y) = −y does not contribute to the angular integral. Taking
the sub-leading factors into account by expanding to second order in y,
f1,2 (ϵ, y) ∼
= −T̃ (ϵ) y[1 + ϵ2 ](−2η1 +2+d)/2
h
i
+ y 2 ϵ[1 + ϵ2 ](−2η1 +d)/2 2η1 − 2 − d − (η1 − 2 − d)(1 + ϵ2 )(−η1 +2+d)/2 T̃ (ϵ)
n
h
i
+ ϵ−2η1 +d+1 /(S1 S2 ) 1 − ϵy 1 + (η1 − 2 − d)(1 + ϵ2 )(−η1 +d)/2 T̃ (ϵ)
+ ϵ2 y 2 (η1 − 2 − d)(1 + ϵ2 )(−η1 −2+d)/2 T̃ (ϵ)
h
i o
× (η1 − 2 − d)(1 + ϵ)(−η1 +2+d)/2 T̃ (ϵ) + (d − η1 )/2 + 1 + ϵ2
,
(C.23)
we find that two terms are competing, giving rise to a further case distinction: If η1 <
d/2, the contributions proportional to ϵ−2η1 +d+1 are sub-leading and the quadratic term
102
C. Re-scaled Flow Integrals
in y dominates. In turn, if η1 > d/2, the term that does not depend on y dominates.
Both must be taken into account if η1 = d/2. As a result,
 2
η1 < d/2
 y η1 S1 S2
ϵ 
2
∼
1 + y dS1 S2 /2 η1 = d/2
f1,2 (ϵ → 0, y) = − y −
,
(C.24)
S1 S2 
 −2η1 +d
ϵ
d/2 < η1 < (d + 1)/2
R
and, after angular integration, Ω y 2 = Ωd /d,
 −η
1 η S S /d
η1 < d/2

1 1 2
 p̂
Ω
d
−2
−d/2
∼
p̂
F1,2 (p̂ → ∞) = −
.
(C.25)
p̂
(1 + S1 S2 /2) η1 = d/2

S1 S2
 η1 −d
p̂
d/2 < η1 < (d + 1)/2
(b) For 2η1 = d + 1, both terms under the integral (C.21) are equally important. We
obtain γ1,2 = −(d + 3)/2 and
ϵy − 1
2
1/2
− (1 + ϵ − 2ϵy) y T̃ (ϵ).
f1,2 (ϵ, y) =
(C.26)
S1 S2
The relevant contribution is f1,2 (0, y) = −y − (S1 S2 )−1 while the terms of order y 2 are
sub-dominant. As a result, the asymptotics (C.25) is supplemented with
Ωd −(d+3)/2
F1,2 (p̂ → ∞) ∼
p̂
=−
S1 S2
if η1 = (d + 1)/2.
(C.27)
(c) For (d + 1)/2 < η1 < d + 2, the leading terms are interchanged. From Eq. (C.21),
one finds γ1,2 = η1 − d − 2 and
ϵy − 1
2η1 −d−1 2
(d+2−2η1 )/2
f1,2 (p̂) =
−ϵ
(ϵ + 1 − 2ϵy)
y T̃ (ϵ).
(C.28)
S1 S2
We find f1,2 (0, y) = −(S1 S2 )−1 , and, together with relation (C.27), the last case of the
asymptotics (C.25) reads
Ωd −2 η1 −d
F1,2 (p̂ → ∞) ∼
p̂ p̂
,
=−
S1 S2
for d/2 < η1 < d + 2.
(C.29)
2. η1 ≥ d + 2, T (q̂ → ∞) ∼ const : In this case, q̂ −η1 +2+d does not diverge for ϵ → 0,
such that no powers of 1/p̂ arise from T (q̂). Again, two competing terms in Eq. (C.20)
require the distinction of three sub-cases. However, for η1 ≥ d + 2, the term proportional
to ϵ−d−1+2η1 is always sub-dominant and can be neglected. The term proportional to
(S1 S2 )−1 in Eq. (C.20) is dominant, such that γ1,2 = 0 and
ϵy − 1
2η1 −d−1
2
(d+2−2η1 )/2
−ϵ
(1 + ϵ − 2ϵy)
y T (q̂).
(C.30)
f1,2 (ϵ, y) =
S1 S2
103
C. Re-scaled Flow Integrals
Taking the limit f1,2 (ϵ → 0, y) and performing the angular integral one obtains the final
asymptotics
 −η
p̂ 1 η1 S1 S2 /d
η1 < d/2




−d/2

(1 + S1 S2 /2) η1 = d/2
 p̂
Ωd −2  η1 −d
∼
F1,2 (p̂ → ∞) = −
p̂
(C.31)
p̂
d/2 < η1 < d + 2 .

S1 S2

2
−1

p̂ (1 + S1 /S2 )
η1 = d + 2



 2
p̂ S2 /S1
d + 2 < η1
Using analogous arguments we find
 η −2d
p̂ 1 S1 S2 (η1 + 2 + d)/(2d)





p̂−3d/2 [1 + S1 S2 (3d + 4)/(4d)]


Ωd −6
p̂3η1 −3d
p̂
F2,1 (p̂) ∼
=

S1 S2



p̂6 (1 + S1 /S2 )−3


 6
p̂ (S2 /S1 )3
 η −2d
p̂ 1
S1 S2 (5η1 − 8 − 4d)/d




−3d/2

[1 − S1 S2 (16 + 3d)/(2d)]
 p̂
Ωd −6  3η1 −3d
∼
p̂
p̂
F2,2 (p̂) =

S1 S2



p̂6 (1 + S1 /S2 )−3


 6
p̂ (S2 /S1 )3
η1 < d/2
η1 = d/2
d/2 < η1 < d + 2 ,
(C.32)
η1 = d + 2
d + 2 < η1
η1 < d/2
η1 = d/2
d/2 < η1 < d + 2 .
(C.33)
η1 = d + 2
d + 2 < η1
The resulting expressions for the UV leading behaviour of the integrals Fi,j (p̂) can be
inserted back into Eqs. (3.89) and (3.90) in order to compute the asymptotic behaviour
(2)
of Iˆ1,2 (p̂). Each case needs to be considered separately. With Eq. (3.74), we find that
(2)
Iˆi (p̂ ≫ 1) ∼ p̂βi ,
(C.34)
with


 4 − 2δd1 − 2η1 + d if (6 + 3d − 2δd1 )/5 ≥ η1
3η1 − 2d − 2
if (6 + 3d − 2δd1 )/5 < η1 ≤ d + 2 ,
β1 =


2η1 − d
if η1 > d + 2
(C.35)

3(η1 − d − 2)



 5η − 4d − 6
1
β2 =

3η
1 − 2d − 2



4η1 − 3d − 4 − 2δd1
(C.36)
if η1 ≤ d/2
if d/2 < η1 ≤ d + 2
if d + 2 < η1 ≤ d + 2 + 2δd1
.
if d + 2 + 2δd1 < η1
See Figure (3.3) where β1 and β2 are plotted with respect to η1 and η2 respectively for
d = 1 and d = 3. The integrals on the right-hand side of Eq. (3.103) converge if βi < ηi ,
corresponding to the allowed range (3.107) for η1 .
104
D. Equations for Ai
In this section we write down explicit equations containing the pre-factors of Eqs. (3.92),
Ai . We use a slightly different notation and replace Eqs. (3.92) by
δZi (p̂ → 0) ∼
= ci + ai p̂αi −ηi ,
as p̂ → 0.
(D.1)
This corresponds to the short-hand notation ai = Ai (αi − ηi ). We consider here cases
where either ci = −1 or αi − ηi < 0. Then when we insert Eqs. (D.1) into the re-scaled
flow integrals Eqs. (C.13) an take the limit p̂ → 0 we can neglect all the 1 + ci factors
since they are sub-dominant as compared to the ai p̂αi −ηi . We get,
√
(2)
Iˆ1 (p̂) ∼
= F1,1 + F1,2 a1 a2 p̂(α1 +α2 )/2+1 + F1,3 a1 a2 p̂α1 +α2 +2δd1 −1 ,
√
(2)
Iˆ2 (p̂) ∼
= F2,1 a1 a2 p̂(α1 +α2 )/2+1 + F2,2 a2 p̂α2 +1
+ F2,3 a1 a2 p̂α1 +α2 +2δd1 −1 + F2,4 a22 p̂2α2 +2δd1 −1 .
(D.2)
We find linear combinations of monomials. When ci = −1 or αi − ηi < 0 the exponents
of these monomials only depend on αi . We see that once we have chosen values of αi
from the solutions of Eqs. (3.93) we can compare the different exponents and identify the
leading monomial in the limit p̂ → 0. It then becomes possible to match the pre-factors
on both sides of Eqs. (3.91) and extract equations containing both ai .
Each case of Eqs. (3.94) must be considered separately. Moreover the case d = 1 must
also be considered separately since the exponents are different than for d ̸= 1. We start
with d = 1.
• If (α1 , α2 ) = (1, 5) Eqs. (D.2) can be simplified to
√
(2)
Iˆ2 (p̂) ∼
= F2,1 a1 a2 p̂4 .
(2)
Iˆ1 (p̂) ∼
= F1,1 ,
The terms proportional to F1,1 and F2,1 are dominating. The corresponding equations are
a1 (η1 − 1) = h F1,1 ,
√
a2 (η2 − 5) = h a1 a2 F2,1 .
(D.3)
• If −2 < α1 < 1 and α2 = −2, the dominating terms are the ones that are proportional to F1,3 and F2,4 ,
(2)
Iˆ1 (p̂) ∼
= F1,3 a1 a2 p̂α1 −1 ,
105
(2)
Iˆ2 (p̂) ∼
= F2,4 a22 p̂−3 ,
D. Equations for Ai
and thus
a1 (η1 − α1 ) = ha1 a2 F1,3 ,
a2 (η2 + 2) = ha22 F2,4 .
(D.4)
(2)
• If α1 = −2 and α2 = −2, the term proportional to F1,3 still dominates Iˆ1 (p̂), but
the term proportional to F2,3 is of the same order as the one proportional to F2,4 .
Taking both into account gives
(2)
(2)
Iˆ1 (p̂) ∼
Iˆ2 (p̂) ∼
= F1,3 a1 a2 p̂−3 ,
= F2,3 a1 a2 + F2,4 a22 p̂−3 ,
and
a1 (η1 + 2) = ha1 a2 F1,3 ,
a2 (η2 + 2) = h a1 a2 F2,3 + a22 F2,4 .
(D.5)
• Finally, for α1 = 1 and α2 = −2 the dominating terms are proportional to F1,1 +
a1 a2 F1,3 and F2,4 , i.e.
(2)
Iˆ1 (p̂) ∼
= F1,1 + F1,3 a1 a2 ,
(2)
Iˆ2 (p̂) ∼
= F2,4 a22 p̂−3 ,
and
a1 (η1 − 1) = h [F1,1 + a1 a2 F1,3 ] ,
a2 (η2 + 2) = ha22 F2,4 .
(D.6)
We now discuss the cases d ̸= 1.
• If (α1 , α2 ) = (1, 5), we obtain
√
(2)
Iˆ2 (p̂) ∼
= F2,1 a1 a2 p̂4 ,
(2)
Iˆ1 (p̂) ∼
= F1,1
and
a1 (η1 − 1) = h F1,1 ,
√
a2 (η2 − 5) = h a1 a2 F2,1 .
(D.7)
• If 0 < α1 < 1 and α2 = 0, we get
(2)
Iˆ1 (p̂) ∼
= F1,3 a1 a2 p̂α1 −1 ,
(2)
Iˆ2 (p̂) ∼
= F2,4 a22 p̂−1 ,
and
a1 (η1 − α1 ) = ha1 a2 F1,3 ,
a2 η2 = ha22 F2,4 .
106
(D.8)
D. Equations for Ai
• If α1 = α2 = 0, we find
(2)
Iˆ1 (p̂) ∼
= F1,3 a1 a2 p̂−1 ,
(2)
Iˆ2 (p̂) ∼
= F2,3 a1 a2 + F2,4 a22 p̂−1 .
and the equations are
a1 η1 = ha1 a2 F1,3 ,
a2 η2 = h a1 a2 F2,3 + a22 F2,4 .
(D.9)
• Finally when α1 = 1 and α2 = 0, one obtains
(2)
Iˆ1 (p̂) ∼
= F1,1 + F1,3 a1 a2 ,
(2)
Iˆ2 (p̂) ∼
= F2,4 a22 p̂−1 .
with
a1 (η1 − 1) = h [F1,1 + a1 a2 F1,3 ] ,
a2 η2 = ha22 F2,4 .
(D.10)
Note that all of these equations depend explicitly on Fi,j and h. Fi,j are shown in
Eqs. (C.11) and (C.12) and are non-linear functions of ηi , δZi (1) and δZi′ (1) and h is
related to the full functions δZi (p̂) through Eqs. (3.76). This means that the asymptotic
behaviour of δZi (p̂) which is described by ai is coupled to the rest of the functions. h can
be eliminated from these equations by taking their ratio but ai stay coupled to δZi (1)
and δZi′ (1) through Fi,j .
For the cases of Eqs. (D.4) and (D.8) the ratio of the two equations simplifies greatly
because F1,3 = F2,4 cancels out. This leads to a relation in between ηi and α1 which is
discussed in the main text, at the end of Section (3.3.5.1). ai however drop out as well
so that we can not say that the asymptotic behaviour of δZi (p̂) is decoupled form the
rest of the functions.
107
E. Energy spectrum decomposition
In this section we decompose the kinetic energy spectrum of the ultra-cold Bose gas and
give definitions for its different components,
ϵkin (p) = ϵinc (p) + ϵcomp (p) + ϵquant (p).
(E.1)
See [39] (which we follow closely here) and reference therein. Our starting point is the
kinetic energy density of the Bose gas,
2
√ 2 i
Ekin
1
1 h √
=
⟨∇ψ · ∇ψ † ⟩ =
⟨ n ∇θ ⟩ + ⟨ ∇ n ⟩ ,
(E.2)
V
2m
2m
√
expressed in terms of the density and the phase of the complex field ψ = n eiθ . We
see already that there are two parts to the kinetic energy spectrum. The first contains
√
√
the fluctuations of w = n∇θ and the second of ∇ n. The first can be associated
with classical hydrodynamics because it contains the fluctuations of the phase while the
second is a result of the quantum nature of the system. We define,
Z
√
√
1
ϵquant (p) =
eip·r ⟨∇ n(t, x + r) · n(t, x)⟩.
(E.3)
2m x
In order to further decompose the kinetic energy spectrum the vector field w is decomposed into
w = winc + wcomp ,
with ∇ · winc = 0.
(E.4)
winc is a divergence-less field analogous to the velocity field of incompressible hydrodynamics. wcomp is defined as the difference of w and winc . We define
Z
1
ϵinc (p) =
eip·r ⟨winc (t, x + r) · winc (t, x)⟩,
2m x
Z
1
ϵcomp (p) =
eip·r ⟨wcomp (t, x + r) · wcomp (t, x)⟩.
(E.5)
2m x
Note that there is in principle an additional term arising from the product of winc and
wcomp . This term is expected to be small because it accounts for cross-correlations in
between very different physical processes. See [39] where this was found numerically. Incompressible excitations contain conservative hydrodynamic flows. They do not remove
matter from the system. For example the phase of a pure vortex solutions is a linear
√
function of the angle around the position of the vortex1 , w = m n/r eφ . Then we have
√ 2
∇ · w = n∇ θ = 0 which is incompressible. On the other hand wcomp contains the
rest of the hydrodynamics: pressure dynamics, sound waves, dissipative processes, etc.
1
m is the circulation of the vortex and eφ is the unit vector along the azimuthal direction.
108
F. Notation and conventions
In this section we introduce some notation and conventions that will be useful later on.
This section is meant to give a precise meaning to definitions that may be a little fuzzy
in the main text and to serve as a reminder for the reader. It is not meant to be read
all at once, but rather to be consulted when in need.
• Vectors are noted in boldface v = (v1 , v2 , .., vd ) and their components are in normal
font vi . Lower case indices (i, j, k, ...) represent spatial indices and run from 1 to
d. Repeated indices are to be summed over and the norm of a vector is in normal
font without index,
v
u d
uX
√
v = vi vi = t
vi2 .
(F.1)
i=1
• Real space differential operators are defined as follows
∂2
∂
∂
∂
,
∇2 =
∇=
,
, ..,
,
∆ = ∇2 , ∇2 , .., ∇2 .
∂x1 ∂x2
∂xd
∂xi ∂xi
(F.2)
Note that we distinguish the Laplacian ∇2 , from the vector Laplacian ∆, by the
object on which they act. The former acts on scalars while the latter on vectors.
• We use the following normalisation for the Fourier transformations
Z
Z
dω dd p i(ωt−p·x)
v(t, x) =
e
v(ω, p), v(ω, p) = dt dd x e−i(ωt−p·x) v(t, x).
(2π)d+1
(F.3)
• We use the short-hand notation
Z
Z
= dt dd x,
Z
t,x
ω,p
1
=
(2π)d+1
Z
dω dd p.
(F.4)
Moreover when the integral is the same whether its integrand is expressed in real
or Fourier space we write, e.g.
Z
Z
Z
v(t, x) · J (t, x) =
v(ω, p) · J (−ω, −p) = v · J .
(F.5)
t,x
ω,p
109
F. Notation and conventions
Finally the symbol Ω is used for angular integrations in Fourier space,
Z
Z
Z ∞
d−1
.
=
dp p
p
Note that
R
Ω
(F.6)
Ω
0
contains a 1/(2π)d factor.
• Angular brackets are used to denote averages with respect either to a stochastic
forcing, to quantum fluctuations or both depending on the context. The c index
denotes connected correlation functions. If the moments of the field v(t, x) are
generated by
Z[J ] ≡ ⟨e
R
J ·v
⟩,
⟨vi1 (t1 , x1 ) · .. · vin (tn , xn )⟩ =
δ n Z[J = 0]
,
δJi1 (t1 , x1 )..δJin (tn , xn )
(F.7)
then the connected correlation functions are generated by its logarithm,
W [J ] ≡ log (Z[J ]) ,
⟨vi1 (t1 , x1 ) · .. · vin (tn , xn )⟩c =
δ n W [J = 0]
.
δJi1 (t1 , x1 )..δJin (tn , xn )
(F.8)
In particular we have
⟨vi (t, x)vj (t′ , x′ )⟩c = ⟨vi (t, x)vj (t′ , x′ )⟩ − ⟨vi (t, x)⟩⟨vj (t′ , x′ )⟩.
(F.9)
• We will encounter angular brackets with a k index ⟨X⟩k . This stands for a correlation function that is computed from the flowing effective action Γk [v] (see Section
(2.1)). It only represents a physical correlation function in the limit where the
cut-off is removed,
⟨X⟩k→0 = ⟨X⟩.
(F.10)
• We will always consider systems that are invariant under space and time translations. The correlation functions will therefore only depend on relative spatiotemporal arguments. In Fourier space this translates to
⟨vi1 (ω1 , p1 )..vin (ωn , pn )⟩ = δ(ω1 + .. + ωn )δ(p1 + pn )(2π)d+1
× ⟨vi1 (ω1 , p1 )..vin (−(ω1 + .. + ωn ), −(p1 + .. + pn ))⟩.
(F.11)
We have slightly abused the notation here. Strictly speaking
⟨vi1 (ω1 , p1 )..vin (−(ω1 + .. + ωn ), −(p1 + .. + pn ))⟩
(F.12)
is not well defined since it is proportional to δ(0). We use instead Eq. (F.11) as a
definition of
⟨vi1 (ω1 , p1 )..vin (−(ω1 + .. + ωn ), −(p1 + .. + pn ))⟩.
Note that we have extracted a factor of (2π)d+1 .
110
(F.13)
F. Notation and conventions
• We deal with a lot of functional differentiation. We write the Taylor expansion of
a functional Γ[v] as
∞ Z
X
vi1 (ω1 , p1 ) · .. · vin (ωn , pn ) (n)
Γ[v] =
Γi1 i2 ..in (ω1 , p1 ; ..; ωn , pn ),
n!
ω1 ,p1 ;..;ωn ,pn
n=1
(F.14)
which is consistent with
(n)
Γi1 i2 ..in (ω1 , p1 ; ..; ωn , pn )
n(d+1)
= (2π)
δnΓ
.
δvi1 (ω1 , p1 )..δvin (ωn , pn ) v=0
(F.15)
• We use the short-hand notation δ(x) = Πdi=1 δ(xi ) for delta distributions with
vectors as arguments.
• As for correlation functions most of the functional derivatives that we deal with are
invariant under space and time translations. This implies the following property
for the derivatives of physical quantities
(n)
Γi1 i2 ..in (ω1 , p1 ; ..; ωn , pn ) = (2π)d+1 δ (p1 + .. + pn ) δ (ω1 + .. + ωn )
(n)
× Γi1 i2 ..in (ω1 , p1 ; ..; ωn−1 , pn−1 ).
(F.16)
Note that we used the same notation Γ(n);i1 ,..,in , in Eqs. (F.14) and (F.16). There is
no risk of confusion since the number of spatial variable is n and n − 1 respectively.
• Straightforward definitions for functional operators and operations in between
them are taken in real space, For operators A and B with matrix elements that
depend on space and time we have,
Z
′
′
AB(t, x; t , x ) =
A(t, x; t′′ , x′′ )B(t′′ , x′′ ; t′ , x′ ),
t′′ ,x′′
Z
AA−1 (t, x; t′ , x′ ) =
A(t, x; t′′ , x′′ )A−1 (t′′ , x′′ ; t′ , x′ ) ≡ δ(t − t′ )δ(x − x′ ),
′′
′′
t ,x
Z
Tr (A) =
A(t, x; t, x).
(F.17)
t,x
The corresponding operations in Fourier space are defined by requiring them to be
consistent with Eqs. (F.3),
Z
′ ′
′ ′
′ ′
A(ω, p, ω , p ) ≡
e−i(ωt+ω t −p·x−p ·x ) A(t, x; t′ , x′ ).
(F.18)
t,x;t′ ,x′
We have,
′
′
Z
AB(ω, p; ω , p ) =
A(ω, p; −ω ′′ , −p′′ )B(ω ′′ , p′′ ; ω ′ , p′ ),
ω ′′ ,p′′
AA−1 (ω, p; ω ′ , p′ ) = (2π)d+1 δ(ω + ω ′ )δ(p + p′ ),
Z
Tr (A) =
A(ω, p; −ω − p).
ω,p
111
(F.19)
F. Notation and conventions
Note that A−1 (ω, p; ω ′ , p′ ) is defined as
Z
′ ′
′ ′
A−1 (ω, p, ω ′ , p′ ) ≡
e−i(ωt+ω t −p·x−p ·x ) A−1 (t, x; t′ , x′ ),
(F.20)
t,x;t′ ,x′
while A−1 (t, x; t′ , x′ ) is defined through AA−1 (t, x; t′ , x′ ) = δ(t − t′ )δ(x − x′ ). The
asymmetry is only apparent since Eq. (F.20) is equivalent to AA−1 (ω, p; ω ′ , p′ ) =
(2π)d+1 δ(ω + ω ′ )δ(p + p′ ).
In particular
δ 2 Γk [v = 0]
δ(ω + ω ′ )δ (p + p′ )
(2)
,
≡ δij Γk (ω, p)
′
′
δvi (ω , p )δvj (ω, p)
(2π)d+1
(F.21)
implies
δ 2 Γk [v = 0]
δvi (ω ′ , p′ )δvj (ω, p)
−1
=
δ(ω + ω ′ )δ (p + p′ )
.
(2)
(2π)d+1
Γ (ω, p)
δij
(F.22)
k
Note that we use two different notations for the inverse of an operator. Compare
Eqs. (F.20) and (F.22). They are completely equivalent.
• The identity operator is given by
1(t, x; t′ , x′ ) = δ(t − t′ )δ(x − x′ ) δij ,
1(ω, p; ω ′ , p′ ) = (2π)d+1 δ(ω + ω ′ )δ(p + p′ ) δij .
(F.23)
• The two-point correlation function of the stochastic forcing f (t, x) is defined as
(F.24)
⟨fi (t, x)fj (t′ , x′ )⟩ = δij δ(t − t′ )F x − x′ .
We define its inverse Flj−1 (t, x; t′ , x′ ) such that
Z
⟨fi (t, x)fl (t′′ , x′′ )⟩ Flj−1 (t′′ , x′′ ; t′ , x′ ) = δij δ(t − t′ )δ(x − x′ ),
(F.25)
t′′ ,x′′
and get
Fij−1 (t′ , x; t′ , x′ ) = δij δ(t − t′ )F −1 x − x′ .
(F.26)
F −1 (x) is then defined through
Z
F x − x′′ F −1 x′′ − x′ = δ(x − x′ ).
(F.27)
x′′
Note that this definition does not imply F −1 (x) = 1/F (x) in real space. Invariance
under spatial translation makes this however true in Fourier space,
Z
1
(F.28)
F (p) ≡
eip·x F (x) = −1 .
F (p)
x
112
F. Notation and conventions
• At an Renormalisation Group (RG) fixed point observables assume a scaling form,
O(λz t; λx) = λη O(t, x),
(F.29)
for a generic space-time dependent observable O(t, x) and λ > 0. We are then free
to choose λx = 1 and write
t
t
η
η
,1 ≡ x g
,1 .
(F.30)
O(t, x) = x O
xz
xz
We use g(a) as a generic scaling function. It may have a different physical meaning
depending on the context but is always defined as g(a) = O (a, 1) for an appropriate
observable, O(t, x).
• The time ordering operator is defined in a standard way. Applied to a product of
time dependent operators it orders them from left to right with decreasing times.
In particular we have
T [ψi (t1 )ψj (t2 )] = ψi (t1 )ψj (t2 ) θ(t1 − t2 ) + ψj (t2 )ψi (t1 ) θ(t2 − t1 ).
(F.31)
• A primed function of a single argument is its derivative F ′ (x) = dF (x)/dx,
δZi′ (1) = dδZi /dp̂|p̂=1 .
• The vorticity of a 3d vector field v(t, x) is given by the pseudo-vector
w(t, x) = ∇ × v(t, x).
(F.32)
It can be physically interpreted as the local rotation of the vector field. If w(t, x) ̸=
0, a fluid element that is following v(t, x) and located at x rotates around an axis
which passes through x and is parallel to w(t, x). Its angular velocity is w(t, x)/2.
• The symbol ∼
= means "approximately equal" or "asymptotically equal",
f (x) ∼
= g(x) as x → a ⇔ lim f (x)/g(x) = 1,
x→a
(F.33)
depending on the context.
• The symbol ∼ means "scales as",
f (x) ∼ xα as x → a ⇔ lim f (x) x−α = const ̸= 0.
x→a
(F.34)
f (x) ∼ xα differs from f (x) ∼
= xα by the fact that with ∼ we do not say anything
about the pre-factor of the power law.
113
G. List of Abbreviations
G.1. Acronyms
•
•
•
•
•
•
•
•
•
•
•
•
•
•
1 Particle Irreducible (1PI)
2 Particle Irreducible (2PI)
Blaizot–Mendez–Wschebor (BMW)
Driven-Dissipative Gross–Pitaevskii Equation (DDGPE)
Exciton–Polariton Condensate (EPC)
Functional Renormalisation Group (FRG)
Gross–Pitaevskii equation (GPE)
Infrared (IR)
Kardar–Parisi–Zhang (KPZ)
Martin–Siggia–Rose/Janssen–de Dominicis (MSR/JD)
Navier–Stokes (NS)
Non-Thermal Fixed Point (NTFP)
Renormalisation Group (RG)
Ultraviolet (UV)
G.2. Short-hand notations
•
R
•
R
tx
=
ω,p
R
dtdd x
= (2π)
• Fij = Fij (0)
R
−d−1
dωdd p
• Si (p̂) =
p
1 + δZi (p̂)
d
• D [v] = Πω,p d v(ω, p)
• Si = Si (1)
Πdi=1 δ(xi )
• δ(x) =
R
• Ωd = Ω = dπ d/2 [(2π)d Γ(d/2 + 1)]−1
• T (q̂) = k 2
• q = p − ker
• ai = Ai (αi − ηi )
• ν̃p = νk (p)p2
• y = p̂ · er /p̂
114
p
z1 /z2 /(ν̃k + ν̃q )
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