Lu Liu, Anders W. Sandvik, WenAn Guo
We discuss the concept of typicality of quantum states at quantum-critical points, using projector Monte Carlo simulations of an $S=1/2$ bilayer Heisenberg antiferromagnet as an illustration. With the projection (imaginary) time $τ$ scaled as $τ=aL^z$, $L$ being the system length and $z$ the dynamic critical exponent (which takes the value $z=1$ in the bilayer model studied here), a critical point can be identified which asymptotically flows to the correct location and universality class with increasing $L$, independently of the prefactor $a$ and the initial state. Varying the proportionality factor $a$ and the initial state only changes the cross-over behavior into the asymptotic large-$L$ behavior. In some cases, choosing an optimal factor $a$ may also lead to the vanishing of the leading finite-size corrections. The observation of typicality can be used to speed up simulations of quantum criticality, not only within the Monte Carlo approach but also with other numerical methods where imaginary-time evolution is employed, e.g., tensor network states, as it is not necessary to evolve fully to the ground state but only for sufficiently long times to reach the typicality regime.
Zhe Fu, Wenan Guo, Henk W. J. Blöte
We explore the phase diagram of the O($n$) loop model on the square lattice in the $(x,n)$ plane, where $x$ is the weight of a lattice edge covered by a loop. These results are based on transfer-matrix calculations and finite-size scaling. We express the correlation length associated with the staggered loop density in the transfer-matrix eigenvalues. The finite-size data for this correlation length, combined with the scaling formula, reveal the location of critical lines in the diagram. For $n>>2$ we find Ising-like phase transitions associated with the onset of a checkerboard-like ordering of the elementary loops, i.e., the smallest possible loops, with the size of an elementary face, which cover precisely one half of the faces of the square lattice at the maximum loop density. In this respect, the ordered state resembles that of the hard-square lattice gas with nearest-neighbor exclusion, and the finiteness of $n$ represents a softening of its particle-particle potentials. We also determine critical points in the range $-2\leq n\leq 2$. It is found that the topology of the phase diagram depends on the set of allowed vertices of the loop model. Depending on the choice of this set, the $n>2$ transition may continue into the dense phase of the $n \leq 2$ loop model, or continue as a line of $n \leq 2$ O($n$) multicritical points.
Songbo Jin, Arnab Sen, Wenan Guo, Anders W. Sandvik
We consider the thermal phase transition from a paramagnetic to stripe-antiferromagnetic phase in the frustrated two-dimensional square-lattice Ising model with competing interactions J1<0 (nearest neighbor, ferromagnetic) and J2 >0 (second neighbor, antiferromagnetic). The striped phase breaks a Z4 symmetry and is stabilized at low temperatures for g=J2/|J1|>1/2. Despite the simplicity of the model, it has proved difficult to precisely determine the order and the universality class of the phase transitions. This was done convincingly only recently by Jin et al. [PRL 108, 045702 (2012)]. Here, we further elucidate the nature of these transitions and their anomalies by employing a combination of cluster mean-field theory, Monte Carlo simulations, and transfer-matrix calculations. The J1-J2 model has a line of very weak first-order phase transitions in the whole region 1/2<g<g*, where g* = 0.67(1). Thereafter, the transitions from g above g* are continuous and can be fully mapped, using universality arguments, to the critical line of the well known Ashkin-Teller model from its 4-state Potts point to the decoupled Ising limit. We also comment on the pseudo-first-order behavior at the Potts point and its neighborhood in the Ashkin-Teller model on finite lattices, which in turn leads to the appearance of similar effects in the vicinity of the multicritical point g* in the J1-J2 model. The continuous transitions near g* can therefore be mistaken to be first-order transitions, and this realization was the key to understanding the paramagnetic-striped transition for the full range of g>1/2. Most of our results are based on Monte Carlo calculations, while the cluster mean-field and transfer-matrix results provide useful methodological bench-marks for weakly first-order behaviors and Ashkin-Teller criticality.
Yougang Wang, Wenan Guo, Henk W. J. Blöte
We explore the physical properties of the completely packed O($n$) loop model on the square lattice, and its generalization to an Eulerian graph model, which follows by including cubic vertices which connect the four incoming loop segments. This model includes crossing bonds as well. Our study of the properties of this model involve transfer-matrix calculations and finite-size scaling. The numerical results are compared to existing exact solutions, including solutions of special cases of a so-called coloring model, which are shown to be equivalent with our generalized loop model. The latter exact solutions correspond with seven one-dimensional branches in the parameter space of our generalized loop model. One of these branches, describing the case of nonintersecting loops, is already known to correspond with the ordering transition of the Potts model. We find that another exactly solved branch, which describes a model with nonintersecting loops and cubic vertices, corresponds with a first-order Ising-like phase transition for $n>2$. For $1<n<2$, this branch can be interpreted in terms of a low-temperature O($n$) phase with corner-cubic anisotropy. For $n>2$ this branch is the locus of a first-order phase boundary between a phase with a hard-square lattice-gas like ordering, and a phase dominated by cubic vertices. The first-order character of this transition is in agreement with a mean-field argument.
Chengxiang Ding, Youjin Deng, Wenan Guo, Henk W. J. Blöte
We study a percolation problem based on critical loop configurations of the O($n$) loop model on the honeycomb lattice. We define dual clusters as groups of sites on the dual triangular lattice that are not separated by a loop, and investigate the the bond-percolation properties of these dual clusters. The universal properties at the percolation threshold are argued to match those of Kasteleyn-Fortuin random clusters in the critical Potts model. This relation is checked numerically by means of cluster simulations of several O($n$) models in the range $1\leq n \leq 2$. The simulation results include the percolation threshold for several values of $n$, as well as the universal exponents associated with bond dilution and the size distribution of the diluted clusters at the percolation threshold. Our numerical results for the exponents are in agreement with existing Coulomb gas results for the random-cluster model, which confirms the relation between both models. We discuss the renormalization flow of the bond-dilution parameter $p$ as a function of $n$, and provide an expression that accurately describes a line of unstable fixed points as a function of $n$, corresponding with the percolation threshold. Furthermore, the renormalization scenario indicates the existence, in $p$ versus $n$ diagram, another line of fixed points at $p=1$, which is stable with respect to $p$.
Yancheng Wang, Wenan Guo, Anders W. Sandvik
We present a quantum Monte Carlo study of the "quantum glass" phase of the 2D Bose-Hubbard model with random potentials at filling $ρ=1$. In the narrow region between the Mott and superfluid phases the compressibility has the form $κ\sim {\rm exp}(-b/T^α)+c$ with $α<1$ and $c$ vanishing or very small. Thus, at $T=0$ the system is either incompressible (a Mott glass) or nearly incompressible (a Mott-glass-like anomalous Bose glass). At stronger disorder, where a glass reappears from the superfluid, we find a conventional highly compressible Bose glass. On a path connecting these states, away from the superfluid at larger Hubbard repulsion, a change of the disorder strength by only $10\%$ changes the low-temperature compressibility by more than four orders of magnitude, lending support to two types of glass states separated by a phase transition or a sharp cross-over.
Zhe Wang, Shang-Qiang Ning, Zenan Liu, Junchen Rong, Yan-Cheng Wang, Zheng Yan, Wenan Guo
We design a (2+1))-dimensional [(2+1)D] quantum spin model in which spin-1/2 ladders are coupled through antiferromagnetic Ising interactions. The model hosts a quantum phase transition in the (2+1)D $Z_2$ universality class from the Haldane phase to the antiferromagnetic Ising ordered phase. We focus on studying the surface properties of three different surface configurations when the Ising couplings are tuned. Different behaviors are found on different surfaces. We find ordinary and two different extraordinary surface critical behaviors (SCBs) at the bulk critical point. The ordinary SCBs belong to the surface universality class of the classical 3D Ising bulk transition. One extraordinary SCBs is induced by the topological properties of the Haldane phase. Another extraordinary SCBs at the bulk critical point is induced by an unconventional surface phase transition where the surface develops an Ising order before the bulk. This surface transition is realized by coupling a (1+1)-dimensional [(1+1)D] $SU(2)_1$ CFT boundary to a (2+1)D bulk with $Z_2$ symmetry. We find that the transition is neither a (1+1)D $Z_2$ transition, expected based on symmetry consideration, nor a Kosterlitz-Thouless-like transition, violating the previous theoretical prediction. This new surface phase transition and related extraordinary SCBs deserve further analytical and numerical exploration.
Wanzhou Zhang, Laixi Li, Wenan Guo
We study the zero temperature phase diagram of hardcore bosons on the dual of the bowtie lattice. Two types of striped diagonal long-range order (striped order I and striped order II) are discussed. A state with type-II striped order and superfluidity is found, even without nearest-neighbor repulsion. The emergence of such a state is due to the inhomogeneity and the anisotropy of the lattice structure. However, neither the translational symmetry nor the symmetry between sublattices of the original lattice is broken. In this paper, we restrict a 'solid state' of lattice bosons as a diagonal long-range ordered state breaking either the translational symmetry of the original lattice or the symmetry of different sublattices. We thus name such a phase a striped superfluid phase (SSF). In the presence of a nearest-neighbor repulsion, we find two striped charge density wave phases(SCDW I and II) with boson density $ρ=1/2$ (with striped order I) and $ρ=2/3$ (with striped order II) respectively, when the hopping amplitude is small compared with the repulsion. The SCDW I state is a solid, in which the translational symmetry of the original lattice is broken. We observe a rather special first-order phase transition showing an interesting multi-loop hysteresis phenomenon between the two SCDW phases when the hopping term is small enough. This can be accounted for by the special degeneracy of the ground states near the classical limit. The SSF re-appears outside the two SCDW phases. The transition between the SCDW I and SSF phases is first order, while the transition between SCDW II and SSF phases is continuous. We find that the superfluid stiffness is anisotropic in the SSF states with and without repulsion. In the SSF with repulsion, the superfluid stiffness is subject to different types of anisotropy in the region near half filling and above 2/3-filling.
Chengxiang Ding, Zhe Fu, Wenan Guo, F. Y. Wu
In a recent paper (arXiv:0911.2514), one of us (FYW) considered the Potts model and bond and site percolation on two general classes of two-dimensional lattices, the triangular-type and kagome-type lattices, and obtained closed-form expressions for the critical frontier with applications to various lattice models. For the triangular-type lattices Wu's result is exact, and for the kagome-type lattices Wu's expression is under a homogeneity assumption. The purpose of the present paper is two-fold: First, an essential step in Wu's analysis is the derivation of lattice-dependent constants $A, B, C$ for various lattice models, a process which can be tedious. We present here a derivation of these constants for subnet networks using a computer algorithm. Secondly, by means of a finite-size scaling analysis based on numerical transfer matrix calculations, we deduce critical properties and critical thresholds of various models and assess the accuracy of the homogeneity assumption. Specifically, we analyze the $q$-state Potts model and the bond percolation on the 3-12 and kagome-type subnet lattices $(n\times n):(n\times n)$, $n\leq 4$, for which the exact solution is not known. To calibrate the accuracy of the finite-size procedure, we apply the same numerical analysis to models for which the exact critical frontiers are known. The comparison of numerical and exact results shows that our numerical determination of critical thresholds is accurate to 7 or 8 significant digits. This in turn infers that the homogeneity assumption determines critical frontiers with an accuracy of 5 decimal places or higher. Finally, we also obtained the exact percolation thresholds for site percolation on kagome-type subnet lattices $(1\times 1):(n\times n)$ for $1\leq n \leq 6$.
Zhe Wang, Fan Zhang, Wenan Guo
Using Quantum Monte Carlo simulations, we study the spin-1/2 Heisenberg model on a two-dimensional lattice formed by coupling diagonal ladders. The model hosts an antiferromagnetic Néel phase, a rung singlet product phase, and a topological none trivial Haldane phase, separated by two quantum phase transitions. We show that the two quantum critical points are all in the three-dimensional O(3) universality class. The properties of the two gapped phases, including the finite-size behavior of the string orders in the Haldane phase, are studied. We show that the surface formed by the ladders ends is gapless, while the surface exposed along the ladders is gapful, in the Haldane phase. Conversely, in the gapped rung singlet phase, the former surface is gapped, and the latter is gapless. We demonstrate that, although mechanisms of the two gapless modes are different, nonordinary surface critical behaviors are realized at both critical points on the gapless surfaces exposed by simply cutting bonds without fine-tuning the surface coupling required to reach a multicritical point in classical models. We also show that, on the gapped surfaces, the surface critical behaviors are in the ordinary class.
Xuan Zou, Shuo Liu, Wenan Guo
Using Monte Carlo simulations and finite-size scaling analysis, we show that the $q$-state clock model with $q=6$ on the simple cubic lattice with open surfaces has a rich phase diagram; in particular, it has an extraordinary-log phase, besides the ordinary and extraordinary transitions at the bulk critical point. We prove numerically that the presence of the intermediate extraordinary-log phase is due to the emergence of an O(2) symmetry in the surface state before the surface enters the $Z_{q}$ symmetry-breaking region as the surface coupling is increased at the bulk critical point, while O(2) symmetry emerges for the bulk. The critical behaviors of the extraordinary-log transition, as well as the ordinary and the special transition separating the ordinary and the extraordinary-log transition are obtained.
Xintian Wu, Nickolay Izmailian, Wenan Guo
Using the bond-propagation algorithm, we study the Ising model on a rectangle of size $M \times N$ with free boundaries. For five aspect ratios $ρ=M/N=1,2,4,8,16$, the critical free energy, internal energy and specific heat are calculated. The largest size reached is $M \times N=64\times 10^6$. The accuracy of the free energy reaches $10^{-26}$. Basing on these accurate data, we determine exact expansions of the critical free energy, internal energy and specific heat. With these expansions, we extract the bulk, surface and corner parts of free energy, internal energy and specific heat. The fitted bulk free energy density is given by $f_{\infty}=0.92969539834161021499(1)$, comparing with Onsager's exact result $f_{\infty}=0.92969539834161021506...$. We prove the conformal field theory(CFT) prediction of the corner free energy, in which the central charge of the Ising model is found to be $c=0.5\pm 1\times 10^{-10}$ comparing with the CFT result $c=0.5$. We find that not only the corner free energy but also the corner internal energy and specific heat are geometry independent, i.e., independent of aspect ratio. The implication of this finding on the finite scaling is discussed. In the second order correction of the free energy, we prove the geometry dependence predicted by CFT and find out a geometry independent constant beyond CFT. High order corrections are also obtained.
Wanzhou Zhang, Yancheng Wang, Wenan Guo
We systematically study an extended Bose-Hubbard model with atom hopping and atom-pair hopping in the presence of a three-body constraint on the triangular lattice. By means of large-scale Quantum Monte Carlo simulations, the ground-state phase diagram are studied. We find a continuous transition between the atomic superfluid phase and the pair superfluid when the ratio of the atomic hopping and the atom-pair hopping is adapted. We then focus on the interplay among the atom-pair hopping, the on-site repulsion and the nearest-neighbor repulsion. With on-site repulsion present, we observe first order transitions between the Mott Insulators and pair superfluid driven by the pair hopping. With the nearest-neighbor repulsion turning on, three typical solid phases with 2/3, 1 and 4/3-filling emerge at small atom-pair hopping region. A stable pair supersolid phase is found at small on-site repulsion. This is due to the three-body constraint and the pair hopping, which essentially make the model a quasi hardcore boson system. Thus the pair supersolid state emerges basing on the order-by-disorder mechanism, by which hardcore bosons avoid classical frustration on the triangular lattice. The transition between the pair supersolid and the pair superfluid is first order, except for the particle-hole symmetric point. We compare the results with those obtained by means of mean-field analysis.
WenJing Zhu, Chengxiang Ding, Long Zhang, Wenan Guo
We study the surface behavior of the two-dimensional columnar dimerized quantum antiferromagnetic XXZ model with easy-plane anisotropy, with particular emphasis on the surface critical behaviors of the (2+1)-dimensional quantum critical points of the model that belong to the classical three-dimensional O(2) universality class, for both $S=1/2$ and $S=1$ spins using quantum Monte Carlo simulations. We find completely different surface behaviors on two different surfaces of geometrical settings: the dangling-ladder surface, which is exposed by cutting a row of weak bonds, and the dangling-chain surface, which is formed by cutting a row of strong bonds along the direction perpendicular to the strong bonds of a periodic system. Similar to the Heisenberg limit, we find an ordinary transition on the dangling-ladder surface for both $S=1$ and $S=1/2$ spin systems. However, the dangling-chain surface shows much richer surface behaviors than in the Heisenberg limit. For the $S=1/2$ easy-plane model, at the bulk critical point, we provide evidence supporting an extraordinary surface transition with a long-range order established by effective long-range interactions due to bulk critical fluctuations. The possibility that the state is an extraordinary-log state seems unlikely. For the $S=1$ system, we find surface behaviors similar to that of the three-dimensional classical XY model with sufficiently enhanced surface coupling, suggesting an extraordinary-log state at the bulk critical point.
Hui Shao, WenAn Guo, Anders W. Sandvik
We present a projector quantum Monte Carlo study of the topological properties of the valence-bond-solid ground state in the $J$-$Q_3$ spin model on the square lattice. The winding number is a topological number counting the number of domain walls in the system and is a good quantum number in the thermodynamic limit. We study the finite-size behaviour and obtain the domain wall energy density for a topological nontrivial valence-bond-solid state.
H. W. J. Blöte, WenAn Guo, M. P. Nightingale
We study a self-dual generalization of the Baxter-Wu model, employing results obtained by transfer matrix calculations of the magnetic scaling dimension and the free energy. While the pure critical Baxter-Wu model displays the critical behavior of the four-state Potts fixed point in two dimensions, in the sense that logarithmic corrections are absent, the introduction of different couplings in the up- and down triangles moves the model away from this fixed point, so that logarithmic corrections appear. Real couplings move the model into the first-order range, away from the behavior displayed by the nearest-neighbor, four-state Potts model. We also use complex couplings, which bring the model in the opposite direction characterized by the same type of logarithmic corrections as present in the four-state Potts model. Our finite-size analysis confirms in detail the existing renormalization theory describing the immediate vicinity of the four-state Potts fixed point.
Yancheng Wang, Wenan Guo, Bernard Nienhuis, Henk W. J. Blöte
We define a percolation problem on the basis of spin configurations of the two dimensional XY model. Neighboring spins belong to the same percolation cluster if their orientations differ less than a certain threshold called the conducting angle. The percolation properties of this model are studied by means of Monte Carlo simulations and a finite-size scaling analysis. Our simulations show the existence of percolation transitions when the conducting angle is varied, and we determine the transition point for several values of the XY coupling. It appears that the critical behavior of this percolation model can be well described by the standard percolation theory. The critical exponents of the percolation transitions, as determined by finite-size scaling, agree with the universality class of the two-dimensional percolation model on a uniform substrate. This holds over the whole temperature range, even in the low-temperature phase where the XY substrate is critical in the sense that it displays algebraic decay of correlations.
Bernard Nienhuis, Wenan Guo, Henk W. J. Blöte
We show that the exactly solved low-temperature branch of the two-dimensional O($n$) model is equivalent with an O($n$) model with vacancies and a different value of $n$. We present analytic results for several universal parameters of the latter model, which is identified as a tricritical point. These results apply to the range $n \leq 3/2$, and include the exact tricritical point, the conformal anomaly and a number of scaling dimensions, among which the thermal and magnetic exponent, the exponent associated with crossover to ordinary critical behavior, and to tricritical behavior with cubic symmetry. We describe the translation of the tricritical model in a Coulomb gas. The results are verified numerically by means of transfer-matrix calculations. We use a generalized ADE model as an intermediary, and present the expression of the one-point distribution function in that language. The analytic calculations are done both for the square and the hexagonal lattice.
Zhehui Deng, Jinshan Wu, Wenan Guo
The $n$-index Rényi mutual information and transfer entropy for the two-dimensional kinetic Ising model with arbitrary single-spin dynamics in the thermodynamic limit are derived as functions of thermodynamic quantities. By means of Monte Carlo simulations with the Wolff algorithm, we calculate the information flows in the Ising model with the Metropolis dynamics and the Glauber dynamics. We find that, not only the global Rényi transfer entropy, but also the pairwise Rényi transfer entropy peaks in the disorder phase. Therefore, the Rényi information flows may be used as better tools than the Shannon counterparts in the study of phase transitions in complex dynamical systems.
Wanzhou Zhang, Wenan Guo, Kaj H Hoglund, Ling Wang, Anders W. Sandvik
We study the distribution of local magnetic susceptibilities in the two-dimensional antiferromagnetic S=1/2 Heisenberg model on various random clusters, in order to determine whether effects of edge disorder could be detected in NMR experiments (through the line shape, as given by the distribution of local Knight shifts). Although the effects depend strongly on the nature of the edge and the cluster size, our results indicate that line widths broader than the average shift should be expected even in clusters as large as $\approx 1000$ lattice spacing in diameter. Experimental investigations of the NMR line width should give insights into the magnetic structure of the edges.