Variational Principle of Bogoliubov and Generalized

Variational Principle of Bogoliubov and Generalized
arXiv:1507.00563v1 [cond-mat.stat-mech] 2 Jul 2015
Variational Principle of Bogoliubov and Generalized
Mean Fields in Many-Particle Interacting Systems∗
A. L. Kuzemsky
Bogoliubov Laboratory of Theoretical Physics,
Joint Institute for Nuclear Research,
141980 Dubna, Moscow Region, Russia.
E-mail: [email protected]
http://theor.jinr.ru/˜kuzemsky
Abstract
The approach to the theory of many-particle interacting systems from a unified standpoint,
based on the variational principle for free energy is reviewed. A systematic discussion is given
of the approximate free energies of complex statistical systems. The analysis is centered around
the variational principle of N. N. Bogoliubov for free energy in the context of its applications to
various problems of statistical mechanics and condensed matter physics. The review presents a
terse discussion of selected works carried out over the past few decades on the theory of manyparticle interacting systems in terms of the variational inequalities. It is the purpose of this
paper to discuss some of the general principles which form the mathematical background to
this approach, and to establish a connection of the variational technique with other methods,
such as the method of the mean (or self-consistent) field in the many-body problem, in which
the effect of all the other particles on any given particle is approximated by a single averaged
effect, thus reducing a many-body problem to a single-body problem. The method is illustrated
by applying it to various systems of many-particle interacting systems, such as Ising and
Heisenberg models, superconducting and superfluid systems, strongly correlated systems, etc.
It seems likely that these technical advances in the many-body problem will be useful in
suggesting new methods for treating and understanding many-particle interacting systems.
This work proposes a new, general and pedagogical presentation, intended both for those who
are interested in basic aspects, and for those who are interested in concrete applications.
Keywords: Statistical physics; mathematical physics; Helmholtz free energy; variational
methods; variational principle of N. N. Bogoliubov; Bogoliubov inequality; many-particle interacting systems; cooperative phenomena; Bogoliubov theory of superfluidity; BCS-Bogoliubov
microscopic theory of superconductivity; generalized mean fields; model Hamiltonians of manyparticle interacting systems; Ising, Heisenberg and Hubbard models.
PACS: 05.30.-d, 05.30.Fk, 05.30.Jp, 05.70.-a, 05.70.Fh, 02.90.+p
∗
International Journal of Modern Physics B, 29, 1530010 (63 pages) (2015); DOI: 10.1142/S0217979215300108
1
Contents
1 Introduction
2
2 The Variational Principles of Quantum Theory
2
3 The Helmholtz Free Energy and Statistical Thermodynamics
5
4 Approximate Calculations of Helmholtz Free Energy
9
5 The Mean Field Concept
12
6 Symmetry Broken Solutions
18
7 The Mathematical Tools
23
8 Variational Principle of Bogoliubov
26
9 Applications of the Bogoliubov Variational Principle
30
10 The Variational Schemes and Bounds on Free Energy
33
11 The Hartree-Fock-Bogoliubov Mean Fields
36
12 Method of an Approximating Hamiltonian
43
13 Conclusions
44
14 Acknowledgements
46
1
1
Introduction
The fundamental works of N.N. Bogoliubov on many-body theory and quantum field theory [1, 2,
3, 4], on the theory of phase transitions, and on the general theory of interacting systems provided
a new perspective in various fields of mathematics and physics. The variational principle of N. N.
Bogoliubov [1, 2, 3, 4, 5] is a useful working tool and has been widely applied to many problems
of physical interest. It has a well-established place in the many-body theory and condensed
matter physics [6, 7, 8, 9, 10, 11, 12, 13, 14]. The variational principle of N. N. Bogoliubov has
led to a better understanding of various physical phenomena such as superfluidity [1, 2, 3, 4],
superconductivity [1, 2, 3, 4, 15], phase transitions [1, 2, 3, 4, 15, 16] and other cooperative
phenomena [5, 15, 17, 18], etc.
Variational methods in physics and applied mathematics were formulated long ago [19, 20, 21, 22,
23, 24, 25, 26, 27, 28]. It was Maupertuis [25], who wrote in 1774 the celebrated statement:
Nature, in the production of its effects, does so always by simplest means.
Since that time variational methods have become an increasingly popular tool in mechanics, hydrodynamics, theory of elasticity, etc. Moreover, the variational methods are useful and workable
tools for many areas of the quantum theory of atoms and molecules [21, 29, 30, 31, 32], statistical
many-particle physics and condensed matter physics. The variational methods have been applied
widely in quantum-mechanical calculations [21, 32, 29, 30, 31, 32], in theory of many-particle interacting systems [6, 7, 8, 9, 10, 11, 12, 13, 14] and theory of transport processes [33, 34]. As a result
of these efforts many important and effective methods were elaborated by various researchers.
From the other hand the study of the quasiparticle excitations in many-particle systems has been
one of the most fascinating subjects for many years [15, 5, 17, 18]. The quantum field-theoretical
techniques have been widely applied to the statistical treatment of a large number of interacting
particles. Many-body calculations are often done for model systems of statistical mechanics using
the perturbation expansion. The basic procedure in many-body theory is to find the relevant
unperturbed Hamiltonian and then take into account the small perturbation operator. This procedure, which works well for the weakly interacting systems, needs the suitable reformulation for
the many-body systems with complicated spectra and strong interaction.
The considerable progress in studying the spectra of elementary excitations and thermodynamic
properties of many-body systems has been for the most part due to the development of the
temperature-dependent Green functions methods [15, 5, 17, 18]. The very important concept of
the whole method is the concept of the generalized mean field [17, 35, 36, 37]. These generalized
mean fields have a complicated structure for the strongly correlated case and are not reduced
to the functional of the mean densities of the electrons. The concept of the generalized mean
fields and the relevant algebra of operators from which the corresponding Green functions are
constructed are the central ones to our treatment of the strongly interacting many-body systems.
It is the purpose of this paper to discuss some of the general principles which form the physical
and mathematical background to the variational approach, and to establish the connection of the
variational technique with other methods in the theory of many-body problem.
2
The Variational Principles of Quantum Theory
It is well known that in quantum mechanics the eigenfunction ψi of the lowest state of any system
has the property of making the integral
Z
ψi∗ Hψi d3 r
(1)
2
a minimum. The value of integral is the corresponding eigenvalue Ei of the Hamiltonian H of a
system. These circumstances lead to a specific approximate method (the variational method) of
finding ψi and Ei by minimizing integral (1) among a restricted class of functions.
The variational method [21, 32, 29, 30, 31] enables one to make estimates of energy levels by using
trial wave functions ψT
R ∗
ψ HψT d3 r
ET = R T ∗
.
(2)
ψT ψT d3 r
The ground state E0 gives the lowest possible energy the system can have. Hence, for the approximation of the ground state energy one would like to minimize the expectation value of the energy
with respect to a trial wave function.
In other words the variational principle states that the ground-state energy of a quantum mechanical system is less than or equal to the expectation value of the Hamiltonian with an arbitrary
wave function. Given a trial wave function with adjustable parameters, the best values of the
parameters are those which minimize the expectation value of the Hamiltonian. The variational
principle consists in adjusting the available parameters, so as to maximize this lower bound.
An important method of finding approximate ground state energies and wave functions is called
by the Rayleigh-Ritz variational principle [21, 29, 30, 31]. The Rayleigh-Ritz variational principle
for the ground state energy is the starting point of many computations and approximations in
quantum mechanics and quantum chemistry of atoms and molecules. This principle states that
the expectation value of H in any state |ψi is always greater than or equal to the ground state
energy, E0
hψ|H|ψi
≥ E0 ,
(3)
hψ|ψi
or
hHi ≥ hψ|H|ψi ≥ E0 .
(4)
Here |ψi ∈ G is arbitrary pure quantum state and H is a Hamiltonian acting on a Hilbert space
G. This relation becomes equality only when ψ = ψ0 . Thus this principle gives the upper bound
to the ground state energy.
It will be instructive also to remind how the variational principle of quantum mechanics complements the perturbation theory [38, 39]. For this aim let us consider the Rayleigh-Schrödinger
perturbation expansion. The second-order level-shift E20 of the ground state of a system have the
form
X hψ 0 |V |ψj ihψj |V |ψ 0 i X |V0j |2
=
,
(5)
E20 =
(E 0 − Ej )
(E 0 − Ej )
j6=0
j6=0
where V0j = hψ 0 |V0j |ψj i and |ψ 0 i is the unperturbed ground state. It is clear then that E20 is
always negative.
The variational principle of quantum mechanics states that the ground state energy E 0 for the
total Hamiltonian H is the minimum of the energy functional
E{Ψ} = hΨ|H|Ψi,
(6)
where Ψ is a trial wave function. It should be noted that it is possible to establish that the sum
of all the higher-order level shifts En0 , starting with n = 2, will be negative, providing that the
relevant perturbation series will converges to E 0 .
To confirm this statement let us consider again the Hamiltonian
H = H0 + λV.
3
(7)
It is reasonable to suppose that the ground state energy E 0 = E 0 (λ) and the ground state
Ψ = Ψ(λ) of the Hamiltonian H are analytic functions (at least for small λ). Note that when
one considers the many-body problem, the concept of relative boundedness is of use, where a
perturbation λV is small compared to H0 in the sense that
(H0 )2 ≥ (λ2 V 2 ).
This means simply that the eigenvalues of the operator ((H0 )2 − (λ2 V 2 )) are non-negative.
Then the corresponding perturbation expansion may be written in the form
(0)
(1)
(2)
(3)
E0 = E0 + λE0 + λ2 E0 + λ3 E0 +
...,
(8)
where E00 = hψ 0 |H|ψ 0 i and E01 = hψ 0 |V |ψ 0 i.
The variational approach states that
E0 = min (hΨ|H0 + λV |Ψi) .
(9)
Thus we obtain
(2)
(3)
(0)
(1)
λ2 E0 + λ3 E0 + . . . = E0 − (E0 + λE0 ) =
min{hΨ|H0 + λV |Ψi} − hψ 0 |H0 + λV |ψ 0 i .
In this expression the second part must satisfy the condition
0
0
min{hΨ|H0 + λV |Ψi} − hψ |H0 + λV |ψ i ≤ 0.
(10)
(11)
In addition, in general case the relevant ground state Ψ which yields a minimum will not coincide
with ψ 0 . Thus we obtain
(2)
(3)
λ2 E0 + λ3 E0 + . . . < 0.
(12)
The last inequality can be rewritten as
(2)
(3)
(4)
E0 < (λE0 + λ2 E0 +
(2)
. . .).
(13)
In the limit λ → 0 we have that E0 < 0. Thus the variational principle of quantum mechanics
confirms the results of the perturbation theory [40].
It is worth mentioning that the Rayleigh-Ritz variational method has a long and interesting history [41, 42, 43]. Rayleigh’s classical book Theory of Sound was first published in 1877. In it
are many examples of calculating fundamental natural frequencies of free vibration of continuum
systems (strings, bars, beams, membranes, plates) by assuming the mode shape, and setting the
maximum values of potential and kinetic energy in a cycle of motion equal to each other. This
procedure is well known as Rayleigh’s Method. In 1908, Ritz laid out his famous method for determining frequencies and mode shapes, choosing multiple admissible displacement functions, and
minimizing a functional involving both potential and kinetic energies. He then demonstrated it in
detail in 1909 for the completely free square plate. In 1911, Rayleigh wrote a paper congratulating
Ritz on his work, but stating that he himself had used Ritz’s method in many places in his book
and in another publication.
Subsequently, hundreds of research articles and many books have appeared which use the method,
some calling it the ”Ritz method” and others the ”Rayleigh-Ritz method.” The article [41] examined the method in detail, as Ritz presented it, and as Rayleigh claimed to have used it. A. W.
Leissa [41] concluded that, although Rayleigh did solve a few problems which involved minimization of a frequency, these solutions were not by the straightforward, direct method presented by
Ritz and used subsequently by others. Therefore, Rayleigh’s name should not be attached to the
method. Additional informative comments were carried out in Refs. [42, 43]
4
3
The Helmholtz Free Energy and Statistical Thermodynamics
Variational methods in thermodynamics and statistical mechanics have been used widely since J.
W. Gibbs groundbreaking works [26, 27, 28]. According to Gibbs approach a workable procedure
for the development of the statistical mechanical ensemble theory is to introduce the Gibbs entropy
postulate. Hence, as a result of the Gibbs ensemble method, the entropy S can be expressed in
the form of an average for all the ensembles, namely,
1
X
1
= kB ln Ω(N, V, E),
(14)
ln
pi ln pi = −kB Ω
S(N, V, E) = −kB
Ω Ω
i
where the summation over i denotes a general summation over all states of the system and pi is
the probability of observing state i in the given ensemble and kB is the Boltzmann constant. This
relation links entropy S and probability pi .
It can be said that in this context the entropy is a state function which is according to the second
law [27, 44] is defined by the relation
dS = β(dE − dF ).
(15)
The energy E and the Helmholtz free energy F are the state functions [27, 44]. The proportionality
coefficient β was termed as the thermodynamic temperature (β = 1/kB T ) of the surrounding with
which the system exchanges by heat Q and work W .
Thus the postulate of equal probabilities in the microcanonical ensemble [45] and the Gibbs
entropy postulate can be considered as a convenient starting points for the development of the
statistical mechanical ensemble theory in a standard approach [27].
After postulating the entropy by means of Eq.(14), the thermodynamic equilibrium ensembles are
determined by the following criterion for equilibrium:
(δS)E,V,N = 0.
(16)
This variational scheme is used for each ensemble (microcanonical, canonical and grand canonical)
with different constraints for each ensemble. In addition, this procedure introduces Lagrange
multipliers which, in turn, must be identified with thermodynamic intensive variables (T, P ). From
the other hand, the procedure of introducing Lagrange multipliers and the task of identifying them
with the thermodynamic intensive properties can be clarified by invoking a more general criterion
for thermodynamic equilibrium.
From the Gibbs
entropy postulate, Eq.(14), the definitions of average and the normalization
P
constraint i pi = 1 one obtains
X
(1 + ln pi )δpi ,
(17)
δS = −kB
i
δE =
X
Ei δpi ,
(18)
Vi δpi ,
(19)
δpi = 0.
(20)
i
δV =
X
i
X
i
Using a Lagrange multiplier λ together with the variational condition, we obtain
X
(Ei + P Vi + λ + kB T + kB T ln pi )δpi ≥ 0.
i
5
(21)
Here all δpi are considered as the independent variables. Thus we deduce that
pi = exp −βλ − 1 − β(P Vi + Ei ) , β = (kB T )−1 .
(22)
The Lagrange multiplier λ, can be determined directly from the definition of entropy (14).
E + P V + λ + kB T
X Ei + P Vi + λ + kB T =
.
(23)
pi
S = −kB
kB T
T
i
Thus we arrive at
λ + kB T = T S − E − P V = −G,
pi = exp β G − P Vi − Ei .
(24)
(25)
Here G is the Gibbs energy (or Gibbs free energy). It may also be defined with the aid of the
Helmholtz free energy G = H − T S. Here H(S, P, N ) is the enthalpy [44].
The usefulness of the thermodynamic potentials G and F may be clarified within the statistical
thermodynamics [27]. For the microcanonical ensemble one should substitute Ei = E and Vi = V ,
which are fixed for every system and since G − P V − E = S, Eq.(25) becomes
pi = e−S/kB .
(26)
For the canonical ensemble one should substitute Vi = V , which is given for each system and in
this case Eq.(25) can be written as
pi = eβ(F −Ei ) .
(27)
Here F = G − P V denotes the Helmholtz free energy. Thus the free energy F is defined by
F = E − T S.
(28)
The Helmholtz free energy describes an energy which is available in the form of useful work.
The second law of thermodynamics asserts that in every neighborhood of any state A in an
adiabatically isolated system there exist other states that are inaccessible from A. This statement
in terms of the entropy S and heat Q can be formulated as
dS = dQ/T + dσ.
(29)
Thus the only states available in an adiabatic process (dQ = 0, or dS = dσ) are those which
lead to an increase of the entropy S. Here dσ ≥ 0 defines the entropy production σ due to the
irreversibility of the transformation.
It is of use to analyze the expression
X
dF = dE − T dS − SdT = −SdT − T dσ − P dV +
µ i Ni .
(30)
Free energy change ∆F of a system during a transformation of a system describes a balance of the
work exchanged with the surroundings. If ∆F > 0, ∆F represents the minimum work that must
be incurred for the system to carry out the transformation. In the case ∆F < 0, |∆F | represents
the maximum work that can be obtained from the system during the transformation. It is obvious
that
dF = dE − T dσ − SdT.
(31)
6
In a closed system without chemical reaction
P and in the absence of any other energy exchange,
the variation ∆F = −SdT − T dS − P dV + µi Ni can be rewritten in the form
dF = −T dσ ≤ 0.
(32)
It means that function F decreases and tends towards a minimum corresponding to equilibrium.
Thus the Helmholtz free energy is the thermodynamic potential of a system subjected to the
constraints constant T, V, Ni .
The Gibbs free energy (free enthalpy) is defined by
G = H − T S = F + P V.
(33)
The physical meaning of the Gibbs free energy is clarified when considering evolution of a system
from a certain initial state to a final state. The Gibbs free energy change ∆G then represents
the work exchanged by the system with its environment and the work of the pressure
P forces,
during a reversible transformation of the system. Here H = E + V P = T S + V P + µi Ni is
the thermodynamic potential of a system termed by enthalpy [44]. The Gibbs free energy is the
thermodynamic potential of a system subjected to the constraints constant T, P, Ni . In this case
dG = −T dσ ≤ 0.
(34)
Thus in the closed system without chemical reaction and in the absence of any other energy
exchange at constant temperature, pressure and amount of substance the function G can only
decrease and reach a minimum at equilibrium.
It will be of use to mention another class of thermodynamic potentials, termed by the MassieuPlanck functions. These objects may be deduced from the fundamental relations in the entropy
representations, S = S(E, V, N ). The corresponding differential form may be written as
dS =
P
µ
1
dE + dV − dN.
T
T
T
(35)
Thus the suitable variables for a Legendre transform will be 1/T , P/T and µ/T . In some cases
working with these variables is more convenient.
It is worth noting that in terms of the Gibbs ensemble method the free energy is the thermodynamic potential of a system subjected to the constraints constant T, V, Ni . Moreover, the
thermodynamic potentials should be defined properly in the thermodynamic limit. The problem
of the thermodynamic limit in statistical physics was discussed in detail by A. L. Kuzemsky [45].
To clarify this notion, let us consider the logarithm of the partition function Q(θ, V, N )
F (θ, V, N ) = −θ ln Q(θ, V, N ).
(36)
This expression determines the free energy F of the system on the basis of canonical distribution. The standard way of reasoning in the equilibrium statistical mechanics do not requires the
knowledge of the exact value of the function F (θ, V, N ). For real system it is sufficient to know
the thermodynamic (infinite volume) limit [15, 27, 45, 46]
lim
N →∞
F (θ, V, N )
|V /N =const = f (θ, V /N ).
N
(37)
Here f (θ, V /N ) is the free energy per particle. It is clear that this function determines all the
thermodynamic properties of the system.
Let us summarize the criteria for equilibrium briefly. In a system of constant V and S, the
internal energy has its minimum value, whereas in a system of constant E and V , the entropy has
7
its maximum value.
It should be noted that the pair of independent variables (V, S) is not suitable one because of the
entropy is not convenient to measure or control. Hence it would be of use to have fundamental
equations with independent variables that is easier to control. The two convenient choices are
possible. First we take the P and T pair. From the practical point of view this is a convenient
pair of variables which are easy to control (measure). For systems with constant pressure the best
suited state function is the Gibbs free energy (also called free enthalpy)
G = H − T S.
(38)
Second relevant pair is V and T . For systems with constant volume (and variable pressure), the
suitable suited state function is the Helmholtz free energy
F = E − T S.
(39)
Any state function can be used to describe any system (at equilibrium, of course), but for a given
system some are more convenient than others. The change of the Helmholtz free energy can be
written as
dF = dE − T dS − SdT.
(40)
Combining this equation with dU = T dS − P dV we obtain the relation of the form
dF = −P dV − SdT.
(41)
In terms of variables (T, V ) we find
dF =
∂F ∂F dT +
dV.
∂T V
∂V T
Comparing the equations one can see that
∂F S=−
,
∂T V
P =
∂F .
∂V T
(42)
(43)
At constant T and V the equilibrium states corresponds to the minimum of Helmholtz free energy
(dF = 0). From F = E − T S we may suppose that low values of F are obtained with low values
of E and high values of S.
In terms of a general statistical-mechanical formalism [3, 4, 15], a many-particle system with
Hamiltonian H in contact with a heat bath at temperature T in a state described by the statistical
operator ρ has a free energy
F = Tr(ρH) + kB T Tr(ρ ln ρ).
(44)
The free energy takes its minimum value
Feq = −kB T ln Z
(45)
in the equilibrium state characterized by the canonical distribution
ρeq = Z −1 exp(−Hβ);
Z = Tr exp(−Hβ).
(46)
Before turning to the next topic, an important remark about the free energy will not be out of place
here. I. Novak [47] attempted to give a microscopic description of Le Chatelier’s principle [48] in
statistical systems. Novak have carried out interesting analysis based on microscopic descriptors
(energy levels and their populations) that provides visualization of free energies and conceptual
rationalization of Le Chatelier’s principle. The misconception ”nature favors equilibrium” was
8
highlighted. This problem is a delicate one and requires a careful discussion [49]. Dasmeh et al.
showed [49] that Le Chatelier’s principle states that when a system is disturbed, it will shift its
equilibrium to counteract the disturbance. However for a chemical reaction in a small, confined
system, the probability of observing it proceed in the opposite direction to that predicted by
Le Chatelier’s principle, can be significant. Their study provided a molecular level proof of Le
Chatelier’s principle for the case of a temperature change. Moreover, a new, exact mathematical
expression was derived that is valid for arbitrary system sizes and gives the relative probability
that a single experiment will proceed in the endothermic or exothermic direction, in terms of a
microscopic phase function. They showed that the average of the time integral of this function
is the maximum possible value of the purely irreversible entropy production for the thermal
relaxation process. The results obtained were tested against computer simulations of the unfolding
of a polypeptide. It was proven that any equilibrium reaction mixture on average responds to a
temperature increase by shifting its point of equilibrium in the endothermic direction.
4
Approximate Calculations of Helmholtz Free Energy
Statistical mechanics provides effective and workable tools for describing the behavior of the systems of many interacting particles. One of such approaches for describing systems in equilibrium
consists in evaluation the partition function Z and then the free energy.
Now we must take note of the different methods for obtaining the approximate Helmholtz free energy in the theory of many-particle systems. Roughly speaking, there are two approaches, namely
the perturbation method and the variational method.
Thermodynamic perturbation theory [50, 51, 52, 53] may be applied to systems that undergo a
phase transition. It was shown [54] that certain conditions are necessary in order that the application of the perturbation does not change the qualitative features of the phase transition. Usually,
the shift in the critical temperature are determined to two orders in the perturbation parameter.
Let us consider here the perturbation method [54] very briefly.
In paper [54] authors considered a system with Hamiltonian H0 , that undergoes a phase transition
at critical temperature TC0 . The task was to determine for what class of perturbing potentials V
will the system with Hamiltonian H0 + V have a phase transition with qualitatively the same
features as the unperturbed system. In their paper authors [54] studied that question using thermodynamic perturbation theory [50, 51, 52]. They found that an expansion for the perturbed
thermodynamic functions can be term by term divergent at the critical temperature TC0 for a class
of potentials V . Under certain conditions the series can be resummed, in which case the phase
transition remains qualitatively the same as in the unperturbed system but the location of the
critical temperature is shifted.
The starting point was the partition function Z0 for a system whose Hamiltonian is H0
Z0 = Tr exp(−H0 β).
(47)
For a system with Hamiltonian H0 + λV , the partition function Z is given by
Z = Tr exp[−(H0 + λV )β].
(48)
It is possible to obtain formally an expansion for Z in terms of the properties of the unperturbed
system by expanding that part of the exponential containing the perturbation in the following
way [54] when V and H0 commute:
∞
∞
X
X
1
1
n n
= Z0
[−λβ] V
[−λβ]n hV n i0 ,
Z = Tr exp(−H0 β)
n!
n!
n
n
9
(49)
where
Z0 hV n i0 = Tr exp[−H0 β]V n .
(50)
Then the expression for Z can be written as
∞
X
1
Z
= exp ln[1 +
(−λβ)n hV n i0 ]
Z0
n!
n
(51)
The free energy per particle f is given by
βfp = βf0 −
∞
X
1 1
ln 1 +
(−λβ)n hV n i0 ,
N
n!
n
(52)
where fp and f0 are the perturbed and unperturbed free energy per particle, respectively, and N
is the number of particles in the system. The standard way to proceed consists of in expanding
the logarithm in powers of λ. As a result one obtains [54]
λ2 β 2 1 2
λβ
hV i0 − hV i20
hV i0 −
βfp = βf0 +
N
2! N
λ3 β 3 1 3
+
hV i0 − 3hV 2 i0 hV i0 + hV i30 + . . .
3! N
(53)
To proceed, it is supposed usually that the thermodynamics of the unperturbed system is known
and the perturbation series (if they converge) may provide us with suitable corrections. If the
terms in the expansion diverge, they may, in principle, be regularized under some conditions. For
example, perturbation expansions for the equation of state of a fluid whose intermolecular potential can be regarded as consisting of the sum of a strong and weak part give reasonable qualitative
results [55, 56].
In paper by Fernandes [53] author investigated the application of perturbation theory to the
canonical partition function of statistical mechanics. The Schwinger and Rayleigh-Schrödinger
perturbation theory were outlined and plausible arguments were formulated that both should give
the same result. It was shown that by introducing adjustable parameters in the unperturbed or
reference Hamiltonian operator, one can improve the rate of convergence of Schwinger perturbation theory. The same parameters are also suitable for Rayleigh-Schrödinger perturbation theory.
Author discussed also a possibility of variational improvements of perturbation theory and gave
a simpler proof of a previously derived result about the choice of the energy shift parameter. It
was also shown that some variational parameters correct the anomalous behavior of the partition
function at high temperatures in both Schwinger and Rayleigh-Schrödinger perturbation theories.
It should be stressed, however, that the perturbation method is valid for small perturbations only.
The variational method is more flexible tool [38, 39, 57, 58, 59, 60, 61, 62] and in many cases is
more appropriate in spite of the obvious shortcomings. But both the methods are interrelated
deeply [39] and enrich each other.
R. Peierls [51, 52, 63] pointed at the circumstance that for a many-particle system in thermal
equilibrium there is a minimum property of the free energy which may be considered as a generalization of the variational principle for the lowest eigenvalue in quantum mechanics. Peierls
attracted attention to the fact that the free energy has a specific property which can be formulated in the following way. Let us consider an arbitrary set of orthogonal and normalized functions
{ϕ1 , ϕ2 , . . . ϕn , . . .}. The expectation value of the Hamiltonian H for nth of them will be written
as
Z
Hnn = ϕ∗n Hϕn dr.
(54)
10
The statement is that for any temperature T the function
X
F̃ = −kB T log Z̃ = −kB T log
exp[−Hnn β]
(55)
n
which would represent the free energy if the Hnn were the true eigenvalues, is higher than the
true free energy
X
F0 = −kB T log Z0 = −kB T log
exp[−En β],
(56)
n
or
F̃ ≥ F0 .
(57)
This is equivalent to saying that the partition function, as formed by means of the expectation
values Hnn :
X
Z̃ =
exp[−Hnn β]
(58)
n
is less than the true partition function
Z0 =
X
exp[−En β],
(59)
n
or
Z0 =
X
n
exp[−En β] ≥ Z̃ =
X
exp[−Hnn β].
(60)
n
Peierls [63] formulated the more general statement, namely, that if f (E) is a function with the
properties
d2 f
df
< 0,
> 0,
(61)
dE
dE 2
the expression
X
f=
f (Hnn )
(62)
n
is less than
f0 =
X
f (En ).
(63)
n
To summarize, Peierls has proved a kind of theorem a special case of which gives a lower bound to
the partition sum and hence an upper bound to the free energy of a quantum mechanical system
X
X
exp[−Ek β] ≥
exp[−Hnn β].
(64)
n
k
When β → ∞ the theorem is obvious, reducing to the fundamental inequality Ek ≤ Hnn for all
n. However for finite β it is not so obvious since higher eigenvalues of H do not necessarily lie
lower than corresponding diagonal matrix elements Hnn . T. D. Schultz [64] skilfully remarked
that, in fact, the Peierls inequality does not depend on the fact that exp[−Eβ] is a monotonically
decreasing function of E, as might be concluded from the original proof. It depends only on the
fact that the exponential function is concave upward. Schultz [64] proposed a simple proof of the
theorem under this somewhat general condition.
Let ϕn be a complete orthonormal set of state vectors and let A be an Hermitean operator which
for convenience is assumed to have a pure point spectrum with eigenvalues ak and eigenstates ψk .
Let f (x) be a real valued function such that
d2 f
>0
dx2
11
(65)
in an interval including the whole spectrum of ak . Then if Trf (A) exists it can be proven that
the following statement holds
X
Trf (A) ≥
f (ann ),
(66)
n
where ann = hn|A|ni. The equality holds if and only if the ϕn are the eigenstates of A.
Since
X
Trf (A) =
hn|f (A)|ni
(67)
n
it is sufficient for the proof to point out that the relation (66) follows from
hn|f (A)|ni ≥ f (ann ),
(68)
which is valid for all n. The inequalities (68) were derived from
f (ak ) ≥ f (ann ) + (ak − ann )f ′ (ann ),
(69)
which is a consequence of equation (65), the right hand side for fixed n being the line tangent
to f (ak ) at ann . Multiplying (69) by |hn|ki|2 and summing on k one obtains (68). Schultz [64]
observed further that the equality in (68) holds if and only if |hn|ki|2 = 0 unless ak = ann , i.e. if
and only if ϕn is an eigenstate of A.
If f (A) is positive definite, then the set ϕn need not be complete,
since the theorem is true even
P
more strongly if positive terms are omitted from the sum n f (ann ).
With the choice f (A) = exp(−A) and A = Hβ, the original theorem of Peierls giving an upper
bound to the free energy is reproduced. With A = (H − µN)β we have an analogous theorem for
the grand potential. The theorem proved by Schultz [64] is a generalization in that it no longer
requires f (x) to be monotonic; it requires only that Trf (A) be finite which can occur even if f (x)
is not monotonic provided A is bounded.
Peierls variational theorem was discussed and applied in a number of papers (see, e.g. Refs. [64, 65,
66, 67]). It has much more generality than, say, the A. Lidiard [68] consideration on a minimum
property of the free energy. Lidiard [68] derived the approximate free energy expression in a way
which shows a strong analogy with the approximate Hartree method of quantum mechanics. By
his derivation he refined the earlier calculations made by Koppe an Wohlfarth in the context of
description of the influence of the exchange energy on the thermal properties of free electrons in
metals.
5
The Mean Field Concept
In general case, a many-particle system with interactions is very difficult to solve exactly, except
for special simple cases. Theory of molecular (or mean) field permits one to obtain an approximate
solution to the problem. In condensed matter physics, mean field theory (or self-consistent field
theory) studies the behavior of large many-particle systems by studying a simpler models. The
effect of all the other particles on any given particle is approximated by a single averaged effect,
thus reducing a many-body problem to a single-body problem.
It is well known that molecular fields in various variants appear in the simplified analysis of many
different kinds of many-particle interacting systems. The mean field concept was originally formulated for many-particle systems (in an implicit form) in Van der Waals [69, 70] dissertation
”On the Continuity of Gaseous and Liquid States”. Van der Waals conjectured that the volume
correction to the equation of state would lead only to a trivial reduction of the available space
for the molecular motion by an amount b equal to the overall volume of the molecules. In reality,
12
the measurements lead him to be much more complicated dependence. He found that both the
corrections should be taken into account. Those were the volume correction b, and the pressure
correction a/V 2 , which led him to the Van der Waals equation [70]. Thus, Van der Waals came
to conclusion that ”the range of attractive forces contains many neighboring molecules”. The
equation derived by Van der Waals was similar to the ideal gas equation except that the pressure
is increased and the volume decreased from the ideal gas values. Hence, the many-particle behavior was reduced to effective (or renormalized) behavior of a single particle in a medium (or a
field). The later development of this line of reasoning led to the fruitful concept, that it may be
reasonable to describe approximately the complex many-particle behavior of gases, liquids, and
solids in terms of a single particle moving in an average (or effective) field created by all the other
particles, considered as some homogeneous (or inhomogeneous) environment.
Later, these ideas were extended to the physics of magnetic phenomena [17, 5, 71, 72], where
magnetic substances were considered as some kind of a specific liquid. This approach was elaborated in the physics of magnetism by P. Curie and P. Weiss. The mean field (molecular field)
replaces the interaction of all the other particles to an arbitrary particle [73]. In the mean field
approximation, the energy of a system is replaced by the sum of identical single particle energies
that describe the interactions of each particle with an effective mean field.
Beginning from 1907 the Weiss molecular-field approximation became widespread in the theory of
magnetic phenomena [17, 5, 71, 72], and even at the present time it is still being used efficiently.
Nevertheless, back in 1965 it was noticed that [74]:
”The Weiss molecular field theory plays an enigmatic role in the statistical mechanics
of magnetism”.
In order to explain the concept of the molecular field on the example of the Heisenberg ferromagnet
one has to transform the original many-particle Hamiltonian
X
X
Siz .
(70)
J(i − j)Si Sj − gµB H
H=−
i
ij
into the following reduced one-particle Hamiltonian
H = −2µ0 µB ~S · ~h
(mf )
.
The coupling coefficient J(i − j) is the measure of the exchange interaction between spins at the
lattice sites i and j and is defined usually to have the property J(i - j = 0) = 0. This transformation
was achieved with the help of the identity [17, 5, 71, 72]
~ · S~ ′ = S
~ · hS~ ′ i + hSi
~ · S~′ − hSi
~ · hS~ ′ i + C.
S
~ − hSi)
~ · (S~ ′ − hS~ ′ i) describes spin correlations. The usual molecularHere, the constant C = (S
field approximation is equivalent to discarding the third term in the right hand side of the above
~ · S~′ i − hSi
~ · hS~ ′ i. for the constant C.
equation, and using the approximation C ∼ hCi = hS
There is large diversity of the mean-field theories adapted to various concrete applications [17, 5,
71, 72].
Mean field theory has been applied to a number of models of physical systems so as to study
phenomena such as phase transitions [75, 76]. One of the first application was Ising model [17, 5,
71, 72]. Consider the Ising model on an N -dimensional cubic lattice. The Hamiltonian is given
by
X
X
H = −J
Si Sj − h
Si ,
(71)
i
hi,ji
13
P
where the hi,ji indicates summation over the pair of nearest neighbors hi, ji , and Si = ±1 and
Sj are neighboring Ising spins. A. Bunde [77] has shown that in the correctly performed molecular
field approximation for ferro- and antiferromagnets the correlation function hS(q)S(−q)i should
fulfill the sum rule
X
hS(q)S(−q)i = 1.
(72)
N −1
q
The Ising model of the ferromagnet was considered [77] and the correlation function hS(q)S(−q)i
was calculated
i−1
h
X
1
1
hS(q)S(−q)i = N −1
,
(73)
1 − βJ(q)
1 − βJ(q)
q
which obviously fulfills the above sum rule. The Ising model and the Heisenberg model were two
the most explored models for the applications of the mean field theory.
It is of instruction to mention that earlier molecular-field concepts described the mean-field in
terms of some functional of the average density of particles hni (or, using the magnetic terminology,
the average magnetization hM i), that is, as F [hni, hM i]. Using the modern language, one can say
that the interaction between the atomic spins Si and their neighbors can be equivalently described
by effective (or mean) field h(mf ) . As a result one can write down
(ext)
Mi = χ0 [hi
(mf )
+ hi
].
The mean field h(mf ) can be represented in the form (in the case T > TC )
X
h(mf ) =
J(Rji )hSi i.
(74)
(75)
i
Here, hext is the external magnetic field, χ0 is the system’s response function, and J(Rji ) is the
interaction between the spins. In other words, in the mean-field approximation a many-particle
system is reduced to the situation, where the magnetic moment at any site aligns either parallel
or anti-parallel to the overall magnetic field, which is the sum of the applied external field and
the molecular field.
Note that only the ”averaged” interaction with i neighboring sites was taken into account, while
the fluctuation effects were ignored. We see that the mean-field approximation provides only a
rough description of the real situation and overestimates the interaction between particles. Attempts to improve the homogeneous mean-field approximation were undertaken along different
directions [17, 5, 71, 72, 35, 36, 37].
An extremely successful and quite nontrivial approach was developed by L. Neel [17, 5, 71, 72],
who essentially formulated the concept of local mean fields (1932). Neel assumed that the sign
of the mean-field could be both positive and negative. Moreover, he showed that below some critical temperature (the Neel temperature) the energetically most favorable arrangement of atomic
magnetic moments is such, that there is an equal number of magnetic moments aligned against
each other. This novel magnetic structure became known as the antiferromagnetism [5, 17].
It was established that the antiferromagnetic interaction tends to align neighboring spins against
each other. In the one-dimensional case this corresponds to an alternating structure, where an
”up” spin is followed by a ”down” spin, and vice versa. Later it was conjectured that the state
made up from two inserted into each other sublattices is the ground state of the system (in the
classical sense of this term). Moreover, the mean-field sign there alternates in the ”chessboard”
(staggered) order.
The question of the true antiferromagnetic ground state is not completely clarified up to the
present time [17, 5, 71, 72, 35, 36, 37]. This is related to the fact that, in contrast to ferromagnets, which have a unique ground state, antiferromagnets can have several different optimal states
14
with the lowest energy. The Neel ground state is understood as a possible form of the system’s
wave function, describing the antiferromagnetic ordering of all spins. Strictly speaking, the ground
state is the thermodynamically equilibrium state of the system at zero temperature. Whether the
Neel state is the ground state in this strict sense or not, is still unknown. It is clear though,
that in the general case, the Neel state is not an eigenstate of the Heisenberg antiferromagnet’s
Hamiltonian. On the contrary, similar to any other possible quantum state, it is only some linear
combination of the Hamiltonian eigenstates. Therefore, the main problem requiring a rigorous
investigation is the question of Neel state stability [17]. In some sense, only for infinitely large
lattices, the Neel state becomes the eigenstate of the Hamiltonian and the ground state of the system. Nevertheless, the sublattice structure is observed in experiments on neutron scattering [17],
and, despite certain worries, the actual existence of sublattices is beyond doubt.
Once Neel’s investigations were published, the effective mean-field concept began to develop at
a much faster pace. An important generalization and development of this concept was proposed
in 1936 by L. Onsager [78] in the context of the polar liquid theory. This approach is now called
the Onsager reaction field approximation. It became widely known, in particular, in the physics
of magnetic phenomena [79, 80, 81]. In 1954, Kinoshita and Nambu [82] developed a systematic
method for description of many-particle systems in the framework of an approach which corresponds to the generalized mean-field concept. N.D. Mermin [83] has analyzed the thermal
Hartree-Fock approximation [84] of Green function theory giving the free energy of a system not
at zero temperature.
Kubo and Suzuki [85] studied the applicability of the mean field approximation and showed that
the ordinary mean-field theory is restricted only to the region kB T ≥ zJ, where J denotes the
strength of typical interactions of the relevant system and z the number of nearest neighbours.
Suzuki [86] has proposed a new type of fluctuating mean-field theory. In that approach the true
critical point T̃C differs from the mean-field value and the singularities of response functions are,
in general, different from those of the Weiss mean-field theory [17, 71].
Lei Zhou and Ruibao Tao [87] developed a complete Hartree-Fock mean-field method to study
ferromagnetic systems at finite temperatures. With the help of the complete Bose transformation, they renormalized all the high-order interactions including both the dynamic and the kinetic
ones based on an independent Bose representation, and obtained a set of compact self-consistent
equations. Using their method, the spontaneous magnetization of an Ising model on a square
lattice was investigated. The result is reasonably close to the exact one. Finally, they discussed
the temperature dependences of the coercivities for magnetic systems and showed the hysteresis
loops at different temperatures.
Later, various schemes of ”effective mean-field theory taking into account correlations” were proposed (see Refs. [17, 37]). We will see below that various mean-field approximations can be in
principle described in the framework of the variation principle in terms of the Bogoliubov inequality [1, 3, 5, 10, 15]:
F = −β −1 ln(Tre−βH ) ≤
Tre−βHmod (H − Hmod )
.
−β −1 ln(Tre−βHmod ) +
Tre−βHmod
(76)
Here, F is the free energy of the system under consideration, whose calculation is extremely
involved in the general case. The quantity Hmod is some trial Hamiltonian describing the effectivefield approximation. The inequality (76) yields an upper bound for the free energy of a manyparticle system.
It is well known, that the study of Hamiltonians describing strongly-correlated systems is an
exceptionally difficult many-particle problem, which requires applications of various mathematical
methods [17, 88, 89, 90, 91]. In fact, with the exception of a few particular cases, even the ground
15
state of the Hubbard model is still unknown. Calculation of the corresponding quasiparticle
spectra in the case of strong inter-electron correlations and correct definition of the mean fields
also turned out to be quite a complicated problem.
The Hamiltonian of the Hubbard model [17] is given by:
X
X
tij a†iσ ajσ + U/2
niσ ni−σ .
(77)
H=
ijσ
iσ
The above Hamiltonian includes the repulsion of the single-site intra-atomic Coulomb U , and tij ,
the one-electron hopping energy describing jumps from a j site to an i site. As a consequence of
correlations electrons tend to ”avoid one another”. Their states are best modeled by atom-like
~ j )]. The Hubbard model’s Hamiltonian can be characterized by
Wannier wave functions [φ(~r − R
two main parameters: U , and the effective band width of tightly bound electrons
X
|tij |2 )1/2 .
∆ = (N −1
ij
The band energy of Bloch electrons ǫ(~k) is given by
X
~i − R
~ j ],
ǫ(~k) = N −1
tij exp[−i~k(R
~k
where N is the total number of lattice sites. Variations of the parameter γ = ∆/U allow one to
study two interesting limiting cases, the band regime (γ ≫ 1) and the atomic regime (γ → 0).
There are many different approaches to construction of generalized mean-field approximations;
however, all of them have a special-case character. The method of irreducible Green functions [17, 35, 36, 37] allows one to tackle this problem in a more systematic fashion.
The efficiency of the method of the irreducible Green functions for description of normal and
superconducting properties of systems with a strong interaction and complicated character of the
electron spectrum was demonstrated in the papers [17, 35, 36, 37]. Let us consider the Hubbard
model (77). The properties of this Hamiltonian are determined by the relationship between the
two parameters: the effective band’s width ∆ and the electron’s repulsion energy U . Drastic
transformations of the metal-dielectric phase transition’s type take place in the system as the
ratio of these parameters changes. Note that, simultaneously, the character of the system description must change as well, that is, we always have to describe our system by the set of relevant
variables. In the case of weak correlation [17, 35, 36, 37] the corresponding set of relevant variables
contains the ordinary second-quantized Fermi operators and a†iσ aiσ , as well as the number of
particles operator niσ = a†iσ aiσ . In the case of strong correlation [17, 35, 36, 37] the problem is
highly complicated.
The Green function in the generalized mean-field’s approximation has the following very complicated functional structure [17, 35, 36, 37]:
F
GM
kσ (ω)
−
ω − (n+
−σ E− + n−σ E+ ) − λ(k)
.
=
+
−
+
(ω − E+ − n−
−σ λ1 (k))(ω − E− − n−σ λ2 (k)) − n−σ n−σ λ3 (k)λ4 (k)
(78)
Here, the quantities λi (k) are the components of the generalized mean field, which cannot be
reduced to the functional of the mean particle’s densities. The expression for Green function (78)
can be written down in the form of the following generalized two-pole solution
−1
−1
n−
n+
−σ (1 + cb )
−σ (1 + da )
+
≈
a − db−1 c
b − ca−1 d
n+
n−
−σ
−σ
+
,
−
+
†
ω − E− − n−σ Wk−σ
ω − E+ − n−
−σ Wk−σ
F
GM
kσ (ω) =
16
(79)
where
±
−
−1
n+
−σ n−σ Wk−σ = N
X
ij
tij exp[−ik(Ri − Rj )] ×
(80)
∓ †
(ha†i−σ n±
iσ aj−σ i + hai−σ niσ aj−σ i) +
†
†
†
†
±
(hn±
n
i
+
ha
a
a
a
i
−
ha
a
a
a
i)
.
iσ
j−σ
iσ
i−σ
j−σ i−σ
i−σ
jσ
j−σ jσ
Green function (79) is the most general solution of the Hubbard model within the generalized mean
field approximation. Equation (80) is nothing else but the explicit expression for the generalized
mean field. As we see, this mean field is not a functional of the mean particle’s densities. The
solution (79) is more general than the solution ”Hubbard III ” [17] and other two-pole solutions.
Hence it was shown in the papers [17, 35, 36, 37] that the solution ”Hubbard I ” [17] is a particular
case of the solution (79), which corresponds to the additional approximation
X
±
±
−1
−
tij exp[−ik(Ri − Rj )]hn±
(81)
W
(k)
≈
N
n
n+
−σ −σ
j−σ ni−σ i.
ij
Assuming hnj−σ ni−σ i ≈ n2−σ , we obtain the approximation ”Hubbard I ” [17, 35, 36, 37]. Thus, we
have shown that in the cases of systems of strongly correlated particles with a complicated character of quasiparticle spectrums the generalized mean fields can have quite a nontrivial structure,
which is difficult to establish by using any kind of independent considerations. The method of
irreducible Green functions allows one to obtain this structure in the most general form.
One should note that the BCS-Bogoliubov superconductivity theory [1, 3, 5, 10, 15] is formulated
in terms of a trial (approximating) Hamiltonian Hmod , which is a quadratic form with respect
to the second-quantized creation and annihilation operators, including the terms responsible for
anomalous (or non-diagonal) averages. For the single-band Hubbard model the BCS-Bogoliubov
functional of generalized mean fields can be written in the following form [92, 93, 94, 95]
!
†
ha
a
i
−ha
a
i
iσ i−σ
i−σ i−σ
Σcσ = U
.
(82)
−ha†i−σ a†iσ i −ha†iσ aiσ i
The anomalous (or nondiagonal) mean values in this expression fix the vacuum state of the system
exactly in the BCS-Bogoliubov form.
It is worth mentioning that the modern microscopic theory of superconductivity was given a
rigorous mathematical formulation in the classic works of Bogoliubov and co-workers [1, 3, 5, 10,
15] simultaneously with the Bardeen, Cooper, and Sehrieffer (BCS) theory. It was shown that the
equations of superconductivity can be derived from the fundamental electron-ion and electronelectron interactions. The set of equations obtained is known as the Eliashberg equations. They
enable us to investigate the electronic and lattice properties of a metal in both the normal and
superconducting states. Moreover, the Eliashberg equations are appropriate to the description
of strong coupling superconductors, in contrast to the equations, which are valid in the weak
coupling regime and describe the electron subsystem in the superconducting state only.
In paper [92] on the basis of the BCS-Bogoliubov functional of generalized mean fields a system
of equations of superconductivity for the tight-binding electrons in the transition metal described
by the Hubbard Hamiltonian was derived. The electron-phonon interaction was written down for
the ”rigid ion model”. Neglecting the vertex corrections in the self-energy operator the closed
system of equations was obtained.
In paper [93] this approach was extended for the Barisic-Labbe-Friedel model of a transition metal.
The renormalized electron and phonon spectra of the model were derived using the method of
17
irreducible Green functions [17, 35, 36, 37] in a the self-consistent way. For the band and atomic
limits of the Hubbard model the explicit solutions for the electron and phonon energies were
obtained. The energy gap, appearing between electron bands in the strong correlation limit,
persists in that calculations. The Eliashberg-type equations of superconductivity were obtained
also.
The equations of strong coupling superconductivity in disordered transition metal alloys have
been derived in paper [94] by means of irreducible Green functions method and on the basis of the
alloy version of the Barisic-Labbe-Friedel model for electron-ion interaction. The configurational
averaging has been performed by means of the coherent potential approximation. Making some
approximations, the formulas for the superconducting transition temperature TC and the electronphonon coupling constant have been obtained. These depend on the alloy component and total
densities of states, the phonon Green function, and the parameters of the model.
To summarize, various schemes of ”effective mean-field theory” taking into account correlations
were proposed [35, 36, 37, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106]. The main efforts were
directed to the aim to describe suitably the collective behavior of particles in terms of effectivefield distribution which satisfies a self-consistent condition. However, although the self-consistent
field approximation often is a reasonable approximation away from the critical point, it usually
breaks down in its immediate neighborhood.
It is of importance to stress again that from our point of view, in real mean-field theory, the mean
field appearing in the single-site problem should be a scalar or vectorial time-independent
quantity.
6
Symmetry Broken Solutions
The formalism of the previous sections may be extended to incorporate the broken symmetry
solutions [17, 18, 107] of the interacting many particle systems, e.g. the pairing effects present in
superconductors [3, 4, 15], etc. Our purpose in this section is to attract attention to subtle points
which are essential for establishing a connection of the generalized mean field approximation and
the broken symmetry solutions [17, 18, 107].
It is well known that a symmetry can be exact or approximate. Symmetries inherent in the physical laws may be dynamically and spontaneously broken, i.e., they may not manifest themselves
in the actual phenomena. It can be as well broken by certain reasons [108, 109].
Within the literature the term broken symmetry is used both very often and with different meanings. There are two terms, the spontaneous breakdown of symmetries and dynamical symmetry
breaking, which sometimes have been used as opposed but such a distinction is irrelevant. However, the two terms may be used interchangeably. It should be stressed that a symmetry implies
degeneracy. In general there are a multiplets of equivalent states related to each other by congruence operations. They can be distinguished only relative to a weakly coupled external environment
which breaks the symmetry. Local gauged symmetries, however, cannot be broken this way because such an extended environment is not allowed (a superselection rule), so all states are singlets,
i.e., the multiplicities are not observable except possibly for their global part.
It is known that when the Hamiltonian of a system is invariant under a symmetry operation, but
the ground state is not, the symmetry of the system can be spontaneously broken. Symmetry
breaking is termed spontaneous when there is no explicit term in a Lagrangian which manifestly
breaks the symmetry.
Peierls [110, 111] gave a general definition of the notion of the spontaneous breakdown of symmetries which is suited equally well for the physics of particles and condensed matter physics.
According to Peierls [110, 111], the term broken symmetries relates to situations in which symmetries which we expect to hold are valid only approximately or fail completely in certain situations.
18
The intriguing mechanism of spontaneous symmetry breaking is a unifying concept that lie at
the basis of most of the recent developments in theoretical physics, from statistical mechanics to
many-body theory and to elementary particles theory [108, 109]. The existence of degeneracy
in the energy states of a quantal system is related to the invariance or symmetry properties of
the system. By applying the symmetry operation to the ground state, one can transform it to a
different but equivalent ground state. Thus the ground state is degenerate, and in the case of a
continuous symmetry, infinitely degenerate. The real, or relevant, ground state of the system can
only be one of these degenerate states. A system may exhibit the full symmetry of its Lagrangian,
but it is characteristic of infinitely large systems that they also may condense into states of lower
symmetry.
It should be pointed out that Bogoliubov’s method of quasiaverages [3, 4, 15] gives the deep
foundation and clarification of the concept of broken symmetry. It makes the emphasis on the
notion of degeneracy and plays an important role in equilibrium statistical mechanics of manyparticle systems. According to that concept, infinitely small perturbations can trigger macroscopic
responses in the system if they break some symmetry and remove the related degeneracy (or quasidegeneracy) of the equilibrium state. As a result, they can produce macroscopic effects even when
the perturbation magnitude is tend to zero, provided that happens after passing to the thermodynamic limit [45]. This approach has penetrated, directly or indirectly, many areas of the
contemporary physics.
The article [18] examines the Bogoliubov’s notion of quasiaverages, from the original papers [4],
through to modern theoretical concepts and ideas of how to describe both the degeneracy, broken
symmetry and the diversity of the energy scales in the many-particle interacting systems. Current
trends for extending and using Bogoliubov’s ideas to quantum field theory and condensed matter
physics problems were discussed, including microscopic theory of superfluidity and superconductivity, quantum theory of magnetism of complex materials, Bose-Einstein condensation, chirality
of molecules, etc. Practical techniques covered include quasiaverages, Bogoliubov theorem on the
singularity of 1/q 2 , Bogoliubov’s inequality, and its applications to condensed matter physics.
It was demonstrated there that the profound and innovative idea of quasiaverages formulated by
N.N. Bogoliubov, gives the so-called macro-objectivation of the degeneracy in domain of quantum
statistical mechanics, quantum field theory and in the quantum physics in general.
The quasiaverages may be obtained from the ordinary averages by using the cluster property
which was formulated by Bogoliubov [3, 4, 15]. This was first done when deriving the Boltzmann equations from the chain of equations for distribution functions, and in the investigation of
the model Hamiltonian in the theory of superconductivity [3, 4, 15]. To demonstrate this let us
consider averages (quasiaverages) of the form
F (t1 , x1 , . . . tn , xn ) = h. . . Ψ† (t1 , x1 ) . . . Ψ(tj , xj ) . . .i,
(83)
where the number of creation operators Ψ† may be not equal to the number of annihilation operators Ψ. We fix times and split the arguments (t1 , x1 , . . . tn , xn ) into several clusters (. . . , tα , xα , . . .), . . . ,
(. . . , tβ , xβ , . . .). Then it is reasonably to assume that the distances between all clusters |xα − xβ |
tend to infinity. Then, according to the cluster property, the average value (83) tends to the product of averages of collections of operators with the arguments (. . . , tα , xα , . . .), . . . , (. . . , tβ , xβ , . . .)
lim
|xα −xβ |→∞
F (t1 , x1 , . . . tn , xn ) = F (. . . , tα , xα , . . .) . . . F (. . . , tβ , xβ , . . .).
(84)
For equilibrium states with small densities and short-range potential, the validity of this property
can be proved [3, 4, 15]. For the general case, the validity of the cluster property has not yet
been proved. Bogoliubov formulated it not only for ordinary averages but also for quasiaverages,
i.e., for anomalous averages, too. It works for many important models, including the models of
19
superfluidity and superconductivity [3, 4, 15].
In his work The Theory of Superfluidity [112], Bogoliubov gave a microscopic explanation of the
phenomenon of superfluidity [2, 113]. Before his works, there were the phenomenological theories
which were based on an assumption about the form of the spectrum of elementary excitations.
Bogoliubov has started from the general Hamiltonian for Bose systems and assumed that a macroscopic number of particles are found in the ground state with zero momentum, and therefore the
creation and annihilation operators of particles with zero momentum are c-numbers [114]. As a
result a definite approximating Hamiltonian was obtained, consisting from a quadratic form of the
creation and annihilation operators. The usual perturbation theory proved to be inapplicable to it
because of the strong interaction of particles with opposite momenta. Therefore the Hamiltonian
was diagonalized with the help of the canonical transformations (the Bogolyubov u − v transformations). This permitted one to calculate the spectrum of elementary perturbations outside the
framework of perturbation theory. Decomposing the field operators into c-numerical and operator
parts, Bogoliubov in fact introduced into quantum theory the method of spontaneous symmetry
breakdown for systems with degenerate ground state. This method was rediscovered in quantum
field theory a decade later [18].
To illustrate these statements consider Bogoliubov’s theory of a Bose-system with separated condensate, which is given by the Hamiltonian [3, 4, 15]
Z
Z
∆
†
(85)
)Ψ(x)dx − µ Ψ† (x)Ψ(x)dx
Ψ (x)(−
HΛ =
2m
Λ
Λ
Z
1
Ψ† (x1 )Ψ† (x2 )Φ(x1 − x2 )Ψ(x2 )Ψ(x1 )dx1 dx2 .
+
2 Λ2
This Hamiltonian can be written also in the following form
Z
∆
)Ψ(q)dq
Ψ† (q)(−
HΛ = H0 + H1 =
2m
Λ
Z
1
Ψ† (q)Ψ† (q ′ )Φ(q − q ′ )Ψ(q ′ )Ψ(q)dqdq ′ .
+
2 Λ2
(86)
Here, Ψ(q), and Ψ† (q) are the operators of annihilation and creation of bosons. They satisfy the
canonical commutation relations
[Ψ(q), Ψ† (q ′ )] = δ(q − q ′ );
[Ψ(q), Ψ(q ′ )] = [Ψ† (q), Ψ† (q ′ )] = 0.
(87)
The system of bosons is contained in the cube A with the edge L and volume V . It was assumed
that it satisfies periodic boundary conditions and the potential Φ(q) is spherically symmetric and
proportional to the small parameter. It was also assumed that, at temperature zero, a certain
macroscopic number of particles having a nonzero density is situated in the state with momentum
zero.
The operators Ψ(q), and Ψ† (q) are represented in the form
√
√
(88)
Ψ(q) = a0 / V ; Ψ† (q) = a†0 / V ,
where a0 and a†0 are the operators of annihilation and creation of particles with momentum zero.
To explain the phenomenon of superfluidity [4, 112], one should calculate the spectrum of the
Hamiltonian, which is quite a difficult problem. Bogoliubov suggested the idea of approximate
calculation of the spectrum of the ground state and its elementary excitations based on the physical
nature of superfluidity. His idea consists of a few assumptions. The main assumption is that at
20
temperature zero the macroscopic number of particles (with nonzero√density) has
the momentum
† √
zero. Therefore, in the thermodynamic limit [45], the operators a0 / V and a0 / V commute
lim
V →∞
h
√
√ i
1
→0
a0 / V , a†0 / V =
V
(89)
and are c-numbers. Hence, the operator of the number of particles N0 = a†0 a0 is a c-number, too.
D. Ya. Petrina [115] shed an additional light on the problem of an approximation of general
Hamiltonians by Hamiltonians of the theories of superconductivity and superfluidity. In his highly
interesting paper [115] Petrina pointed out that the model Hamiltonian of the theory of superconductivity [3, 15] can be obtained from the general Hamiltonian for Fermi systems if the Kronecker
symbol, which expresses the law of conservation of momentum in the interaction Hamiltonian, is
replaced by two Kronecker symbols so that only particles with opposite momenta interact. The
model Hamiltonian of the theory of superfluidity can be obtained from the general Hamiltonian
for Bose systems if we replace the Kronecker symbol, which expresses the law of conservation of
momentum, by several Kronecker symbols, preserving only the terms that contain at least two
operators with momenta zero in the interaction Hamiltonian. This list of model systems can be
continued [116].
The concept of quasiaverages was introduced by Bogoliubov on the basis of an analysis of manyparticle systems with a degenerate statistical equilibrium state. Such states are inherent to various
physical many-particle systems. Those are liquid helium in the superfluid phase, metals in the
superconducting state, magnets in the ferromagnetically ordered state, liquid crystal states, the
states of superfluid nuclear matter, etc.
In many-body interacting systems, the symmetry is important in classifying different phases and
in understanding the phase transitions between them. According to Bogoliubov’s ideas [3, 15,
4, 112, 107] in each condensed phase, in addition to the normal process, there is an anomalous
process (or processes) which can take place because of the long-range internal field, with a corresponding propagator. Additionally, the Goldstone theorem [18] states that, in a system in which
a continuous symmetry is broken ( i.e. a system such that the ground state is not invariant under
the operations of a continuous unitary group whose generators commute with the Hamiltonian),
there exists a collective mode with frequency vanishing, as the momentum goes to zero. For manyparticle systems on a lattice, this statement needs a proper adaptation. In the above form, the
Goldstone theorem is true only if the condensed and normal phases have the same translational
properties. When translational symmetry is also broken, the Goldstone mode appears at a zero
frequency but at nonzero momentum, e.g., a crystal and a helical spin-density-wave ordering (see
for discussion Refs. [17, 117]).
The antiferromagnetic state is characterized by a spatially changing component of magnetization
which varies in such a way that the net magnetization of the system is zero. The concept of antiferromagnetism of localized spins which is based on the Heisenberg model and the two-sublattice
Neel ground state is relatively well founded contrary to the antiferromagnetism of delocalized
or itinerant electrons. The itinerant-electron picture is the alternative conceptual picture for
magnetism [118]. In the antiferromagnetic many-body problem there is an additional ”symmetry broken” aspect [17, 117]. For an antiferromagnet, contrary to ferromagnet, the one-electron
Hartree-Fock potential can violate the translational crystal symmetry. The period of the antiferromagnetic spin structure L is greater than the lattice constant a. The Hartree-Fock is the
simplest approximation but neglects the important dynamical part. To include the dynamics one
should take into consideration the correlation effects.
The anomalous propagators for an interacting many-fermion system corresponding to the ferromagnetic (FM) , antiferromagnetic (AFM), and superconducting (SC) long-range ordering are
21
given by
F M : Gf m ∼ hhakσ ; a†k−σ ii,
(90)
AF M : Gaf m ∼ hhak+Qσ ; a†k+Q′ σ′ ii,
SC : Gsc ∼ hhakσ ; a−k−σ ii.
In the spin-density-wave case, a particle picks up a momentum Q − Q′ from scattering against
the periodic structure of the spiral (nonuniform) internal field, and has its spin changed from σ to
σ ′ by the spin-aligning character of the internal field. The Long-Range-Order (LRO) parameters
are:
X †
F M : m = 1/N
hakσ ak−σ i,
(91)
kσ
X †
AF M : MQ =
hakσ ak+Q−σ i,
kσ
SC : ∆ =
X †
ha−k↓ a†k↑ i.
k
It is of importance to note that the long-range order parameters are functions of the internal field,
which is itself a function of the order parameter. There is a more mathematical way of formulating
this assertion. As it was stressed earlier [18], the notion symmetry breaking means that the state
fails to have the symmetry that the Hamiltonian has.
In terms of the theory of quasiaverages, a true breaking of symmetry can arise only if there are infinitesimal ”source fields”. Indeed, for the rotationally and translationally invariant Hamiltonian,
suitable source terms should be added:
X †
F M : εµB Hx
akσ ak−σ ,
(92)
AF M : εµB H
X
kσ
a†kσ ak+Q−σ ,
kQ
X †
SC : εv
(a−k↓ a†k↑ + ak↑ a−k↓ ),
k
where ε → 0 is to be taken at the end of calculations.
For example, broken symmetry solutions of the spin-density-wave type imply that the vector Q is
a measure of the inhomogeneity or breaking of translational symmetry.
In this context the Hubbard model is a very interesting tool for analyzing the broken symmetry
concept [35, 36, 37]. It is possible to show that antiferromagnetic state and more complicated states
(e.g. ferrimagnetic) can be made eigenfunctions of the self-consistent field equations within an
”extended” (or generalized) mean-field approach, assuming that the anomalous averages ha†iσ ai−σ i
determine the behaviour of the system on the same footing as the ”normal” density of quasiparticles ha†iσ aiσ i. It is clear, however, that these “spin-flip” terms break the rotational symmetry of
the Hubbard Hamiltonian. For the single-band Hubbard Hamiltonian, the averages ha†i−σ ai,σ i = 0
because of the rotational symmetry of the Hubbard model. The inclusion of anomalous averages
leads to the following approximation
ni−σ aiσ ≈ hni−σ iaiσ − ha†i−σ aiσ iai−σ .
22
(93)
Thus, in addition to the standard Hartree-Fock term, the new so-called “spin-flip” terms are
retained [117]. This example clearly shows that the structure of mean field follows from the
specificity of the problem and should be defined in a proper way. So, one needs a properly
defined effective Hamiltonian Heff . In paper [117] we thoroughly analyzed the proper definition
of the irreducible Green functions which includes the “spin-flip” terms for the case of itinerant
antiferromagnetism of correlated lattice fermions. For the single-orbital Hubbard model [35, 36,
37, 117], the definition of the ”irreducible” part should be modified in the following way:
(ir)
hhak+pσ a†p+q−σ aq−σ |a†kσ iiω = hhak+pσ a†p+q−σ aq−σ |a†kσ iiω −
δp,0 hnq−σ iGkσ − hak+pσ a†p+q−σ ihhaq−σ |a†kσ iiω .
(94)
From this definition it follows that this way of introduction of the irreducible Green functions
broadens the initial algebra of operators and the initial set of the Green functions. This means
that the “actual” algebra of operators must include the spin-flip terms from the beginning, namely:
(aiσ , a†iσ , niσ , a†iσ ai−σ ). The corresponding initial Green function will be of the form
!
hhaiσ |a†jσ ii
hhaiσ |a†j−σ ii
.
(95)
hhai−σ |a†jσ ii hhai−σ |a†j−σ ii
With this definition, one introduces the so-called anomalous (off-diagonal) Green functions which
fix the relevant vacuum and select the proper symmetry broken solutions. In fact, this approximation was investigated earlier by Kishore and Joshi [119]. They clearly pointed out that they
assumed a system to be magnetized in the x direction instead of the conventional z axis.
The problem of finding the superconducting, ferromagnetic and antiferromagnetic ”symmetry broken” solutions of the correlated lattice fermion models within irreducible Green functions method
was investigated in Refs. [17, 35, 36, 37, 117]. A unified scheme for the construction of generalized mean fields (elastic scattering corrections) and self-energy (inelastic scattering) in terms
of the Dyson equation was generalized in order to include the ”source fields”. The ”symmetry
broken” dynamic solutions of the Hubbard model which correspond to various types of itinerant
antiferromagnetism were discussed as well [17, 35, 36, 37, 117]. This approach complements previous studies of microscopic theory of the Heisenberg antiferromagnet [120] and clarifies the concepts
of Neel sublattices for localized and itinerant antiferromagnetism and ”spin-aligning fields” of correlated lattice fermions.
We shall see shortly that in order to discuss the mean field theory (and generalized mean fields) on
the firm ground the Bogoliubov inequality provides the formal basis and effective general approach.
7
The Mathematical Tools
Before entering fully into our subject, we must recall some basic statements. This will be necessary
for the following discussion.
The number of inequalities in mathematical physics is extraordinary plentiful and the literature on
inequalities is vast [121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133]. The physicists
are interested mostly in intuitive, physical forms of inequalities rather than in their most general
versions. Often it is easier to catch the beauty and importance of original versions rather than
decoding their later, abstract forms.
Many inequalities are of a great use and directly related with the notion of entropy, especially
with quantum entropy [124, 134]. The von Neuman entropy of ρ ∈ Sn , S(ρ), is defined by
S(ρ) = −Tr(ρ log ρ).
23
(96)
The operator ρ log ρ is defined using the spectral theorem [124]. Here Sn denotes the set of density
matrices ρ on Cn .
In fact, S(ρ) depends on ρ only through its eigenvalues.
S(ρ) = −
n
X
λj log λj .
(97)
j=1
Otherwise put, the von Neumann entropy is unitarily invariant; i.e.,
S(U ρU ∗ ) = S(ρ).
(98)
− S(ρ) = − log(n).
(99)
The convexity condition leads to [124]
This equality valid if and only if each λj = 1/n Thus, one may arrive at [124]
0 ≤ S(ρ) ≤ log n
(100)
for all ρ ∈ Sn , and there is equality on the left if ρ is a pure state, and there is equality on the
right if ρ = (1/n)I. Actually, S(ρ) is not only a strictly concave function of the eigenvalues of ρ,
it is strictly concave function of ρ itself.
The notions of convexity and concavity of trace functions [124] are of great importance in mathematical physics [135, 136]. Inequalities for quantum mechanical entropies and related concave
trace functions play a fundamental role in quantum information theory as well [124, 134].
A function f is convex in a given interval if its second derivative is always of the same sign in that
interval. The sign of the second derivative can be chosen as positive (by multiplying by (−1) if
necessary). Indeed, the notion of convexity means that if d2 f /dx2 > 0 in a given interval, xj are
a set of points in that interval, pj are a set of weights such that pj ≥ 0, which have the property
P
j pj = 1, then
X
X
(101)
p j xj .
pj f (xj ) ≥ f
j
j
P
The equality will be valid only if xj = hxi = j pj xj . In other words, a real-valued function
f (x) defined on an interval is called convex (or convex downward or concave upward) if the line
segment between any two points on the graph of the function lies above the graph, in a Euclidean
space (or more generally a vector space) of at least two dimensions. Equivalently, a function is
convex if its epigraph (the set of points on or above the graph of the function) is a convex set.
A real-valued function f on an interval (or, more generally, a convex set in vector space) is said
to be concave if, for any x1 and x2 in the interval and for any α in [0, 1],
(102)
f (1 − α)x1 + (α)x2 ≥ (1 − α)f (x1 ) + (α)f (x2 ).
A function f (x) is concave over a convex set if and only if the function −f (x) is a convex function
over the set.
As an example we mentioned above briefly a reason this concavity matters, pointing to the inequality (100) that was deduced from the concavity of the entropy as a function of the eigenvalues
of ρ.
It is of importance to stress that in quantum statistical mechanics, equilibrium states are determined by maximum entropy principles [124], and the fact that
= log n,
(103)
sup S(ρ)
ρ∈Sn
24
reflects the famous Boltzmann formulae
S = kB log W.
(104)
It follows from Boltzmann definition that the entropy is larger if ρ is smeared out, where ρ is the
probability density on phase space. The microscopic definition of entropy given by Boltzmann
does not, by itself, explain the second law of thermodynamics. even in classical physics. The task
to formulate these questions in a quantum framework was addressed by Oskar Klein in his seminal
paper [137] of 1931. He found a fundamentally new way for information to be lost hence entropy
to increase, special to quantum mechanics. This result was called Klein lemma [136, 138, 137].
M. B. Ruskai [138] has reviewed many fundamental properties of the quantum entropy [134] including one important class of inequalities relates the entropy of subsystems to that of a composite
system. That article presented self-contained proofs of the strong subadditivity inequality for von
Neumann quantum entropy, S(ρ), and some related inequalities for the quantum relative entropy,
most notably its convexity and its monotonicity under stochastic maps. The approach to subadditivity and relative entropy presented was used to obtain conditions for equality in properties of
relative entropy, including its joint convexity and monotonicity. In addition, the Klein inequality
was presented there in detail.
Indeed, the fact that the relative entropy is positive [138], i.e., H(ρ1 , ρ2 ) ≥ 0 when Trρ1 = Trρ2 ,
is an immediate consequence of the following fundamental convexity result due to Klein [137, 139,
140]. The corresponding theorem [138] states that for A, B > 0
Tr A(log A − log B) ≥ Tr(A − B),
(105)
with equality if and only if (A = B).
In more general form [124] the Klein inequality may be formulated in the following way.
For all A, B ∈ Hn , and all differentiable convex functions f : R → R, or for all A, B ∈ H†n and all
differentiable convex functions f : (0, ∞) → R
Tr f (A) − f (B) − (A − B)f ′ (B) ≥ 0.
(106)
In either case, if f is strictly convex, there is equality if and only if A = B.
A few more words about Oskar Klein and his inequality will not be out of place here. Oskar
Klein (1894 - 1977) was famous Swedish theoretical physicist [] who worked on a wide variety
of subjects [141]. For example, the Klein-Gordon equation was the first relativistic wave equation. Oskar Klein was also a collaborator of Niels Bohr in Copenhagen. It is interesting to note
that Oskar Klein defended his thesis and was awarded his doctoral degree in 1921 for his work
in physical chemistry about strong electrolytes. In 1931 Oskar Klein [137, 139, 140, 141], using
his experience in both quantum and statistical mechanics, succeeded in solving the problem of
whether the quantum statistics on molecular level can explain how the entropy increases with
time in accordance with the second law of thermodynamics. The problem in classical statistical
mechanics had been already noticed by Gibbs earlier. Klein proof [137, 139, 140], that used the
statement that only the diagonal elements in the density matrix for the phase space of the particles
are relevant for the entropy, has led him to the Klein lemma. With Klein lemma, the entropy can
increase according to the formula of Boltzmann microscopic definition, where it is described with
the number of states in the phase space. A useful and informative discussion of the Klein paper
and Klein lemma was carried out in the book of R. Jancel [142].
According to M. B. Ruskai [138], the closely related Peierls-Bogoliubov inequality is sometimes
used instead of Klein inequality. Golden-Thompson and Peierls-Bogoliubov inequalities were extended to von Neumann algebras, which have traces, by Ruskai [128] (see also Ref. [143]). H.
25
Araki [129] extended them to a general von Neumann algebra. This kind of investigations is
particularly valuable since the Bogoliubov inequality is remarkable because it have significant
applications in statistical quantum mechanics [3, 10, 11, 12, 144, 145]. It provides insight into a
number of other interesting questions as well.
It will be of use to write down the mathematical formulation of Peierls-Bogoliubov inequalities
which was provided by Carlen [124].
Let us consider A ∈ Hn , and let f be any convex function on R. Let {u1 , . . . , un } be any orthonormal base of Cn . Then
n X
huj , Auj i ≤ Tr[f (A)].
(107)
j=1
There is equality if each uj is an eigenvector of A, and if f is strictly convex, only in this case.
Now consider the formulation of thegeneralizedPeierls-Bogoliubov inequality [124]. For every
natural number n, the map A 7→ log Tr[exp(A)] is convex on Hn . As a consequence one may
deduce [124] that
Tr[exp(A + B)] Tr[B exp(A)]
log
≥
(108)
Tr[exp(A)]
Tr[exp(A)]
Frequently this relation, which has many uses, is referred to as the Peierls-Bogoliubov inequality.
It is worth noting that according to tradition the term Gibbs-Bogoliubov inequality [8] is used for a
classical statistical mechanical systems and term Peierls-Bogoliubov inequality [124] for quantum
statistical mechanical systems.
At the very least, it must have been meant to indicate that Peierls inequality does not have a
classical analog, whereas Bogoliubov inequality has.
8
Variational Principle of Bogoliubov
It is known that there are several variational principles which provide upper bounds for the
Helmholtz free-energy function. With these instruments, it is possible to construct various approximations to the statistical thermodynamic behavior of systems. For any variational formulation,
its effectiveness as a minimal principle will be enhanced considerably if there is a workable tool
for determining lower bounds to the Helmholtz free energy function. Bogoliubov inequality for
the free-energy functional is an inequality that gives rise to a variational principle of statistical
mechanics. It is used [1, 2, 3, 4, 5] to obtain the exact thermodynamic limit [18] solutions of
model problems in statistical physics, in studies using the method of molecular fields, in proving
the existence of the thermodynamic limit [45], and also in order to obtain physically important
estimates for the free energies of various many-particle interacting systems.
A clear formulation of the variational principle of Bogoliubov and Bogoliubov inequality for the
free-energy functional was carried out by S. V. Tyablikov [5]. We shall follow close to that formulation. Tyablikov [5] used the theorems relating to the minimum values of the free energy. As a
result, it was possible to formulate a principle which then was used to deduce the molecular field
equations.
Principle of the free energy minimum is based on the following arguments. Let us consider an
arbitrary complete system of orthonormalized functions {ϕn }, which are not the eigenfunctions
of the Hamiltonian H of a system. Then it is possible to write down the inequality
F (H) ≤ Fmod (H).
(109)
Here F (H) is the intrinsic free energy of the system
F (H) = −θ ln Z,
Z=
X
ν
26
exp(−Eν /θ),
(110)
θ = kB T , Eν are eigenfunctions of the Hamiltonian H, Fmod (H) is the model free energy, which
gives approximately the upper limit of the intrinsic free energy:
X
(111)
Fmod (H) = −θ ln Zmod , Zmod =
exp(−Hnn /θ), Hnn = ϕ∗n , Hϕn .
n
The inequality (109) may be written also in the following way:
Z ≥ Zmod .
(112)
The relationships represented by the equality sign in Eqs. (109) and (112) applies if ϕn are eigenfunctions of the Hamiltonian of the system. It should be noted that for finite values of the number
(N )
of partial sums Z (N ) , the quantity Fmod does not reach its maximum for any system of functions
ϕ1 , . . . , ϕN . In fact, the inequality will be satisfied really [5, 45] in the limit N → ∞.
Using these results, it is possible to formulate a variational principle for the approximate determination of the free energy of a system [5]. To proceed, let us suppose that the functions {ϕn }
depend on some arbitrary parameter λ. It was established above that
X
F (H) ≤ Fmod (H) = −θ ln
exp(−Hnn (λ)/θ).
(113)
n
It is clear that the best approximation for the upper limit of the free energy F is obtained by
selecting the values of the parameter λ in accordance with the condition for the minimum of the
model free energy Fmod . Indeed, let the Hamiltonian of the system, H, be written in the form
H = H0 (λ) + ∆H(λ) ≡ H0 (λ) + H − H0 (λ) ,
(114)
where H0 (λ) is some operator depending on the parameter λ. The concrete form of the operator
H0 (λ) should be selected on the basis of convenience in calculations. We shall use notation En0
and ϕn for the eigenvalues and the eigenfunctions of the operator H0 . To denote the diagonal
matrix elements of the operator ∆H in terms of the functions ϕn we shall use the notation ∆Hnn .
For a generality, we shall assume that ϕn are not the eigenfunctions of the total Hamiltonian H.
Clearly, En0 and ∆Hnn are also some functions of the parameter λ. In this sense, the system of
functions {ϕn } plays a role of a trial system of functions. Then we may write that
(115)
Hnn = En0 + ∆Hnn ≡ En0 + Hnn − En0 .
As a consequence, the free energy will satisfy the inequality
1
X
exp − En0 + ∆Hnn .
F (H) ≤ −θ ln
θ
n
(116)
Now let us suppose that the operator ∆H can be considered as a small perturbation compared
with the operator H. We obtain then [5], to within quantities of the first order of smallness with
respect to ∆H
Tr ∆H exp(−H0 /θ)
.
(117)
F (H) ≤ F (H0 ) +
Tr exp(−H0 /θ)
Note that in this case, the best approximation to the upper limit of the free energy is obtained
by selecting the value of the parameter λ from the condition for the minimum of the right-hand
side of Eq.(117). The formulation of the variational principle of Eq.(117) is more restricted than
27
the initial formulation of Eq.(109).
The variational principle in the form of Eq.(117) can be strengthened, following to the Bogoliubov
suggestion [5], by removing the limitation of the smallness of the operator ∆H. As a result we
obtain
F (H) ≤ Fmod (H).
(118)
Here
Tr ∆H exp(−H0 /θ)
,
Fmod (H) = F (H0 ) +
Tr exp(−H0 /θ)
F (H0 ) = −θ ln Tr exp(−H0 /θ).
(119)
(120)
Hence one may write down also that for a system with the Hamiltonian
H = H0 + ∆H
(121)
the free energy has a certain upper bound. Bogoliubov inequality states that:
F ≤ F0 + hH − H0 i0 ,
(122)
F ≤ F0 hHi0 − T S0 ,
(123)
or
where S0 is the entropy and where the average is taken over the equilibrium ensemble of the
reference system with Hamiltonian H0 . Usually H0 contains one or more variational parameters
which are chosen such as to minimize the right hand side of Eq.(122). In the special case that
the reference Hamiltonian is that of a non-interacting system and can thus be written as a sum
single-particles Hamiltonians [5]
N
X
hi .
(124)
H0 =
i=1
Then it is possible to improve the upper bound by minimizing the right hand side of the inequality
(122). The minimizing reference system is then the trial approximation to the true system using
non-correlated degrees of freedom, and is known as the mean field approximation.
Starting with the one-particle model Hamiltonian that can be exactly solved in the Bogoliubov
variational method, one may get a self-consistent result such as the molecular field theory in the
ferromagnet and the Hartree-Fock approximation in many-particle problems. Since the variational
method yields a result which is always greater than the correct answer, the mathematical meaning
for improving upon the approximation in the variational method is strictly defined by lowering
the upper bound of the free energy. But these variational methods, the molecular field theory
and the Hartree-Fock approximation, have such a feature that the correlation effects cannot be
taken into account correctly. In general case [5] the Hamiltonian of a system contains interparticle interactions. Thus Bogoliubov variational principle can be considered as the mathematical
foundation of the mean field approximation in the theory of many-particle interacting systems.
Using the Klein inequality (106) it is possible to write down a general form of the Bogoliubov
inequality for the free energy functional. The following inequality is valid for any Hermitian
operators and H1 and H2
N −1 hH1 − H2 iH1 ≤ f (H1 ) − f (H2 ) ≤ N −1 hH1 − H2 iH2 ,
(125)
where
f (H) = −θN −1 ln Tr exp(−H/θ).
28
(126)
This expression has the meaning of the free energy density for a system with Hamiltonian H and
the extensive parameter N may be treated as the number of particles or the volume, depending
on the system.
Derrick [146] established a simple variational bound to the entropy S(E) of a system with energy
E
(127)
S(E) ≥ −kB ln TrU 2
for all Hermitian matrices U (with no negative eigenvalues) for which TrU = 1 and Tr(HU ) = E,
where H is the Hamiltonian.
This principle has the advantage that U 2 is in general much easier to evaluate than U ln U which
appears in the conventional bound given by von Neumann
(128)
S(E) ≥ −kB TrU ln U .
There are numerous methods for proving of the Bogoliubov inequality [5, 10, 11, 12, 39, 48, 147,
148]. A. Oguchi [149] proposed an approach for determination of an upper bound and a lower
bound of the Helmholtz free energy in the statistical physics. He used as a basic tool the Klein
lemma [124, 138, 137]. He obtained a new approximate expression of the free energy. This approximate value of the free energy was conjectured to be greater than the lower bound and less than
the upper bound. An approach which can be extended to improve the approximation was formulated. The upper bound and the lower bound of the approximate free energy converge to the true
free energy as the successive approximation proceeds. The method was first applied to the Ising
ferromagnet and then applied to the Heisenberg ferromagnet. In the simplest approximation the
results agrees with the Bethe-Peierls approximation for the Ising model and the constant coupling
approximation for the Heisenberg model. In his subsequent paper, A. Oguchi [150] formulated a
new variational method for the free energy in statistical physics. According to his calculations, the
value of the free energy was obtained by using this new variational method was lower than that
of the Bogolyubov variational method. Author concluded that the new variational free energy
satisfies the thermodynamic stability criterion.
However, J. Stolze [151] by careful examination of the papers [149, 150], has shown that the calculation in Ref. [150] contains a mistake which invalidates the result. He also pointed out several
errors seriously affecting the results of an earlier paper [149]. Oguchi assumed that the Hamiltonian H0 contains a variational parameter ”a” distributed according to a probability density P (a).
Stolze derived a corrected inequality which clearly states that the new upper bound on the free
energy suggested by Oguchi [149, 150] cannot be better (i.e., lower) than the Peierls-Bogoliubov
bound, no matter how cleverly P (a) was chosen. This shows clearly that no advantage over the
Peierls-Bogolyubov bound was obtained.
The standard proof was given in Callen second edition book on thermodynamics [48] for the
case when the unperturbed Hamiltonian and the perturbation commute. Another proof (for the
general case), was carried out in Feynman book on statistical mechanics [147]. Feynman used
Baker-Campbell-Hausdorff expansion [123, 124] for the exponential of a sum of two non commuting operators. Prato and Barraco [148] presented a proof of the Bogoliubov inequality that does
not require of the Baker-Campbell-Hausdorff expansion.
Several variational approaches for the free energy have been proposed [152, 153] as attempts to improve results obtained through the well established Bogoliubov principle. This principle requires
the use of a trial Hamiltonian depending on one or more variational parameters. The only way to
improve the Bogoliubov principle by itself is to choose a more complete trial Hamiltonian, closing
it to the exact one, but in almost all cases the possibilities are soon exhausted. The usual mean
field approximation may be obtained using the above principle utilizing a sum of single spins in
an effective field (the variational parameter) as the trial Hamiltonian.
29
Lowdin [154] and Lowdin and Nagel [155] studied a generalization of the Gibbs-Bogoliubov inequality F ≤ F0 + hH − H0 i0 for the free energy F which leads to a variation principle for this
quantity that may be of importance in certain computational applications to quantum systems.
This approach is coupled with a study of the perturbation expansion of the free energy for a
canonical ensemble with H = H0 + λV in the general case when H0 and V do not commute. A
simple proof was given for the thermodynamic inequality F − F0 − hH − H0 i0 < 0 in the case when
the two Hamiltonian H0 and V do not commute. The second- and high-order derivatives of the
free energy with respect to the perturbation parameter λ were calculated. From the second-order
term was finally obtained a second-order correction to the previous variational minimum for the
free energy.
A. Decoster [39] established a sequence of inequalities which generalize the the Gibbs-Bogoliubov
inequality in classical statistical mechanics and the Peierls and Bogoliubov inequalities in quantum
mechanics; they can be presented as rearrangements of perturbation expansions, which provide
exact bounds which are used in variational calculations.
W. Kramarczyk [156] argued that the Bogoliubov variational principle may be shown to be equivalent to the minimizing of the information gained while replacing the exact state by an approximate
one. Consequently, the quasiparticles introduced in the thermal Hartree-Fock approximation may
be redefined information-theoretically.
9
Applications of the Bogoliubov Variational Principle
Bogoliubov variational principle has been successfully applied to a wide range of problems in
the theory of many-particle systems. The first application of Bogoliubov inequality to concrete
many-particle problem was carried out in the work by I. A. Kvasnikov [157] on the application of
a variational principle to the Ising model of ferromagnetism.
Ising model [17, 158] is defined by the following Hamiltonian H (i.e. energy functional of variables;
in this case the ”spins” Si = ±1 on the N sites of a regular lattice in a space of dimension d)
N
N
X
1 X
Si
Jij Si Sj − µB H
H=−
2
(129)
i=1
i<j=1
Here, Jij play a role of ”exchange constants”, H is a (normalized) magnetic field, involving an
P
interpretation of the model to describe magnetic ordering in solids (M = N
i=1 Si is ”magnetization”; µB HM is the Zeeman energy, i.e. is the energy gained due to application of the field).
The main task is to calculate statistical sum Z
X
Z=
exp − H/θ .
(130)
Si
Kvasnikov [157] considered the approximation of nearest neighbours, i.e. Jij = J for nearest
neighbours < i, j >.
According to Bogoliubov variational principle one can write
F ≤ F0 + hH − H0 i0 .
(131)
The upper bound for the free energy Fsup is given by
Fsup = −θ ln Zinf ,
30
(132)
where
Z0 = Tr exp − H0 /θ ;
S = (Z0 )−1 Tr ∆H exp −[H0 /θ] .
Zinf = Z0 exp −(S/θ),
(133)
(134)
The parameters of partition, which were introduced into H0 and ∆H, and, hence, into Z0 and S,
should be determined from the condition of the minimum of Fsup . Thus we obtain
N
−
X
H0
Si ,
= µB (B − χ)
θ
(135)
i=1
N
N
i=1
i6=j
X
1X
∆H
Si +
−
= µB χ
Kij Si Sj ,
θ
2
(136)
where B = H/θ, K = J/θ, χ is some parameter. Then, according to relation (132), one finds
−1
Zinf (χ)
=
(137)
N
o
n
1 X
Kij tanh2 µB (B − χ) .
2 coth µB (B − χ) exp µB χ tanh µB (B − χ) +
2N
i6=j
Parameter χ is determined by the equation
N
µB χ,
tanh µB (χ − B) = P
Kij
1−
N
1 X
Kij + P
(µB χ)2 > 0.
N
Kij
(138)
When the approximation of nearest neighbours is considered in the above equation the following
substitution should be done
N
X
Kij = zKN,
(139)
i6=j
where z is the number of nearest neighbours.
Hence Fsup is an approximate expression for the free energy and Zinf is the approximate statistical
sum of the model. It will be of instruction to compare these values with those which were calculated
by other methods. To proceed, let us consider the regions of low and high temperatures. In the
first case we will have that θ ≪ zJ. The low-temperature approximation is expressed as a series
expansion in terms of the small parameter exp(−K). The iterative solution of the Eq.(138) will
have the form
µB χ = −zK 1 − 2 exp 2(−Kz − µB B) − 8zK exp 4(−Kz − µB B) + . . . .
(140)
It is sufficient to confine oneself to the values of the order exp(−2Kz). The result is
−1
Zinf
= exp(Kz/2 + µB B) 1 + exp 2(−Kz − B) + . . . .
(141)
This result is in accordance with the other low-temperature expansions [5, 158]
Nz
exp 4[−K(z − 1) − µB B]
Z = exp 2(Kz/2 + µB B)N 1 + N exp 2(−Kz − µB B) +
2
N (N − 1) N z
−
} exp 4(−Kz − µB B) + . . . .
+{
2
2
31
(142)
In the case of the high temperature, when θ ≥ zJ, the approximate solution of the Eq.(138) will
have the form
tanh µB B
µB χ ≃ −zK
.
(143)
1 − [zK/ cosh2 µB B]
Then after some transformations one can arrive to the expression (up to the terms K 3 ):
1
Zinf ≃ [2 cosh µB B]N 1 + Kz tanh2 µB B +
2
1 2
2
K zN [4z tanh µB B + (N z + 4z) tanh4 µB B] .
8
(144)
This expression is also in accordance with the known high-temperature expansions [5, 158] for
N ≫ z.
Let us consider now the expression for magnetization [159] (the averaged magnetic moment)
M=
1 ∂ ln Zinf
.
N ∂B
(145)
Using Eq.(138), we obtain
H
m
m
,
= tanh µB
+n
µB p
θ
θ
(146)
where p is the number of lattice sites per unit volume, m = M p is the magnetization per unit volume. This result coincides with the result of the phenomenological theory [5]. The corresponding
basic values of the P. Weiss theory, the Curie point θ0 and Weiss parameter w have the form
N
1 X
θ0 =
Jij ;
N
i6=j
P
N −1 Jij
θ0
= 2 .
w=
2
µB p
µB p
(147)
Hence, with the help of the Bogoliubov variational scheme it was possible to calculate the reasonable approximate expression for the statistical sum of the Ising model and describe the macroscopic
properties of ferromagnetic systems in the wide interval of temperatures. It is thus seen that one
may derive directly a consistent mean-field-type theory from a variational principle.
Clearly Bogoliubov variational principle had a deep impact on the field of statistical mechanics of
classical and quantum many-particle systems by making possible the analysis of complex statistical
systems. Many interesting developments can be viewed from the point of a central theme, namely
the Bogoliubov inequality, in particular in quantum theory of magnetism [5, 159, 160, 161, 162, 163]
and interacting many-body systems [164, 165, 166, 167, 168, 169, 170, 171, 91].
Radcliffe [160] carried out a systematic investigation of the approximate free energies and Curie
temperatures that can be obtained by using trial density matrices (which describe various possible
decompositions of the ferromagnet into clusters) in a variational calculation of the free energy.
Single-spin clusters lead to the molecular field model (as is well known) and two-spin clusters yield
the Oguchi pair model [71]. The relation of the constant-coupling method to these approximations
was clarified. A rigorous calculations using three-spin clusters were carried out.
Rudoi [161] investigated the link between Bogoliubov statistical variational principle for free energy, the method of partial diagram summation of the perturbation theory, and the LuttingerWard theorem. On the basis of Matsubara Green function method he solved the nonlinear integral
Dyson equation by approximating the effective potential. As a result, a new implicit equation of
magnetic state was obtained for the Ising model.
Soldatov [172] generalized the Peierls-Bogoliubov inequality. A set of inequalities was derived instead, so that every subsequent inequality in this set approximates the quantity in question with
better precision than the preceding one. These inequalities lead to a sequence of improving upper
32
bounds to the free energy of a quantum system if this system allows representation in terms of
coherent states. It follows from the results obtained that nearly any upper bound to the ground
state energy obtained by the conventional variational principle can be improved by means of the
proposed method.
F. Abubrig [173] studied the mixed spin-3/2 and spin-2 Ising ferrimagnetic system with different
single-ion anisotropies in the absence of an external magnetic field within the mean-field theory based on Bogoliubov inequality for the Gibbs free energy. Second-order critical lines were
obtained in the temperature-anisotropy plane. Tricritical line separating second-order and firstorder lines was found. Finally, the existence and dependence of a compensation points on single-ion
anisotropies was also investigated for the system. It was shown that this mixed-spin model exhibits
one, two or three compensation temperature depending on the values of the anisotropies.
10
The Variational Schemes and Bounds on Free Energy
During last few decades, numerous variational schemes have become an increasingly popular workable tool in quantum-mechanical many-particle theory [5, 10, 11, 14]. Bounds of free energy and
canonical ensemble averages were of considerable interest as well. For many complex systems,
such as Ising and Heisenberg ferromagnets or composite materials, methods of obtaining bounds
are the practical useful tools which are both tractable and informative. A few illustrative topics
will be of instruction to discuss in this context.
MacDonald and Richardson [174] used the density matrix of von Neumann to formulate an exact variational principle for quantum statistics which embodies the principle of maximization of
entropy. In terms of the formalism of second quantization, authors wrote this variational principle for fermions or bosons and then derived from it an approximate variational procedure which
yields the particle states of a system of interacting bosons or fermions as well as the distribution
of particles in these states. These equations, in authors’s opinion, yield the generalization of the
Hartree-Fock equations for nonzero temperature and the corresponding extension to bosons.
W. Schattke [175] found an upper bound for the free energy for superconducting system in magnetic field. Starting from the BCS theory, the free energy was obtained by a combination of a
variational method and perturbation theory. The variational equations obtained were non-local.
The parameters of the perturbation calculation were the vector potential and the spatial variations of the order parameter, which have to be small. Boundary conditions were set for the case
of diffuse reflection and pair-breaking at the surface. As an example, the superconducting plate
was discussed.
Krinsky et. al. used [176] the variational principle to derive a new approximation to a ferromagnet in a magnetic field, below its critical temperature. They considered [176] a ferromagnet in an
external magnetic field with T ≤ TC . Using a variational approximation based on the zero-field
solution, the authors obtained an upper bound on the free energy, an approximate equation of
state, and a lower bound on the magnetization, all having the correct critical indices. Explicit
numerical calculations have been carried out for the two-dimensional Ising model, and it was
found that the results obtained provide a good approximation to the results of series expansions
throughout the region T ≤ TC .
The Gibbs-Bogoliubov inequality [8] was used [177] to develop a first-order perturbation theory
that provides an upper bound on the Helmholtz free energy per unit volume of a classical statistical mechanical system in terms of the free energy and pair distribution function. Charged systems
as well as a system of Lennard-Jones particles were discussed and detailed numerical estimates of
the bounds were presented.
S. Okubo and A. Isihara [178] derived important general inequalities for the derivatives of the partition function of a quantum system with respect to the parameters included in the Hamiltonian.
33
Applications of the inequalities were used to discuss relations for critical initial exponents, kinetic
energy, susceptibility, electrical conductivity and so on. Existence of an inconsistency analogous
to the Schwinger-term difficulty in the quantum field theory was pointed out.
In their second paper [179], S. Okubo and A. Isihara analyzed from a general point of view an
inequality for convex functions in quantum-statistical mechanics. From an inequality for a convex
function of two hermitian operators, the Peierls and Gibbs operators, coarse graining and other
important inequalities were derived in a unified way. Various different forms of the basic inequality
were given. They are found useful in discussing the entropy and other physical problems. Special
accounts were given of functions such as exp(x) and x log x.
A variational method for many-body systems using a separation into a difference of Hamiltonians
was presented by M. Hader and F. G. Mertens [180]. A particular ansatz for the wave function
was considered which leads to an upper bound for the exact ground-state energy. This allowed a
variation with respect to a separation parameter. The method was tested for a one-dimensional
lattice with Morse interactions where the Toda subsystems can be solved by the Bethe ansatz. In
two limiting cases results obtained were exact, otherwise they were in agreement with the quantum transfer integral method.
Yeh [181, 182, 183] proposed a derivation of a lower bound on the free energy; in addition he analyzed the bounds of the average value of a function [183]. He also established [181] a weaker form
of Griffiths theorem for the ferromagnetic Heisenberg model. It was described as follows [182]:
Free energy in the canonical ensemble was taken as
X
F = −β −1 ln
hn| exp(−Hβ)|ni,
(148)
n
where |ni is any complete set of orthonormal states. Bounds of F can be obtained from bounds
of hn| exp(−Hβ)|ni. As we seen, a very simple upper bound of F was given by Peierls [63]; one
way to prove his theorem is by showing that
hψ| exp(−Hβ)|ψi ≥ exp(−βhψ|H|ψi).
(149)
Yeh [182] derived a rather simple lower bound of F by similar method. He considered a a Hamiltonian with a ground-state energy E0 = 0. He considered a real function f (E) = exp(−Eβ),
β > 0. It was shown that for any normalized state |ψi a weaker but simpler upper bound for f
may be written as
exp −βhψ|H|ψip ≥ hψ| exp −Hβ |ψi ≥ exp −βhψ|H|ψi ,
(150)
where
p = exp
(−βhψ|H 2 |ψi) hψ|H|ψi
(151)
Identifying β = (kB T )−1 and H as Hamiltonian, a lower bound of free energy was obtained from
Eq.(150) as
X
F ≥ −β −1 ln
exp −βhψ|H|ψip ,
(152)
ψ
where |ψi is any complete orthonormal set of states. This is a general formula for a lower bound
on the free energy.
Upper and lower bounds of the canonical ensemble average of any operator A can be written down
in terms of hϕn |H|ϕn i, where the ϕn are eigenstates of A. Furthermore, bounds of thermodynamic
derivatives can be obtained by noting that the bounds of
∂ i f¯
∂β i
34
(153)
can be also derived [182] in similar manner. Here
X
f¯ = hψ| exp(−Hβ)|ψi =
ρn exp(−En β);
n
X
ρn = 1.
(154)
n
From Eq.(150), it is clear that all the bounds are more accurate at higher temperatures. These
bounds have been useful in determining the properties of Heisenberg ferromagnets [181].
K. Symanzik [184] proved, refined, and generalized a lower bound given by Feynman for the quantum mechanical free energy of an oscillator. The method, application of a classical inequality to
path integrals, also gives upper bounds for one-temperature Green functions.
M. Heise and R. J. Jelitto [185] formulated the asymptotically exact variational approach to the
strong coupling Hubbard model. They used a generalization of Bogoliubov variational principle,
in order to develop a molecular field theory of the Hubbard model, which becomes asymptotically
exact in the strong coupling limit. In other words, in their paper authors have started from a generalized form of Bogoliubov variational theorem in order to set up a theory of the Hubbard model,
which yields nontrivial results in the strong coupling regime and becomes asymptotically exact in
the strong coupling limit. For this purpose the Hamiltonian was rotated by a unitary two-particle
transformation, before the variational principle was applied. However, the real form of the generalized mean fields for the the Hubbard model in the strong coupling regime was not determined
in complete form. This task was fulfilled by A. L. Kuzemsky in a series of papers [17, 35, 36, 37].
K. Zeile [186] proposed a generalization of Feynman variational principle for real path integrals
in a systematic way. He obtained an asymptotic series of lower bounds for the partition function.
Author claimed that the method was tested on the anharmonic oscillator and showed excellent
agreement with exact results. However, P. Dorre [187] et al. using the equivalence between Feynman and Bogoliubov variational principle, discussed [187] in the formalism of Hamiltonian quantum mechanics an improved upper bound for the free energy which has been given by Zeile [186]
using path integral methods. It was shown that Zeile’s variational principle does not guarantee a
thermodynamically consistent description.
U. Brandt and J. Stolze formulated [188] a new hierarchy of upper and lower bounds on expectation values. Upper and lower bounds were constructed for expectation values of functions of a real
random variable with derivatives up to order (N + 1) which are alternately negative and positive
over the whole range of interest. The bounds were given by quadrature formulas with weights
and abscissas determined by the first (N + 1) moments of the underlying probability distribution.
Application to a simple disordered phonon system yielded sharp bounds on the specific heat.
K. Vlachos [171] proposed a variational method that uses the frequency and the energy shift as
variational parameters. The quantum-mechanical partition function was approximated by a formally simple expression, for a generalized anharmonic oscillator in one and many dimensions. The
numerical calculations for a single quartic and two coupled quartic oscillators have led to nearly
exact values for the free energy, the ground state, and the difference between the ground state
and the first excited state.
C. Predescu [189] presented a generalization of the Gibbs-Bogoliubov-Feynman inequality for
spinless particles and then illustrated it for the simple model of a symmetric double-well quartic
potential. The method gives a pointwise lower bound for the finite-temperature density matrix
and it can be systematically improved by the Trotter composition rule. It was also shown to produce ground state energies better than the ones given by the Rayleigh-Ritz principle as applied
to the ground state eigenfunctions of the reference potentials. Based on this observation, it was
conjectured that the local variational principle may performs better than the equivalent methods
based on the centroid path idea and on the Gibbs-Bogoliubov-Feynman variational principle, especially in the range of low temperatures. However clear evidence for such a statement was not
given.
35
All these points of view acquire significance of the variational principles as a general method of
solution for better insight into the complicated behavior of the many-particle systems.
11
The Hartree-Fock-Bogoliubov Mean Fields
The microscopic theory of superconductivity was created simultaneously by Bardeen, Cooper, and
Schrieffer [190, 191] and Bogoliubov [192, 193, 194, 195, 196, 197, 198]. An important contribution
to the theory of superconductivity were the works of Fröhlich [199], who put forward the idea of
the importance of the electron-phonon interaction for the phenomenon of superconductivity, and
the theory of Schafroth, Butler, and Blatt [200], who conjectured that superconductivity is due
to Bose-Einstein condensation of correlated electron pairs. In their paper, Bardeen, Cooper, and
Sehrieffer determined the ground state energy, and the spectrum of elementary excitations of their
model [190, 191]. The BCS theory was constructed on the basis of a model Hamiltonian that takes
into account only the interaction of electrons with opposite momenta and spins, whereas Bogoliubov theory was based on the Fröhlich Hamiltonian [199] and used the method of compensation
of dangerous diagrams [194]. N. N. Bogoliubov, V. V. Tolmachev and D. V. Shirkov [194] have
generalized to Fermi systems the Bogoliubov method of canonical transformations proposed earlier
in connection with a microscopic theory of superfluidity for Bose systems [112]. This approach
has formed the basis of a new method for investigating the problem of superconductivity. Starting
from Fröhlich Hamiltonian, the energy of the superconducting ground state and the one-fermion
and collective excitations corresponding to this state were obtained. It turns out that the final
formulae for the ground state and one-fermion excitations obtained independently by Bardeen,
Cooper and Schrieffer [190] were correct in the first approximation. The physical picture appears
to be closer to the one proposed by Schafroth, Butler and Blatt. The effect on superconductivity
of the Coulomb interaction between the electrons was analyzed in detail. A criterion for the superfluidity of a Fermi system with a four-line vertex Hamiltonian was established.
Roughly speaking, to explain simply the theory of superconductivity it is possible to say that the
Fermi sea is unstable against the formation of a bound Cooper pair when the net interaction is
attractive; it is reasonable to expect that the pairs will be condense until an equilibrium point will
be reached. The corresponding antisymmetric wave functions for many electrons was constructed
in BCS model [201, 202]. They noted also that their solution may be considered as an exact in
the thermodynamic limit.
The most clear and rigorous arguments in favor of the statement that the BCS model is an exactly solvable model of statistical physics were advanced in the papers of Bogoliubov, Zubarev,
and Tserkovnikov [192, 197, 198]. They showed that the free energy and the correlation functions
of the BCS model and a model with a certain approximating quadratic Hamiltonian are indeed
identical in the thermodynamic limit. In his theory [192, 193, 194, 195, 196, 197, 198], Bogoliubov
gave a rigorous proof that at vanishing temperature the correlation functions and mean values of
the energy of the BCS model and the Bogolyubov-Zubarev-Tserkovnikov model are equal in the
thermodynamic limit. Moreover, Bogoliubov constructed a complete theory of superconductivity
on the basis of a model of interacting electrons and phonons [192, 193, 194, 195, 196, 197, 198].
Generalizing his method of canonical transformations [15, 203, 204] to Fermi systems and advancing the principle of compensation of dangerous graphs [194], he determined the ground state
consisting of paired electrons with opposite moments and spins, its energy, and the energy of
elementary excitations. It was shown also that the phenomenon of superconductivity consists
in the pairing of electrons and a phase transition from a normal state with free electrons to a
superconducting state with pair condensate.
36
The pairing Hamiltonian has the form
X
X
H − µN =
E(k)a†kσ akσ +
V (k, p)a†k↑ a†−k↓ a−p↓ ap↑ ,
kσ
(155)
kp
where µ is the chemical potential and N is the number of particles.
The essential step which was made by Bogoliubov was connected with introducing the anomalous
averages or the generalized mean fields Fp = ha−p↓ ap↑ i. It is reasonably to suppose that because
of the large number of particles involved, the fluctuations of a−p↓ ap↑ about these expectations
values Fp must be small. Hence, it is thus possible to express such products of operators in the
form
a−p↓ ap↑ = Fp + (a−p↓ ap↑ − Fp ).
(156)
It is reasonably to suppose that one may neglects by the quantities which are bilinear in the presumably small fluctuation term in brackets. This way leads to the Bogoliubov model Hamiltonian
of the form
X
kσ
E(k)a†kσ akσ +
X
kp
Hmod − µN =
V (k, p) a†k↑ a†−k↓ Fp + Fk∗ a−p↓ ap↑ − Fk∗ Fp .
(157)
Here the Fk should be determined self-consistently [192, 193, 194, 195, 196, 197, 198].
Thus Bogoliubov created a rigorous theory of superfluidity [112] and superconductivity [198] within
the unified scheme [115] of the non-zero anomalous averages or the generalized mean fields, and
showed that at the physical basis of these two fundamental phenomena of nature lies the process
of condensation of Bose particles [116] and, respectively, pairs of fermions.
Indeed, N. N. Bogoliubov, D. N. Zubarev and Yu. A. Tserkovnikov [192, 197], have shown on
the basis of the model Hamiltonian of BCS-Bogoliubov, that the thermodynamic functions of a
superconducting system, which were obtained by a variation method in BCS, are asymptotically
exact for V → ∞, N/V = const (V is the volume of the system, and N the number of particles). This conclusion was based on the fact that each term of the perturbation-theory series,
by means of which the correction to that solution was calculated, is asymptotically small for
V → ∞. In addition, it was shown that it is possible to satisfy with asymptotic exactness the
entire chain of equations for Green functions constructed on the basis of the model Hamiltonian of
BCS-Bogoliubov. Thus the asymptotic exactness of the known solution for the superconducting
state was proved without the use of perturbation theory. It was shown also that the trivial solution that corresponds to the normal state should be rejected at temperatures below the critical
temperature. In other words, starting with the reduced Hamiltonian of superconductivity theory,
Bogoliubov, Zubarev, and Tserkovnikov [192, 197] proved the possibility of exact calculation of
the free energy per unit volume.
Somewhat later, on the basis of the BCS theory, a similar investigation was made by other authors [205, 206, 207, 208]. B. Muhlschlegel [205] studied an asymptotic expansion of the BCS
partition function by means of the functional method. The canonical operator exp[−β(H − µN )]
associated with the BCS model Hamiltonian of superconductivity was represented as a functional
integral by the use of Feynman’s ordering parameter. General properties of the partition function
in this representation were investigated. Taking the inverse volume of the system as an expansion
parameter, it was possible to calculate the thermodynamic potential including terms independent
of the volume. Muhlschlegel’s theory yielded an additional evidence that the BCS variational
value is asymptotically exact. The behavior of the canonical operator for large volume was described and related to the state of free quasiparticles. A study of the terms of the thermodynamic
potential which were of smaller order in the volume in the low-temperature limit, showed that the
37
ground state energy is nondegenerate and belongs to a number eigenstate.
W. Thirring and A. Wehrl [209] investigated in which sense the Bogoliubov-Haag treatment of
the BCS-Bogoliubov model gives the correct solution in the limit of infinite volume. They found
that in a certain subspace of the infinite tensor product space the field operators show the correct
time behaviour in the sense of strong convergence. Thus a solution of the superconducting type
with a gap in the spectrum of elementary excitations really can exists for the model Hamiltonian
of BCS-Bogoliubov.
In general, the problem of explaining the phenomenon of superconductivity required the solution
of the very difficult mathematical problems associated with the foundation of applied approximations [2, 15]. In connection with this, Bogoliubov investigated [192, 193, 194, 195, 196, 197, 198]
the reduced Hamiltonian, in which the interaction of single electrons is studied, and carried out
for it a complete mathematical investigation for zero temperature. In this connection he laid the
bases of a new powerful method of the approximating Hamiltonian, which allows linearization
of nonlinear quantum equations of motion, and reduction of all nonlinearity to self-consistent
equations for the ordinary functions into which the defined operator expressions translate. This
method was extended later to nonzero temperatures and a wide class of systems, and became one
of most powerful methods of solving nonlinear equations for quantum fields [2, 15].
D. Ya. Petrina contributed much to the further clarification of many complicated aspects of the
BCS-Bogoliubov theory. He performed a close and subtle analysis [15, 115, 210, 211, 212, 213] of
the BCS-Bogoliubov model and various related mathematical problems.
In his paper [210] ”Hamiltonians of quantum statistics and the model Hamiltonian of the theory
of superconductivity” an investigation was made of the general Hamiltonian of quantum statistics
and the model Hamiltonian of the theory of superconductivity in an infinite volume. The Hamiltonians were given a rigorous mathematical definition as operators in a Hilbert space of sequences of
translation-invariant functions. It was established that the general Hamiltonian is not symmetric
but possesses a real spectrum; the model Hamiltonian is symmetric and its spectrum has a gap
between the energy of the ground state and the excited states.
In the following paper [211], the model Hamiltonian of the theory of superconductivity was investigated for an infinite volume and a complete study was made of its spectrum. The grand
partition function was determined and the equation of state was found. In addition, the existence
of a phase transition from the normal to the superconducting state was proved. It was shown
that in the limit V → ∞ the chain of equations for the Green functions of the model Hamiltonian
has two solutions, namely the free Green function and the Green function of the approximating
Hamiltonian.
In his paper [212] D.Petrina has shown that the Bogolyubov result that the average energies (per
unit volume) of the ground states for the BCS-Bogoliubov Hamiltonian and the approximating
Hamiltonian asymptotically coincide in the thermodynamic limit is also valid for all excited states.
He also established that, in the thermodynamic limit, the BCS-Bogoliubov Hamiltonian and the
approximating Hamiltonian asymptotically coincide as quadratic forms.
D.Petrina [213] considered also the BCS Hamiltonian with sources, as it was proposed by Bogoliubov and Bogoliubov, Jr. It was proved that the eigenvectors and eigenvalues of the BCSBogoliubov Hamiltonian with sources can be exactly determined in the thermodynamic limit.
Earlier, Bogoliubov proved that the energies per volume of the BCS-Bogoliubov Hamiltonian
with sources and the approximating Hamiltonian coincide in the thermodynamic limit. These
results clarified substantially the microscopic theory of superconductivity and provided a deeper
mathematical foundation to it.
Raggio and Werner [169] have shown the existence of the limiting free-energy density of inhomogeneous (site-dependent coupling) mean-field models in the thermodynamic limit [45], and derived
a variational formula for this quantity. The formula requires the minimization of an energy term
38
plus an entropy term as a functional depending on a function with values in the one-particle state
space. The minimizing functions describe the pure phases of the system, and all cluster points
of the sequence of finite volume equilibrium states have unique integral decomposition into pure
phases. Some applications were considered; they include the full BCS-model, and random meanfield models.
A detailed and careful mathematical analysis of certain aspects of the BCS-Bogoliubov theory was
carried out by S. Watanabe [214, 215, 216, 217, 218, 219, 220, 221, 222], mainly in the context of
the solutions to the BCS-Bogoliubov gap equation for superconductivity.
BCS-Bogoliubov theory correctly yields an energy gap [223, 224]. The determination of this important energy gap is by solving a nonlinear singular integral equation. An investigation of the
solutions to the BCS-Bogoliubov gap equation for superconductivity was carried out by S. Watanabe [214, 215, 216, 217, 218, 219, 220, 221, 222]. In his works the BCS-Bogoliubov equations were
studied in full generality. Watanabe investigated the gap equation in the BCS-Bogoliubov theory
of superconductivity, where the gap function is a function of the temperature T only. It was shown
that the squared gap function is of class C 2 on the closed interval [ 0, TC ]. Here, TC stands for
the transition temperature. Furthermore, it was shown that the gap function is monotonically
decreasing on [0, TC ] and the behavior of the gap function at T = TC was obtained and some
more properties of the gap function were pointed out.
On the basis of his study Watanabe then gave, by examining the thermodynamical potential, a
mathematical proof that the transition to a superconducting state is a second-order phase transition. Furthermore, he obtained a new and more precise form of the gap in the specific heat
at constant volume from a mathematical point of view. It was shown also that the solution to
the BCS-Bogoliubov gap equation for superconductivity is continuous with respect to both the
temperature and the energy under the restriction that the temperature is very small. Without
this restriction, the solution is continuous with respect to both the temperature and the energy,
and, moreover, the solution is Lipschitz continuous and monotonically decreasing with respect to
the temperature.
D. M. van der Walt, R. M. Quick and M. de Llano [225, 226] have obtained analytic expressions for
the BCS-Bogoliubov gap of a many-electron system within the BCS model interaction in one, two,
and three dimensions in the weak coupling limit, but for arbitrary interaction width ν = ~D/EF ,
0 < ν < ∞. Here ~D is the maximum energy of a force-mediating boson, and EF is the Fermi
energy (which is fixed by the electronic density). The results obtained addressed both phononic
(ν ≪ 1) as well as nonphononic (e.g., exciton, magnon, plasmon, etc.) pairing mechanisms where
the mediating boson energies are not small compared with EF , provided weak electron-boson
coupling prevails. The essential singularity in coupling, sometimes erroneously attributed to the
two-dimensional character of the BCS model interaction with (ν ≪ 1), was shown to appear in
one, two, and three dimensions before the limit ν → 0 is taken.
B. McLeod and Yisong Yang [227] studied the uniqueness and approximation of a positive solution of the BCS-Bogolyubov gap equation at finite temperatures. When the kernel was positive
representing a phonon-dominant phase in a superconductor, the existence and uniqueness of a
gap solution was established in a class which contains solutions obtainable from bounded domain
approximations. The critical temperatures that characterize superconducting-normal phase transitions realized by bounded domain approximations and full space solutions were also investigated.
It was shown under some sufficient conditions that these temperatures are identical. In this case
the uniqueness of a full space solution follows directly. Authors [227] also presented some examples for the nonuniqueness of solutions. The case of a kernel function with varying signs was also
considered. It was shown that, at low temperatures, there exist nonzero gap solutions indicating
a superconducting phase, while at high temperatures, the only solution is the zero solution, representing the dominance of the normal phase, which establishes again the existence of a transition
39
temperature.
In a series of papers [228, 229, 230], M. Combescot, W.V. Pogosov and O. Betbeder-Matibet studied various aspects of the BCS ansatz for superconductivity [190] in the light of the Bogoliubov
approach.
In paper [228] they extended the one-pair Cooper configuration towards BCS-Bogoliubov model of
superconductivity by adding one-by-one electron pairs to an energy layer, where a small attraction
acts. To do it, they solved Richardson’s equations analytically in the dilute limit of pairs on the
one-Cooper pair scale. It was found, through only keeping the first order term in this expansion,
that the N correlated pair energy reads as the energy of N isolated pairs within a N (N − 1) correction induced by the Pauli exclusion principle which tends to decrease the average pair binding
energy when the pair number increases. Quite remarkably, extension of this first-order result to
the dense regime gives the BCS-Bogoliubov condensation energy exactly. These facts may lead
ones to a different interpretation of the BCS-Bogoliubov condensation energy with a pair number
equal to the number of pairs feeling the potential and an average pair binding energy reduced by
Pauli blocking to half the single Cooper pair energy - instead of the more standard but far larger
superconducting.
In the next paper [229] the usual formulation of the BCS-Bogoliubov ansatz for superconductivity in the grand canonical ensemble makes the handling of the Pauli exclusion principle between
paired electrons straightforward. It however masks that many-body effects between Cooper pairs
interacting through the reduced BCS-Bogoliubov potential are entirely controlled by this exclusion. To show it up, one has to work in the canonical ensemble. The proper handling of Pauli
blocking between a fixed number of composite bosons is however known to be quite difficult. To
do it, author have developed a commutator formalism for Cooper pair condensate, along the line
they used for excitons. Authors [229] then rederived, within the N -pair subspace, a few results
of BCS-Bogoliubov theory of superconductivity obtained in the grand canonical ensemble, to evidence their Pauli blocking origin. They ended by reconsidering what should be called Cooper pair
wave function and concluded differently from usual understanding.
In their third paper M. Combescot, W.V. Pogosov and O. Betbeder-Matibet [230] showed that the
Bogoliubov approach to superconductivity provides a strong mathematical support to the wave
function ansatz proposed by Bardeen, Cooper and Schrieffer [190]. However there are some subtle
differences in the both the approaches. Indeed, the BCS ansatz - with all pairs condensed into the
same state - corresponds to the ground state of the Bogoliubov Hamiltonian. From the other hand,
this Hamiltonian only is part of the BCS Hamiltonian. As a result, the BCS ansatz definitely
differs from the BCS Hamiltonian ground state. This can be directly shown either through a perturbative approach starting from the Bogoliubov Hamiltonian, or better by analytically solving
the BCS Schrodinger equation along Richardson-Gaudin exact procedure. Still, the BCS ansatz
leads not only to the correct extensive part of the ground state energy for an arbitrary number
of pairs in the energy layer where the potential acts - as recently obtained by solving RichardsonGaudin equations analytically - but also to a few other physical quantities such as the electron
distribution, as it was shown by authors. The paper [230] also considered arbitrary filling of the
potential layer and evidences the existence of a super dilute and a super dense regime of pairs,
with a gap different from the usual gap. These regimes constitute the lower and upper limits of
density-induced BEC-BCS cross-over in Cooper pair systems. It should be noted, however, that
this theory needs an additional careful examination.
In 1958 N. N. Bogoliubov [231] proposed a new variational principle in the many-particle problem.
This variational principle is the generalization of the Hartree-Fock variational principle [5, 10]. It
is well known [232, 233] that the Hartree-Fock approximation is a variational method that provides the wave function of a many-body system assumed to be in the form of a Slater determinant
for fermions and of a product wave function for bosons. It treats correctly the statistics of the
40
many-body system, antisymmetry for fermions and symmetry for bosons under the exchange of
particles. The variational parameters of the method are the single-particle wave functions composing the many-body wave function.
Bogoliubov [231] considered a model dynamical Fermi system describing by the Hamiltonian with
two-body forces. The Hamiltonian of a nonrelativistic system of identical fermions interacting by
two-body interactions was
X
1 X
J(k, k′ |σ1 σ2 σ2′ σ1′ )a†kσ a†kσ akσ akσ .
(158)
H=
E(k) − EF a†kσ akσ +
2V
′
kσ
k,k ,σ
The a†kσ and akσ are single-particle creation and annihilation operators satisfying the usual anticommutation relations, EF is the Fermi energy level and V is the volume of the system.
The Hamiltonian under consideration is a model Hamiltonian; it takes into account the pair interaction of the particles with opposite momentum only. It can be rewritten in the following
form [231]:
H=
X
1 X
E(k) − EF a†qs aqs +
I(q, q ′ |s1 , s2 , s′2 s′1 )a†qs1 a†qs2 aq′ s′2 aq′ s′1 .
2V
′
qs
(159)
q,q ,s
Here q describes the pair of momentum (k, −k); hence q and −q describe the same pair. Index
s = (σ, ν), where ν = ±1 is an additional index [231] permitting to classify k as (q, ν). N. N.
Bogoliubov [231] shown that the ground state of the system can be found asymptotically exactly
for the limit V → ∞ by following to approach of the paper [192].
This approach found numerous applications in the many-body nuclear theory [232, 233, 234, 235,
236, 237, 238, 239, 240]. The properties of all existing and theoretically predicted nuclei can be
calculated based on various nuclear many-body theoretical frameworks. The classification of nuclear many-body methods can be also done from the point of view of the pair nuclear interaction,
from which the many-body Hamiltonian is constructed. An important goal of nuclear structure
theory is to develop the computational tools for a systematic description of nuclei across the
chart of the nuclides. Nuclei come in a large variety of combinations of protons and neutrons
(≤ 300). Understanding the structure of the nucleus is a major challenge. To study some collective phenomena in nuclear physics, we have to understand the pairing correlation due to residual
short-range correlations among the nucleons in the nucleus. This has usually been calculated
by using the BCS theory or the Hartree-Fock-Bogoliubov theory. The Hartree-Fock-Bogoliubov
theory is suited well for describing the level densities in nuclei. [237, 239]. The theory of level
densities reminds in certain sense the ordinary thermodynamics. The simplest level density of
nucleons calculations were based usually on a model Hamiltonian which included a simple version
of the pairing interaction (between nucleons in states differing only by the sign of the magnetic
quantum number).
J. A. Sheikh and P. Ring [236] derived the symmetry-projected Hartree-Fock-Bogoliubov equations using the variational ansatz for the generalized one-body density-matrix in the Valatin form.
It was shown that the projected-energy functional can be completely expressed in terms of the
Hartree-Fock-Bogoliubov density-matrix and the pairing-tensor. The variation of this projectedenergy was shown to result in Hartree-Fock-Bogoliubov equations with modified expressions for
the pairing-potential and the Hartree-Fock field. The expressions for these quantities were explicitly derived for the case of particle number-projection. The numerical applicability of this
projection method was studied in an exactly soluble model of a deformed single-j shell.
A. N. Behkami and Z. Kargar [237] have determined the nuclear level densities and thermodynamic
functions for light A nuclei, from a microscopic theory, which included nuclear pairing interaction.
41
Nuclear level densities have also been obtained using Bethe formula as well as constant temperature formula. Level densities extracted from the theories were compared with their corresponding
experimental values. It was found that the nuclear level densities deduced by considering various
statistical theories are comparable; however, the Fermi-gas formula [241] becomes inadequate at
higher excitation energies. This conclusion, which has also been arrived at by other investigations,
revealed that a realistic treatment of the statistical nuclear properties requires the introduction
of residual interaction. The effects of the pairing interaction and deformation on nuclear state
densities were illustrated and discussed.
L. M. Robledo and G. F. Bertsch [238] have presented a computer code for solving the equations of the Hartree-Fock-Bogoliubov theory by the gradient method, motivated by the need for
efficient and robust codes to calculate the configurations required by extensions of the HartreeFock-Bogoliubov theory, such as the generator coordinate method. The code was organized with a
separation between the parts that are specific to the details of the Hamiltonian and the parts that
are generic to the gradient method. This permitted total flexibility in choosing the symmetries
to be imposed on the Hartree-Fock-Bogoliubov solutions. The code solves for both even and odd
particle-number ground states, with the choice determined by the input data stream.
M. Lewin and S. Paul [240] shown that the best method for describing attractive quantum systems
is the Hartree-Fock-Bogoliubov theory. This approach deals with a nonlinear model which allows
for the description of pairing effects, the main explanation for the superconductivity of certain
materials at very low temperature. Their paper is a detailed study of Hartree-Fock-Bogoliubov
theory from the point of view of numerical analysis. M. Lewin and S. Paul started by discussing
its proper discretization and then analyzed the convergence of the simple fixed point (Roothaan)
algorithm. Following works for electrons in atoms and molecules, they shown that this algorithm
either converges to a solution of the equation, or oscillates between two states, none of them being
solution to the Hartree-Fock-Bogoliubov equations. They also adapted the Optimal Damping Algorithm to the Hartree-Fock-Bogoliubov setting and also analyzed it. The last part of the paper
was devoted to numerical experiments. Authors considered a purely gravitational system and
numerically discovered that pairing always occurs. They then examined a simplified model for
nucleons, with an effective interaction similar to what is often used in nuclear physics. In both
cases M. Lewin and S. Paul [240] discussed the importance of using a damping algorithm.
Many other applications of the Hartree-Fock-Bogoliubov theory to various many-particle systems
were discussed in Refs. [242, 243, 244, 245, 246] Generalization of Lieb variational principle [166]
to Bogoliubov-Hartree-Fock theory was considered recently by V. Bach et al. [167] In its original
formulation, Lieb variational principle holds for fermion systems with purely repulsive pair interactions. As a generalization authors proved for both fermion and boson systems with semi-bounded
Hamiltonian that the infimum of the energy over quasifree states coincides with the infimum over
pure quasifree states. In particular, the Hamiltonian was not assumed to preserve the number of
particles.
It is instructive to remind that in mathematics, the infimum (abbreviated inf; plural infima) of
a subset S of a partially ordered set T is the greatest element of T that is less than or equal to
all elements of S. Consequently the term greatest lower bound is also commonly used. Infima
of real numbers are a common special case that is especially important in analysis. However,
the general definition remains valid in the more abstract setting of order theory where arbitrary
partially ordered sets are considered.
To shed light on the relation between authors’ result and the usual formulation of Lieb variational
principle in terms of one-particle density matrices, it was also included a characterization of pure
quasifree states by means of their generalized one-particle density matrices.
42
12
Method of an Approximating Hamiltonian
It is worth noting that a complementary method, which was called by the method of an approximating Hamiltonian, was formulated [3, 4, 247, 248, 249] for treating model systems of statistical
mechanics. The essence of the method consists in replacement of the initial model Hamiltonian
H, which is not amenable to exact solution, by a suitable approximating (or trial) Hamiltonian
H appr . The next step consists of proving their thermodynamical equivalence, i.e., proving that
the thermodynamic potentials and the mean values calculated on the basis of H and H appr are
asymptotically equal in the thermodynamic limit [45] N, V → ∞, N/V = const.
When investigating the phenomenon of superconductivity, Bogolyubov suggested the method of
approximating Hamiltonian and justified it for the case of temperatures close to zero. By employing this method, Bogolyubov rigorously solved the BCS model of superconductivity at temperature
zero. This model was defined by the Hamiltonian of interacting electrons with opposite momenta
and spins.
To explain the superconductivity phenomenon, it was necessary to solve very difficult mathematical problems connected with the justification of approximations employed. In this connection,
Bogoliubov considered the reduced Hamiltonian in which only the interaction of electrons was
taken into account. He gave a complete mathematical investigation of this Hamiltonian at temperature zero. Moreover, he laid the foundation of a new powerful method of approximating
Hamiltonian which allows one to linearize nonlinear quantum equations of motion so that the
nonlinearity is preserved only in self-consistent equations for ordinary functions that are obtained
from certain operator expressions. This method was then extended to the case of nonzero temperatures and applied to a broad class of systems. Later, this approach became one of the most
effective methods for solving nonlinear equations for quantum fields.
The method of approximating Hamiltonian is based on the proof of the thermodynamic equivalence
of the model under consideration and approximating Hamiltonian. Thermodynamic equivalence
means here the coincidence of specific free energies and Green functions for model and approximating Hamiltonian in the thermodynamic limit [45] when V and N tends to ∞, N/V = const.
It was shown above that in many cases it may be assumed that the effective Hamiltonian H for the
system of particles may be written as the sum of the Hamiltonian of the reference system H appr ,
plus the rest of the effective Hamiltonian H = H appr + ∆H. Then the Bogoliubov inequality
states that the Helmholtz free energy F of the system is given by
F ≤ F appr + hH − H appr iappr ,
(160)
where F appr notes the free energy of the reference system and the brackets a canonical ensemble
average over the reference system.
N.N. Bogoliubov Jr. elaborated a new method [247, 248, 249, 250, 251] of finding exact solutions
for a broad class of model systems in quantum statistical mechanics – the method of approximating
Hamiltonian. As it was mentioned above this method appeared in the theory of superconductivity [197, 198].
N.N. Bogoliubov Jr. investigated some dynamical models [247] generalizing those of the BCS
type. A complete proof was presented that the well-known approximation procedure leads to an
asymptotically exact expression for the free energy, when the usual limiting process of statistical
mechanics is performed. Some special examples were considered.
A detailed analysis of Bogoliubov approach to investigations of (Hartree-Fock-Bogoliubov) meanfield type approximations for models with a four-fermion interaction was given in the papers [250,
251]. An exactly solvable model with paired four-fermion interaction that is of interest in the
theory of superconductivity was considered. Using the method of approximating Hamiltonian,
it was shown that it is possible to construct an asymptotically exact solution for this model.
43
In addition a theorem was proved that allows us to compute, with asymptotic accuracy in the
thermodynamic limit, the density of the free energy under sufficiently general conditions imposed
on the parameters of the model system. An approximate method for investigating models with
four-fermion interaction of general form was presented. The method was based on the idea of
constructing an approximating Hamiltonian and it allows one to study the dynamical properties
of these models. The method combines the standard approach to the method of the approximating Hamiltonian for the investigation of models with separable interaction and the Hartree-Fock
scheme of approximate computations based on the concept of self-consistency. To illustrate the
efficiency of the approach presented, the BCS model that plays an important role in the theory of
superconductivity was considered in detail. Thus, the effective and workable approach was formulated which allows one to investigate dynamical and thermodynamical properties of models with
four-fermion interaction of general type. The approach combines ideas of the standard Bogoliubov
approximating Hamiltonian method for the models with separable interaction with the method of
Hartree-Fock approximation based on the ideas of self-consistency.
A. P. Bakulev, N. N. Bogoliubov, Jr., and A. M. Kurbatov [252] discussed thoroughly the principle of thermodynamic equivalence in statistical mechanics in the approach of the method of
approximating Hamiltonian. They discussed the main ideas that lie at the foundations of the
approximating Hamiltonian method in statistical mechanics. The principal constraints for model
Hamiltonian to be investigated by approximating Hamiltonian method were considered along with
the main results obtainable by this method. It was shown how it is possible to enlarge the class
of model Hamiltonians solvable by approximating Hamiltonian method with the help of an example of the BCS-type model. Additional rigorous studies of the theory of superconductivity
with Coulomb-like repulsion was carried out by A. P. Bakulev [253]. The traditional method of
the approximating hamiltonian was applied for the investigation of a model of a superconductor
with interaction of the BCS type and Coulomb-like repulsion, the latter being described by unbounded operators. It was shown that the traditional method can be generalized in such a way
that for the model under consideration one can prove the asymptotic (in the thermodynamic limit
V → ∞, N → ∞, N/V = const) coincidence not only of the free energies (per unit volume) but
also of the correlation functions of the model and approximating Hamiltonian.
13
Conclusions
The aim of the present overview was to justify a statement that in many cases the methods of
quantum statistical mechanics, many of which were formulated and developed by N.N. Bogoliubov [1, 2, 3, 4], allow one to develop efficient approaches for solution of complicated problems of
the many-particle interacting systems.
In the present survey we discussed tersely the Bogoliubov variational principle. It was shown in
the preceding sections that this principle provides an extremely valuable treatment of mean-field
methods and their application to the problems in statistical mechanics and many-particle physics
of interacting systems. With its remarkable workability the Bogoliubov variational principle found
many applications as an effective method not only in condensed matter physics but also in many
other areas of physics (see, e.g. Ref. [254]). It is also hoped that this work will lead to greater
insight into the application of variational principles to various many-particle problems.
There is another aspect of the problem under consideration. It is of great importance to determine
correctly the mean-field contribution when one describes the interacting many-particle systems
by the equations-of-motion method [5, 17]. It was mentioned briefly that the method of two-time
temperature Green functions [5, 17] allows one to investigate efficiently the quasiparticle manybody dynamics generated by the main model Hamiltonians from the quantum solid state theory
and the quantum theory of magnetism. The method of quasiaverages allows one to take a deeper
44
look at the problems of spontaneous symmetry breaking, as well as at the problems of symmetry
and dissymmetry in the physics of condensed matter [5, 17, 18].
Summarizing the basic results obtained by N. N. Bogoliubov by inventing the variational principles, method of quasiaverages and results in the area of creation of asymptotic methods of
statistical mechanics, one must especially emphasize that thanks to their deep theoretical content and practical direction, these methods have obtained wide renown everywhere. They have
enriched many-particle physics and statistical mechanics with new achievements in the area of
mathematical physics as well as in the areas of concrete applications to physics, e.g. theories of
superfluidity and superconductivity.
In the papers [17, 35, 36, 37], we have formulated the self-consistent theory of the correlation effects
for many-particle interacting systems using the ideas of quantum field theory for interacting electron and spin systems on a lattice. The workable and self-consistent irreducible Green functions
approach to the decoupling problem for the equation-of-motion method for double-time temperature Green functions has been presented. The main achievement of this formulation was the
derivation of the Dyson equation for double-time retarded Green functions instead of causal ones.
That formulation permitted to unify convenient analytical properties of retarded and advanced
Green functions and the formal solution of the Dyson equation, that, in spite of the required approximations for the self-energy, provides the correct functional structure of single-particle Green
function. The main advantage of the mathematical formalism was brought out by showing how
elastic scattering corrections (generalized mean fields) and inelastic scattering effects (damping
and finite lifetimes) could be self-consistently incorporated in a general and compact manner. We
have presented there the novel method of calculation of quasi-particle spectra for basic spin lattice models, as the most representative examples. Using the irreducible Green functions method,
we were able to obtain a closed self-consistent set of equations determining the electron Green
function and self-energy. For the Hubbard and Anderson models, these equations gave a general
microscopic description of correlation effects both for the weak and strong Coulomb correlation,
and, thus, determined the interpolation solutions of the models. Moreover, this approach gave
the workable scheme for the definition of relevant generalized mean fields written in terms of appropriate correlators.
We hope that these methods of statistical mechanics have been explained with sufficient details
to bring out their scope and power, since we believe that those techniques will have application
to a variety of many-body systems with complicated spectra and strong interaction.
These applications have illustrated some of subtle details of the irreducible Green functions approach and exhibited their physical significance in a representative form.
As it was seen, these treatments has advantages in comparison with the standard methods of
decoupling of higher-order Green functions within the equation-of-motion approach.
The main advantage of the whole method is the possibility of a self-consistent description of quasiparticle spectra and their damping in a unified and coherent fashion.
The most important conclusion to be drawn from the present consideration is that the generalized
mean fields for the case of strong Coulomb interaction in the Hubbard model has quite a nontrivial
structure and cannot be reduced to the mean-density functional.
Recently the problem of the advanced mean field methods in complex systems [255] has attracted
big attention. Our consideration reveals the fundamental importance of the adequate definition of
generalized mean fields at finite temperatures, that results in a deeper insight into the nature of
quasiparticle states of the correlated lattice fermions and spins and other interacting many-particle
systems.
45
14
Acknowledgements
The author recollects with gratefulness discussions of this review topics with N.N. Bogoliubov
(21.08.1909 - 13.02.1992) and D.N. Zubarev (30.11.1917 - 16.07.1992).
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