Resonating plaquette phases in SU(4) Heisenberg antiferromagnet Cenke Xu and Congjun Wu

Resonating plaquette phases in SU(4) Heisenberg antiferromagnet Cenke Xu and Congjun Wu
PHYSICAL REVIEW B 77, 134449 共2008兲
Resonating plaquette phases in SU(4) Heisenberg antiferromagnet
Cenke Xu1 and Congjun Wu2
1Department
of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
of Physics, University of California, San Diego, California 92093, USA
共Received 16 January 2008; revised manuscript received 26 March 2008; published 30 April 2008兲
2Department
Large spin cold atom systems can exhibit magnetic properties that do not appear in usual spin-1 / 2 systems.
We investigate the SU共4兲 resonating plaquette state in the three-dimensional cubic optical lattice with spin-3 / 2
cold fermions. A gauge field formalism is constructed to describe the Rokhsar–Kivelson type of Hamiltonian
and a duality transformation is used to study the phase diagram. Due to the proliferation of topological defects,
the system is generally gapped for the whole phase diagram of the quantum model, which agrees with the
recent numerical studies. The classical plaquette model on the cubic lattice is also studied, and a critical phase
is predicted by tuning one parameter in the low energy field theory.
DOI: 10.1103/PhysRevB.77.134449
PACS number共s兲: 75.10.Jm, 75.40.Mg, 75.45.⫹j
I. INTRODUCTION
Quantum fluctuations and non-Néel ordering magnetic
states in low dimensional spin-1 / 2 antiferromagnets are important topics in strongly correlated physics. The quantum
dimer model 共QDM兲 constructed by Rokhsar–Kivelson 共RK兲
in which each dimer represents an SU共2兲 singlet provides a
convenient way to investigate novel quantum magnetic states
such as the exotic topological resonating valence bond
共RVB兲 states.1 The QDM in the two-dimensional 共2D兲 square
lattice generally exhibits crystalline ordered phase except at
the RK point where the ground state wave function is a superposition of all possible dimer coverings.2 In contrast, a
spin liquid RVB phase has been shown in the triangular lattice in a finite range of interaction parameters by Moessner
et al. and Sondhi.3 The three-dimensional 共3D兲 RVB type of
spin liquid states has also been studied by using the QDM.4,5
Recently, there is a considerable interest on large spin
magnetism with cold atoms in optical lattices,6–11 whose
physics is fundamentally different from its counterpart in
solid state systems. In solid state systems, the large spin on
each site is formed by electrons coupled by Hund’s rule. The
corresponding magnetism is dominated by the exchange of a
single pair of spin-1 / 2 electrons and, thus, quantum fluctuations are suppressed by the large S effect. In contrast, it is a
pair of large spin atoms that is exchanged in cold atom systems; thus, quantum fluctuations can even be stronger than
those in spin-1 / 2 systems. In particular, a hidden and generic
Sp共4兲 symmetry has been proved in spin-3 / 2 systems without fine tuning by Wu et al.8,9 This large symmetry enhances
quantum fluctuations and brings many novel magnetic
physics.9,11–14
Below, we will focus on a special case of spin-3 / 2 fermions at the quarter filling 共one particle per site兲 in the 3D
cubic lattice with an SU共4兲 symmetry, which just means that
all of the four spin components are equivalent to each other.
The exchange model is the SU共4兲 antiferromagnetic Heisenberg model with each site in the fundamental representation.
Its key feature is that at least four sites are required to form
an SU共4兲 singlet two sites, i.e., two sites cannot form such a
singlet. This SU共4兲 model was also constructed in spin-1 / 2
systems with orbital degeneracy.15,16 This model is different
1098-0121/2008/77共13兲/134449共7兲
from the previous large-N version of the SU共N兲 Heisenberg
model defined in the bipartite lattices where two neighboring
sites are with complex-conjugate representations and the
Sp共2N兲 Heisenberg model defined in nonbipartite lattice,17,18
both of which can have singlet dimers. The natural counterpart of the dimer here is the SU共4兲 singlet plaquette state as
1
†
†
†
†
4! ⑀␣␤␥␦␺␣共1兲␺␤共2兲␺␥共3兲␺␦共4兲, where ␣, ␤, ␥, and ␦ take the
3
value of Sz as ⫾ 2 and ⫾ 21 . Recently, the crystalline ordered
SU共4兲 plaquette state has been investigated in quasi-1D ladder and 2D square lattice systems.11,16,19 The resonating
quantum plaquette model 共QPM兲 in three dimensions has
been constructed in Ref. 20 where quantum Monte Carlo
simulation shows that the ground state is solid in the entire
phase diagram. The SU共N兲 plaquette generalizations of the
Affleck–Kennedy–Lieb–Tasaki states21 have also been given
in Ref. 22.
In this paper, we will formulate a gauge field representation to the resonating plaquette model based on the SU共4兲
antiferromagnetic Heisenberg model in 3D cubic lattice. Unlike the QDM in 3D cubic lattice, this QPM is generally
gapped for the whole phase diagram due to the unavoidable
proliferation of topological defects. We study the gauge field
in dual language, where a local description of topological
defects is possible. The classical ensemble of the plaquette
system is also discussed, and unlike its quantum version, our
theory predicts that the classical ensemble can have an algebraic liquid phase by tuning one parameter. Classification of
topological sectors of the QPM is also discussed.
II. QUANTUM PLAQUETTE MODEL
The QPM model in the 3D cubic lattice can be represented as follows. The effective Hilbert space is constructed
by all the plaquette configurations allowed by the constraint:
every site in the cubic lattice is connected to one and only
one plaquette. Three flippable plaquette configurations exist
in each unit cube corresponding to the pairs of faces of left
and right, top and bottom, and front and back denoted as A,
B, and C in Fig. 1, respectively. The RK-type Hamiltonian1
reads
134449-1
©2008 The American Physical Society
PHYSICAL REVIEW B 77, 134449 共2008兲
CENKE XU AND CONGJUN WU
A
B
lel to the XY and YZ planes in a similar way. Then, we define
the boson number n with integer values on every face of the
cubic lattice. n = 1 corresponds to a face with plaquette, and
n = 0 otherwise. A strong local potential term
U共ni+共1/2兲␮ˆ +共1/2兲␯ˆ − 21 兲2 is turned on at every face to guarantee
that the low energy subspace of the boson Hilbert space is
identical to the Hilbert space with all the plaquette configurations. Since every site is connected to one and only one
plaquette, the summation of n over all 12 faces sharing one
site needs to be 1. Next, we define the rank-2 symmetric
traceless tensor electric field on the lattice as
C
冉
Ei,␮␯ = ␩共i兲 ni+共1/2兲␮ˆ +共1/2兲␯ˆ −
FIG. 1. 共Color online兲 Three flippable configurations in one
cube. The resonance is represented in the t term in Eq. 共1兲.
H=−t
兺
兵兩A典具B兩 + 兩B典具C兩 + 兩C典具A兩 + H.c.其
each cube
+V
兺
兵兩A典具A兩 + 兩B典具B兩 + 兩C典具C兩其,
共1兲
H=t
兺
兵兩Q1典具Q1兩 + 兩Q2典具Q2兩其 + 共V − 2t兲
each cube
+ 兩B典具B兩 + 兩C典具C兩其,
兺
兵兩A典具A兩
where ␩共i兲 = 共−1兲ix+iy+iz equals 1 when i belongs to one of the
two sublattices of the cubic lattice and equals −1 otherwise.
It is straightforward to check that the one-site-one-plaquette
local constraint on the Hilbert space can be written compactly as
ⵜxⵜyExy + ⵜyⵜzEyz + ⵜzⵜxEzx = 5␩共i兲,
where 兩Q1典 = 兩A典 + ␻兩B典 + ␻2兩C典, 兩Q2典 = 兩A典 + ␻2兩B典 + ␻兩C典, and
␻ = ei共2␲/3兲. As a result, at V = 2t 共the RK point兲, the ground
state wave function should be annihilated by the projectors
兩Q1典具Q1兩 and 兩Q2典具Q2兩, i.e., the equal weight superposition
between all the plaquette configurations, which can be connected to each other through finite steps of local resonances,
i.e., all the configurations within one topological sector. At
V / t ⬎ 2, all the plaquette configurations without flippable
cubes are eigenstates of the Hamiltonian, one of which is the
staggered plaquette state. The phase diagram of this RK
model has been studied numerically in Ref. 20. In particular,
the classical Monte Carlo simulation performed shows that at
this RK point, a weak crystalline order of resonating cubes is
formed, which forms a cubic lattice with doubled lattice constant. At V / t ⬍ 2, the system starts to favor flippable cubes.
For instance, at −V / t Ⰷ 1, the ground states are 12-fold degenerate with columnar ordering. All the transitions between
different phases are of the first order.
The original RK Hamiltonian for the quantum dimer
model can be mapped to the compact U共1兲 gauge theory,2,23
from which one can show that the 2 + 1 dimensional QDM is
gapped except for one special RK point, while the 3 + 1 dimensional QDM has a deconfined algebraic liquid phase.24
By contrast, the quantum plaquette model in the cubic lattice
can be mapped into a special type of lattice gauge field
theory as follows. We denote all the square faces parallel to
the XZ plane of the cubic lattice by the sites to the left and
bottom corner of the face, i + 21 x̂ + 21 ẑ, and denote faces paral-
共4兲
where ⵜ is the lattice derivative with the usual definition
ˆ 兲 − f共i兲.
ⵜ␮ f = f共i + ␮
The canonical conjugate variable of Ei,␮␯ is denoted as the
vector potential of Ai,␮␯,
Ai,␮␯ = ␩共i兲␪i+共1/2兲␮ˆ +共1/2兲␯ˆ ,
each cube
共2兲
E␮␯ = E␯␮共␮ ⫽ ␯兲,
共3兲
each cube
where t has been shown to be positive in Ref. 20, and we
leave the value of V / t arbitrary for generality. Equation 共1兲
can be represented as
冊
1
,
2
A ␮␯ = A ␯␮
共␮ ⫽ ␯兲.
共5兲
␪i+共1/2兲␮ˆ +共1/2兲␯ˆ is the canonical conjugate variable of boson
number ni+共1/2兲␮ˆ +共1/2兲␯ˆ , which is also the phase angle of boson
creation operator. A␮␯ and E␮␯ satisfy
关Ei,␮␯,A j,␳␴兴 = i␦ij共␦␮␳␦␯␴ + ␦␮␴␦␯␳兲.
共6兲
Because E␮␯ only takes values with an integer step, A␮␯ is a
compact field with period of 2␲. Due to the commutator,
关Ei,␮␯,exp共iA j,␯␴兲兴 = 共␦␮␳␦␯␴ + ␦␮␴␦␯␳兲exp共iA j,␯␴兲,
共7兲
operators, exp共iA j,␯␴兲 changes the eigenvalue of Ei,␮␯ by 1.
As a result, the plaquette flipping process can be represented
as
Ht = − t关cos共ⵜzAxy − ⵜxAyz兲 + cos共ⵜxAyz − ⵜyAzx兲
+ cos共ⵜyAzx − ⵜzAxy兲兴,
共8兲
which is invariant under the gauge transformation of
A ␮␯ → A ␮␯ + ⵜ ␮ⵜ ␯ f ,
共9兲
which is already implied by the local constraint 关Eq. 共4兲兴. f is
an arbitrary scalar function. The low energy Hamiltonian of
the system can be written as
H = Ht + U
+V
2
共E2xy + E2yz + Ezx
兲
兺
each cube
兺
关共ⵜxEyz兲2 + 共ⵜyEzx兲2 + 共ⵜzEzx兲2兴, 共10兲
each cube
which is subject to the constraint in Eq. 共4兲. Besides the
gauge symmetry 关Eq. 共9兲兴, Hamiltonian 共10兲 together with
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RESONATING PLAQUETTE PHASES IN SU共4兲…
constraint 共4兲 share another symmetry as follows:
␮ → − ␮,
␳ → ␳,
␴ → ␴,
共11兲
E ␮␯ → − E ␮␯,
E ␴␮ → − E ␴␮,
E␯␴ → E␯␴ ,
共12兲
A ␮␯ → − A ␮␯,
A ␴␮ → − A ␴␮,
A␯␴ → A␯␴ .
共13兲
␮, ␯, and ␴ are three space coordinates. This symmetry forbids terms such as ExyEyz to be generated under renormalization group flow at low energy.
III. DUALITY TRANSFORMATION
A major question in which we are interested is whether
the Hamiltonians 关Eqs. 共1兲 and 共10兲兴 have an intrinsic liquid
phase, just like the 3D QDM in the cubic lattice.24 A liquid
state here corresponds to a gapless Gaussian state in which
we are allowed to expand the cosine functions in Eq. 共10兲 at
their minima, i.e., a “spin wave” treatment. However, the
Gaussian phase could also be a superfluid phase, which
breaks the conservation of boson numbers 共or effectively the
plaquette numbers兲 with 具exp共i␪兲典 ⫽ 0. In our current
problem, a superfluid phase is not possible because
具exp共i␪兲典 ⫽ 0 necessarily breaks the local gauge symmetry
关Eq. 共9兲兴 of Hamiltonian 共11兲. In other words, a superfluid
state is a coherent state of boson phase ␪ implying a strong
fluctuation of boson numbers, which obviously violates the
local one-site-one-plaquette constraint.
In this type of lattice bosonic models, because bosonic
phase variable A␮␯ is compact, the biggest obstacle of liquid
phase is the proliferation of topological defect, which tunnels
between two minima of the cosine function in Eq. 共8兲. Since
the topological defects are nonlocal, the best way to study
them is go to the dual picture, in which the topological defects become local vertex operators of the dual height variables. Similar duality transformations have been widely used
in studying all types of bosonic rotor models, such as in
proving the intrinsic gap of 2D QDM,2,25 showing the existence of “bose metal phase,”26 as well as the deconfine phase
of 3D QDM,24 and very recently the stable liquid phase of
three-dimensional “graviton” model.27
Besides the topological defects, another convenience one
gains from the dual formalism is the solution of the constraint, i.e., we are no longer dealing with a Hilbert space
with a strict one-site-one-plaquette constraint in Eq. 共4兲. The
dual variables are defined on the dual lattice sites ī, which
are the centers of the unit cubes. In order to completely solve
the constraint, one needs to introduce three components of
the height field h␮ 共␮ = 1 , 2 , 3兲 on every dual site ī, which is
the center of a unit cubic of the original lattice,
Exy = ⵜz共hx − hy兲 + E0xy ,
Eyz = ⵜx共hy − hz兲 + E0yz ,
0
Ezx = ⵜy共hz − hx兲 + Ezx
,
1
2
FIG. 2. 共Color online兲 The duality transformation defined in Eq.
共14兲. On dual sites 1 and 2, there are three components of dual
vector height h␮, and the dual transformation for the shaded face is
Eyz = 共hy − hz兲2 − 共hy − hz兲1 = ⵜx共hy − hz兲.
0
fields that only take discrete integer values. E0xy, E0yz, and Ezx
are background charges satisfying the constraint 关Eq. 共4兲兴.
We can just take the configuration of the columnar phase to
define the value of the background charges as
E0xy共i, j,k兲
冦
=
共− 兲k
2
共when both i and j are even兲
共− 兲i+j+k+1
共otherwise兲,
2
0
E0yz共i, j,k兲 = Ezx
共i, j,k兲 =
共− 兲i+j+k+1
.
2
冧
共15兲
The canonical momenta ␲␮ to the dual fields h␮ on each
dual site are
␲x = ⵜyAzx − ⵜzAxy ,
共16兲
␲y = ⵜzAxy − ⵜxAyz ,
␲z = ⵜxAyz − ⵜyAzx .
共17兲
One can check the commutation relation and see that ␲␮ and
h␮ are a pair of conjugate variables. Then, the dual Hamiltonian of Eq. 共10兲 reads
H=
兺
␮=x,y,z
0 2
− t cos ␲␮ + U 兺 ␨␮␯␳关ⵜ␮共h␯ − h␳兲 − E␯␳
兴
␮␯␳
0
+ V 兺 ␨␮␯␳关ⵜ␮共ⵜ␮共h␯ − h␳兲 − E␯␳
兲兴2 ,
␮␯␳
共18兲
where ␨␮␯␳ is a fully symmetric rank-3 tensor, which equals
zero when any two of its three coordinates are equal, and
equals one otherwise. On each dual lattice site ī, the ␲␮ fields
satisfy the relation that 兺␮=x,y,z␲␮,i¯ = 0.
The symmetry transformations of Hamiltonian 共18兲 can
be extracted from the duality transformation 关Eqs. 共14兲 and
共17兲兴,
共14兲
whose geometric illustration is shown in Fig. 2. hx,y,z are
134449-3
hx → hx + f共x,y,z兲 + g1共x兲,
hy → hy + f共x,y,z兲 + g2共y兲,
PHYSICAL REVIEW B 77, 134449 共2008兲
CENKE XU AND CONGJUN WU
hz → hz + f共x,y,z兲 + g3共z兲,
共19兲
where f is a function of three spatial coordinates and g1,2,3
only depends on one spatial coordinate. This type of symmetry is a quasilocal symmetry, which also exists in the Bose
metal states26 and p-band cold atom systems.28
The main purpose of this paper is to study whether Hamiltonians 共18兲 and 共10兲 have a liquid phase that preserves all
the lattice symmetries, just like the deconfined algebraic liquid phase of 3D QDM. In this kind of algebraic liquid phase,
one can expand the cosine functions in Eq. 共10兲 and relax the
discrete values of the h␮ fields; the long distance physics can
be described by a field theory, which only involves the
coarsed grained mode of h␮. Let us denote the long scale
mode as h̃␮. In this Gaussian phase, one can also define a
continuous tensor electric field Ẽ␮␯ as the coarse grained
mode of E␮␯; the relation between Ẽ␮␯ and h̃␮ is Ẽ␮␯
= ␨␮␯␳⳵␳共h̃␮ − h̃␯兲. A Gaussian field theory of h̃␮ should satisfy
the continuous version of symmetries listed in Eq. 共19兲:
h̃␮ → h̃␮ + f̃共x , y , z兲 + g̃␮共r␮兲; now, h̃␮ as well as functions f̃
and g̃␮ can all take continuous values. A low energy field
theory action is conjectured to be
L=兺
␮
K
K
共⳵␶h̃␮兲2 + 兺 ␨␮␯␳关ⵜ␮共h̃␯ − h̃␳兲兴2 + ¯ , 共20兲
2
2 ␮␯␳
where the h̃x,y,z fields take continuous real values. No other
quadratic terms of h̃␮ with second spatial derivative is allowed by the symmetry in this action. Notice that in Eq. 共20兲,
we have rescaled the space-time coordinates to make the
coefficients of the first and second term equal. The action
关Eq. 共20兲兴 describes a state with enlarged conservation laws
of ␲␮. If there is a state described by the Gaussian action
关Eq. 共20兲兴, ␲x, ␲y, and ␲z are conserved within each YZ, ZX,
and XY plane, respectively. So any operator with nonzero
expectation values at this state has to satisfy the special 2D
planar conservation law of ␲␮.
The Gaussian part of action 共20兲 has one unphysical pure
gauge mode, which corresponds to function f in Eq. 共19兲,
and two gapless physical modes with low energy dispersion,
along each axis in the momentum space, instead of only at
the origin. Similar directional modes are also found in other
systems with quasilocal symmetries.26,28
The ellipses in Eq. 共20兲 contain the non-Gaussian vertex
operators denoted as Lv, which manifest the discrete nature
of h␮. Since h␮ only takes integer values, a periodic potential
cos共2␲h␮兲 can be turned on in the dual lattice Hamiltonian
共18兲. At low energy, the non-Gaussian term Lv generated by
cos共2␲h␮兲 has to satisfy all the symmetries in Eq. 共19兲; the
simplest form it can take is cos关2␲h̃␮兴. However, this vertex
operator only has lattice scale correlation at the Gaussian
fixed point because it violates the gauge symmetry of action
共20兲. Thus, the simplest vertex operator with possible long
range correlation is
Lv =
共21兲
The second mode ␻2 vanishes at every coordinate axis of
reciprocal space 共kx , ky , kz兲. The strong directional nature of
␻2 directly roots in the quasilocal gauge symmetries in Eq.
共19兲. The same modes can be obtained from the continuum
Gaussian limit action of Hamiltonian 共10兲,
1
共22兲
In this action, Ã␮␯ is the coarse grained mode of A␮␯, and Ã␮␯
is no longer a compactified quantity. The fact that ␻2 vanishes at every coordinate axis plays a very important role in
our following analysis since it will create infrared divergence
共23兲
具VN,␮␯共0兲VN,␮␯共r兲典
⬃ exp兵− 共2␲兲2N2具关h̃␮共0兲 − h̃␯共0兲兴关h̃␮共r兲 − h̃␯共r兲兴典其
冋
= ␦r␮␦r␯ exp −
共2␲兲2N2
K
冕
→ ␦r␮␦r␯ const共r → + ⬁兲.
␻22 ⬃ k2x + k2y + kz2 − 冑k4x + k4y + kz4 − k2x k2y − k2y kz2 − k2x kz2 .
关共⳵␶Ã␮␯兲2 − ␨␮␯␳共⳵␮Ã␯␳ − ⳵␯Ã␳␮兲2兴.
兺
2K
␮⫽␯
− ␣ cos关2␲共h̃␮ − h̃␯兲 + B␮␯共ī兲兴,
and B共ī兲 is a function of dual sites, which is interpreted as
Berry’s phase. The specific form of Berry’s phase of the
vertex operators depends on the background charge of the
original gauge field formalism, which determines the crystalline pattern of the gapped phase.25 However, since the liquid
phase is a phase in which the vertex operators are irrelevant,
whether a liquid phase exists or not does not depend on
Berry’s phase; thus, in the current work, we will not give a
complete analysis of Berry’s phase of our problem. In the
continuum limit, the most relevant vertex operators are the
ones with multidefect processes without Berry’s phase and
consistent with symmetries 关Eq. 共19兲兴: cos关2␲N共h̃␮ − h̃␯兲兴; let
us denote this vertex operators as VN,␮␯, and integer N can be
determined from the detailed analysis of Berry’s phase. The
correlation function between two vertex operators with arbitrary N separated in space-time is calculated as follows:
␻21 ⬃ k2x + k2y + kz2 + 冑k4x + k4y + kz4 − k2x k2y − k2y kz2 − k2x kz2 ,
L=
兺
␮⫽␯
ជ
d4k 共2k20 + 3k␮2 + 3k␯2兲eik·rជ
共2␲兲4 共k20 + ␻21兲共k20 + ␻22兲
册
共24兲
The correlation function 具h̃共r兲h̃共r⬘兲典 is evaluated at the
Gaussian fixed point described by the continuum limit
action 共20兲 without Lv. The delta function ␦r␮␦r␯ in
Eq. 共24兲 is due to the continuous quasilocal symmetry of
action 共20兲 or, in other words, the conservation of ␲␮
within each planes. For instance, correlation function
˜
˜
具ei2␲N关hx共0兲−h̃y共0兲兴e−i2␲N关hx共r兲−h̃y共r兲兴典 can only be nonzero when
rx = ry = 0; otherwise, ␲x conservation within every YZ plane
will be violated once rx ⫽ 0.
Since the correlation function calculated in Eq. 共24兲
reaches a finite constant in the long distance limit, the vertex
operators are very relevant at the Gaussian fixed point described by action 共20兲, and the system is generally gapped
with crystalline order in the whole phase diagram. Since this
result is applicable to any N and independent of Berry’s
phase, the same conclusion is applicable to all the QPM with
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RESONATING PLAQUETTE PHASES IN SU共4兲…
The leading order correlation functions are
a definite number of plaquette connected to each site. The
specific crystalline order can be determined from the detailed
analysis of Berry’s phase.
具VN,xy,共0,0,0兲VN,xy,共0,0,z兲典
冋
IV. CLASSICAL ROKHSAR–KIVELSON POINT
At the RK point, the ground state wave function is an
equal weight superposition of all the configurations allowed
by constraint 共4兲. All the static physics of this state is mathematically equivalent to a classical ensemble, with partition
function defined as summation of all the plaquette configurations with equal Boltzmann weights. Since there is no energetic terms in the partition function, all that rules is the
entropy. If we define the tensor electric field as Eq. 共3兲, the
classical ensemble can be written as
Z=
冋
兺 ␦ ␮兺⫽␯ ⵜ␮ⵜ␯E␮␯ − 5␩共i兲
Ei,␮␯
册 冋
exp − U 兺
兺 共Ei,␮␯兲2
i ␮⫽␯
册
.
共25兲
The delta function enforces the constraint, and the term
−U兺␮⫽␯共Ei,␮␯兲2 in the exponential makes sure that all the
low energy E␮␯ configurations are one-to-one mapping of the
plaquette configurations. Now, solving the constraint by introducing dual height field h␮, the classical partition function
can be rewritten as
再
Z = 兺 exp − U 兺
h¯i,␮
兺 ␨␮␯␳关ⵜ␮共h¯i,␯ − h¯i,␳兲 − E␯␳0 兴2
¯i ␮␯␳
冎
.
共26兲
Again, we are mainly interested in whether this classical
ensemble is an algebraic liquid state or by tuning parameters,
one can reach an algebraic liquid phase. We can conjecture a
low energy classical field theory generated by entropy, which
is allowed by symmetry 共19兲. The same strategy has been
used to study the classical six-vertex model, classical threecolor model, and four color model.29 Here, the simplest low
energy effective classical field theory reads
F=
K̃
兺 ␨␮␯␳关ⵜ␮共h̃␯ − h̃␳兲兴2 + ¯ .
␮␯␳ 2
共2␲兲2N2
⬃ exp −
共27兲
The number K̃ cannot be determined from our field theory.
This is the simplest free energy allowed by symmetry. The
physical meaning of this free energy is that the total number
of plaquette configurations 共entropy兲 in a three dimension
volume is larger if the average tensor electric field E␮␯ is
small, i.e., the entropy favors zero average tensor electric
field.
The ellipses in Eq. 共27兲 includes the vertex operators in
Eq. 共23兲. The relevance of the vertex operators can be
checked by calculating the scaling dimensions of the vertex
operators at the Gaussian fixed point action 共27兲. Let us denote vertex operator cos关2␲N共h̃␮ − h̃␯兲兴 as VN,␮␯. Due to symmetry 共19兲, VN,xy can only correlate with itself along the
same ẑ axis, and VN,zx and VN,yz can never have nonzero
correlation between each other when they are separated spatially along the ẑ axis.
=
1
z 4␲N
2/K̃
K̃
冕
共k2x + k2y 兲eikzz
d 3k
共2␲兲3 k2x k2y + k2y kz2 + kz2k2x
册
共28兲
.
In the above calculations, we have chosen the simplest regularization, replacing spatial derivative on the lattice by momentum ikx. It has been shown that the scaling dimensions of
operators in these type of models with extreme anisotropy
can depend on the regularization on the lattice.26 Here, the
scaling dimension of operator VN,␮␯ is regularization independent. These vertex operators are irrelevant if K̃ ⬍ K̃c
= 2␲N2; in this parameter regime, the contribution of VN,xy to
various correlation functions can be perturbatively calculated.
Some other vertex operators can be generated under
renormalization group flow, but these vertex operators all
have algebraic correlations, with a regularization dependent
scaling dimension proportional to 1 / K̃. For instance, vertex
n
共h̃x − h̃y兲兴 has nonzero algebraic correoperator cos关2␲Nⵜmx
lation function in the YZ plane at long distance. Here, lattice
derivative ⵜmx is defined as ⵜmx f共rជ兲 = f共rជ + mx̂兲 − f共rជ兲. If we
regularize the theory by replacing lattice derivative ⵜmx with
i2 sin共mkx / 2兲 in the momentum space, the scaling dimension
n
共h̃x − h̃y兲兴 with n = 1 and arbitrary inof VN,mx,xy = cos关2␲Nⵜmx
2
teger m is 4␲N / K̃, and the scaling dimension is isotropic in
the whole YZ plane,
具VN,mx,xy,共0,0,0兲VN,mx,xy,共0,y,z兲典 ⬃
c共␪兲
共y + z2兲4␲N
2
2/K̃
.
共29兲
Here, c共␪兲 is a positive function of ␪ = arctan共z / y兲. Notice
that the rotation symmetry in the YZ plane is not restored
even at long length scale. The scaling dimension of
n
共h̃x − h̃y兲兴 rapidly increases with number n. Thus,
cos关2␲Nⵜmx
all the vertex operators are irrelevant when K̃ is small
enough, and there is a critical K̃c separating a crystalline
order and the algebraic liquid phase. At the liquid line, the
crystalline order parameter should have algebraic correlation
functions. Coefficient K̃ can be tuned from adding energetic
terms in the system. Recall that now the configurations with
zero average E␮␯ are favored by entropy, if we want to reduce K̃, we can add energetic terms that disfavor zero average E␮␯. For instance, if we give the flippable cubics a
smaller weight than the unflippable cubics, coefficient K̃
should be reduced.
The above results can be roughly understood from a
simple physical argument. Notice that all the flippable cubes
have zero average electric field, so the entropy effectively
favors flippable cubes. If K̃ ⬎ K̃c, the entropy strongly favors
flippable cubes; the system will develop crystalline order that
maximizes the number of flippable cubes. This kind of effect
134449-5
PHYSICAL REVIEW B 77, 134449 共2008兲
CENKE XU AND CONGJUN WU
is usually called “order by disorder.” It is also natural that the
crystalline order tends to be weakened or even melt if we
reduce K. Since the melting transition of the crystalline order
is driven by the proliferation of defect operators, the universality class of this transition is very similar to the Kosterlitz–
Thouless transition of the 2D XY model. Unusual Kosterlitz–
Thouless-like transition in three dimensions or higher
dimensions have also been discussed in other systems with
similar quasilocal symmetries,26 where the dimensionality of
the system is effectively reduced to two dimensions.
Recent Monte Carlo simulation20 shows that the whole
phase diagram of RK Hamiltonian 共1兲 is gapped with crystalline order, including the RK point. Our results based on
duality is consistent with this numerical results, and the
equal weight classical partition function should have
K̃ ⬎ K̃c. Our theory also predicts that if we turn on energetic
terms that favors unflippable cubes, there is a critical line
described by the Gaussian field theory 关Eq. 共27兲. This prediction can be checked by classical Monte Carlo simulations.
Another prediction which, in principle, can be made in our
formalism is the most favored crystalline order when K̃ is
slightly larger than K̃c. This requires a detailed analysis of
Berry’s phase of the vertex operators in the dual theory,
which we leave to future studies.
V. TOPOLOGICAL SECTOR
Now, let us discuss the topological sector, within which
every configuration can be connected to each other through
finite local movings depicted in Fig. 1. Topological sectors
are especially useful when one is dealing with a quantum
liquid state, where Landau’s classification of phases are no
longer applicable. In the original quantum dimer model on
square lattice, the topological sector on a torus is specified by
two integers,1 which can be interpreted as winding numbers
of electric fields. Here, we choose a lattice with even number
of sites in each axis and impose the periodic boundary condition. To specify a topological sector, one needs to know the
conserved quantities under local movings. It is straightforward to check that quantity mix,iy,xy = 兺izEi,xy for any 2D coordinate 共ix , iy兲 is a conserved quantity. Notation 兺iz means
summation over all the sites with the same x and y coordinates 共Fig. 3兲. However, these quantities are not independent.
For instance, by using constraint 共4兲, we have the following
identity:
m0,0,xy − m1,0,xy + m1,1,xy − m0,1,xy
= 兺 ⵜxⵜyExy
z
a
b
FIG. 3. 共Color online兲 共a兲 The conserved quantity mxy is defined
as the summation of all Exy on all the shaded squares along one z
axis. 共b兲 The view of the 3D lattice from the top. If the quantity mxy
is fixed on all the shaded squares shown in this figure, mxy is determined on the whole lattice.
topological sector on a three-dimensional torus, and the number scales with the linear size of the lattice.
VI. SUMMARY AND COMPARISON WITH OTHER
MODELS
This work studies a three-dimensional quantum resonating plaquette model motivated from a special SU共4兲 invariant point in spin-3 / 2 cold atom system. The effective low
energy physics of the problem can be mapped to a special
type of lattice gauge field. Our current QPM together with
previously studied 3D QDM24 and soft-graviton model27 all
have local constraint and low energy gauge field description
without gapless matter fields. Unlike the QDM and the softgraviton model, the QPM almost always suffers from the
proliferation of topological defects, and a generic stable algebraic liquid state as an analog of the photon phase of 3D
QDM does not exist.
The reason of the existence of a stable liquid phase of 3D
QDM, as well as the 3D soft-graviton model, has been discussed in Ref. 27. Both models with stable liquid phases are
self-dual gauge theories, with strong enough gauge symmetries in both the original description of the problem or the
dual theories, i.e., one cannot write down a gauge invariant
vertex operator that gaps out the liquid phase. In our current
QPM, the symmetry of the dual theory does not rule out all
the vertex operators, and gauge invariant vertex operators are
very relevant. Thus, in this type of bosonic quantum rotor
models, large enough gauge symmetries are necessary for
both sides of the duality to guarantee the existence of a stable
liquid phase if gapless matter field is absent.
iz
= 兺 5␩共i兲 − 共ⵜyⵜzEyz + ⵜzⵜxEzx兲 = 0.
共30兲
ACKNOWLEDGMENTS
iz
Thus, as long as one fixes the quantity mxy for one column
and one row in the XY plane, their values for the whole
lattice are determined. Conserved quantities associated with
Ezx and Eyz can be treated in the same way. Thus, we conclude that one needs infinite number of integers to specify a
The authors thank D. Arovas, L. Balents, S. Kivelson, and
S. Sondhi for helpful discussions. C.W. is supported by the
start up funding at the University of California, San Diego;
C.X. is supported by the Milton Funds of Harvard University.
134449-6
PHYSICAL REVIEW B 77, 134449 共2008兲
RESONATING PLAQUETTE PHASES IN SU共4兲…
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