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 134449-2 PHYSICAL REVIEW B 77, 134449 共2008兲 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2h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2h兲 has to satisfy all the symmetries in Eq. 共19兲; the simplest form it can take is cos关2h̃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2N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 + 3k2 + 3k2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ei2N关hx共0兲−h̃y共0兲兴e−i2N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 134449-4 PHYSICAL REVIEW B 77, 134449 共2008兲 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⫽ ⵜⵜE − 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 − E0 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2N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 4N 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 = 2N2; 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2Nⵜmx 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2Nⵜmx 2 teger m is 4N / 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4N 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2Nⵜmx 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… 1 D. S. Rokhsar and S. A. Kivelson, Phys. Rev. Lett. 61, 2376 共1988兲. 2 E. Fradkin and S. A. Kivelson, Mod. Phys. Lett. B 4, 225 共1990兲. 3 R. Moessner and S. L. Sondhi, Phys. Rev. Lett. 86, 1881 共2001兲. 4 D. A. Huse, W. Krauth, R. Moessner, and S. L. Sondhi, Phys. Rev. Lett. 91, 167004 共2003兲. 5 M. Hermele, T. Senthil, and M. P. A. Fisher, Phys. Rev. B 72, 104404 共2005兲. 6 F. Zhou, Int. J. Mod. Phys. B 17, 2643 共2003兲. 7 E. Demler and F. Zhou, Phys. Rev. Lett. 88, 163001 共2002兲. 8 C. Wu, J. P. Hu, and S. C. Zhang, Phys. Rev. Lett. 91, 186402 共2003兲. 9 C. Wu, Mod. Phys. Lett. B 20, 1707 共2006兲. 10 C. Wu, Phys. Rev. Lett. 95, 266404 共2005兲. 11 S. Chen, C. Wu, S. Zhang, and Y. Wang, Phys. Rev. B 72, 214428 共2005兲. 12 C. Wu and S. C. Zhang, Phys. Rev. B 71, 155115 共2005兲. 13 H.-H. Tu, G.-M. Zhang, and L. Yu, Phys. Rev. B 74, 174404 共2006兲. 14 H.-H. Tu, G.-M. Zhang, and L. Yu, Phys. Rev. B 76, 014438 共2007兲. 15 Y.-Q. Li, M. Ma, D.-N. Shi, and F.-C. Zhang, Phys. Rev. B 60, 12781 共1999兲. 16 M. van den Bossche, P. Azaria, P. Lecheminant, and F. Mila, Phys. Rev. Lett. 86, 4124 共2001兲. 17 D. P. Arovas and A. Auerbach, Phys. Rev. B 38, 316 共1988兲. 18 S. Sachdev and N. Read, Int. J. Mod. Phys. B 5, 219 共1991兲. 19 M. V. D. Bossche, F. C. Zhang, and F. Mila, Eur. Phys. J. B 17, 367 共2000兲. 20 S. Pankov, R. Moessner, and S. L. Sondhi, Phys. Rev. B 76, 104436 共2007兲. 21 I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Phys. Rev. Lett. 59, 799 共1987兲. 22 D. P. Arovas, Phys. Rev. B 77, 104404 共2008兲. 23 N. Read and S. Sachdev, Phys. Rev. B 42, 4568 共1990兲. 24 R. Moessner and S. L. Sondhi, Phys. Rev. B 68, 184512 共2003兲. 25 E. Fradkin, D. A. Huse, R. Moessner, V. Oganesyan, and S. L. Sondhi, Phys. Rev. B 69, 224415 共2004兲. 26 A. Paramekanti, L. Balents, and M. P. A. Fisher, Phys. Rev. B 66, 054526 共2002兲. 27 C. Xu, Phys. Rev. B 74, 224433 共2006兲. 28 C. Xu and M. P. A. Fisher, Phys. Rev. B 75, 104428 共2007兲. 29 J. Kondev and C. L. Henley, Nucl. Phys. B 464, 540 共1996兲. 134449-7

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Download PDF

advertisement