Three-state Antiferromagnetic Potts Model Research Articles

Surface criticality of the antiferromagnetic Potts model

We study the three-state antiferromagnetic Potts model on the simple-cubic lattice, paying attention to the surface critical behaviors. When the nearest neighboring interactions of the surface is tuned, we obtain a phase diagram similar to the XY model, owing to the emergent O(2) symmetry of the bulk critical point. For the ordinary transition, we get $y_{h1}=0.780(3)$, $\eta_\parallel=1.44(1)$, and $\eta_\perp=0.736(6)$; for the special transition, we get $y_s=0.59(1)$, $y_{h1}=1.693(2)$, $\eta_\parallel=-0.391(4)$, and $\eta_\perp=-0.179(5)$; in the extraordinary-log phase, the surface correlation function $C_\parallel(r)$ decays logarithmically, with decaying exponent $q=0.60(2)$, however, the correlation $C_\perp(r)$ still decays algebraically, with critical exponent $\eta_\perp=-0.442(5)$. If the ferromagnetic next nearest neighboring surface interactions are added, we find two transition points, the first one is a special point between the ordinary phase and the extraordinary-log phase, the second one is a transition between the extraordinary-log phase and the $Z_6$ symmetry-breaking phase, with critical exponent $y_{\rm s}=0.41(2)$. The scaling behaviors of the second transition is very interesting, the surface spin correlation function $C_\parallel(r)$ and the surface squared staggered magnetization at this point decays logarithmically, with exponent $q=0.37(1)$; however, the surface structure factor with the smallest wave vector and the correlation function $C_\perp(r)$ satisfy power-law decaying, with critical exponents $\eta_\parallel=-0.69(1)$ and $\eta_\perp=-0.37(1)$, respectively.

A comprehensive neural networks study of the phase transitions of Potts model

Using the techniques of neural networks (NN), we study the three-dimensional (3D) five-state ferromagnetic Potts model on the cubic lattice as well as the two-dimensional (2D) three-state antiferromagnetic Potts model on the square lattice. Unlike the conventional approach, here we follow the idea employed by Li et al (2018 Ann. Phys., NY 391 312–331). Specifically, instead of numerically generating numerous objects for the training, the whole or part of the theoretical ground state configurations of the studied models are considered as the training sets. Remarkably, our investigation of these two models provides convincing evidence for the effectiveness of the method of preparing training sets used in this study. In particular, the results of the 3D model obtained here imply that the NN approach is as efficient as the traditional method since the signal of a first order phase transition, namely tunneling between two channels, determined by the NN method is as strong as that calculated with the Monte Carlo technique. Furthermore, the outcomes associated with the considered 2D system indicate even little partial information of the ground states can lead to conclusive results regarding the studied phase transition. The achievements reached in our investigation demonstrate that the performance of NN, using certain amount of the theoretical ground state configurations as the training sets, is impressive.

The critical behavior of the two-dimensional three-state Potts model on a triangular lattice with quenched disorder

The critical behavior of the two-dimensional three-state antiferromagnetic Potts model with quenched disorder on a triangular lattice is investigated by the Monte Carlo method. Static critical exponents for the susceptibility γ, the magnetization β, the specific heat α, and the exponent of the correlation radius ν at spin concentrations p = 0.90; 0.80; 0.70; 0.65 are calculated on the basis of the finite-size scaling theory. The critical exponents are found to be increasing with a rise in disorder without violating the feasibility of scaling equation 2βν+γν=d, while relations γ/ν and β/ν remain unchanged. This behavior of the critical exponents we have come to associate with the weak universality of a critical behavior typical for disordered systems.

Ground State of an Antiferromagnetic Three-State Potts Model on a Triangular Lattice with Competing Interactions

The ground state of the spin structures described by an antiferromagnetic three-state Potts model on a triangular lattice is studied with allowance for the next-nearest neighbors. The numerical data obtained by the Monte Carlo method are used to reveal the ranges of ordered and disordered phases in these structures.

Entanglement entropy and massless phase in the antiferromagnetic three-state quantum chiral clock model

The von Neumann entanglement entropy is used to estimate the critical point $h_c/J \simeq 0.143(3)$ of the mixed ferro-antiferromagnetic three-state quantum Potts model $H = \sum_i [ J ( X_i X_{i+1}^{\,2} + X_i^{\,2} X_{i+1} ) - h\, R_i ]$, where $X_i$ and $R_i$ are standard three-state Potts spin operators and $J>0$ is the antiferromagnetic coupling parameter. This critical point value gives improved estimates for two Kosterlitz-Thouless transition points in the antiferromagnetic ($\beta < 0$) region of the $\Delta$--$\beta$ phase diagram of the three-state quantum chiral clock model, where $\Delta$ and $\beta$ are, respectively, the chirality and coupling parameters in the clock model. These are the transition points $\beta_c \simeq - 0.143(3)$ at $\Delta = \frac12$ between incommensurate and commensurate phases and $\beta_c \simeq - 7.0(1)$ at $\Delta = 0$ between disordered and incommensurate phases. The von Neumann entropy is also used to calculate the central charge $c$ of the underlying conformal field theory in the massless phase $h \le h_c$. The estimate $c \simeq 1$ in this phase is consistent with the known exact value at the particular point $h/J = -1$ corresponding to the purely antiferromagnetic three-state quantum Potts model. The algebraic decay of the Potts spin-spin correlation in the massless phase is used to estimate the continuously varying critical exponent $\eta$.

Phase transitions in the two-dimensional antiferromagnetic Potts model on a triangular lattice

Phase transitions and thermodynamic properties in the two-dimensional three-state antiferromagnetic Potts model on a triangular lattice are investigated using the Monte Carlo method and the histogram analysis of the data. It is shown that pronounced first-order phase transitions are observed in this model for systems with rather large linear dimensions (L > 120). No first-order PTs are observed for systems with L < 120.

Monte Carlo study of the antiferromagnetic three-state Potts model with a staggered polarization field on the square lattice

Using the Wang-Landau Monte Carlo method, we study the antiferromagnetic (AF) three-state Potts model with a staggered polarization field on the square lattice. We obtain two phase transitions; one belongs to the ferromagnetic three-state Potts universality class, and the other to the Ising universality class. The phase diagram obtained is quantitatively consistent with the transfer matrix calculation. The Ising transition in the large nearest-neighbor interaction limit has been made clear by the detailed analysis of the energy density of states.

Ordered phase and phase transitions in the three-dimensional generalized six-state clock model

We study the three-dimensional generalized six-state clock model at values of the energy parameters, at which the system is considered to have the same behavior as the stacked triangular antiferromagnetic Ising model and the three-state antiferromagnetic Potts model. First, we investigate ordered phases by using the Monte Carlo twist method (MCTM). We confirmed the existence of an incompletely ordered phase (IOP1) at intermediate temperature, besides the completely ordered phase (COP) at low-temperature. In this intermediate phase, two neighboring states of the six-state model mix, while one of them is selected in the low temperature phase. We examine the fluctuation the mixing rate of the two states in IOP1 and clarify that the mixing rate is very stable around 1:1. The high temperature phase transition is investigated by using non-equilibrium relaxation method (NERM). We estimate the critical exponents beta=0.34(1) and nu=0.66(4). These values are consistent with the 3D-XY universality class. The low temperature phase transition is found to be of first-order by using MCTM and the finite-size-scaling analysis.

Sine-Gordon description of the scaling three-state Potts antiferromagnet on the square lattice

The scaling limit as T->0 of the antiferromagnetic three-state Potts model on the square lattice is described by the sine-Gordon quantum field theory at a specific value of the coupling. We show that the correspondence follows unambigously from an analysis of the sine-Gordon operator space based on locality, and that the scalar operators carrying solitonic charge play an essential role in the description of the lattice model. We then evaluate the correlation functions within the form factor approach and give a number of universal predictions that can be checked in numerical simulations.

Ferrimagnetic phases of the Blume-Emery-Griffiths model and the Potts model on the diamond lattice

The phase transitions in the Blume-Emery-Griffiths (BEG) model and antiferromagnetic Potts model on the diamond lattice are investigated using the cluster-variation method in the pair approximation. The ferrimagnetic phases are found to be different from those on the simple-cubic lattice. The phase diagrams of the BEG model are also calculated. In the vicinity of the parameter line where the BEG model reduces to the three-state antiferromagnetic Potts model, new types of phase diagram are obtained. The results are different from those of the mean-field theory, which is a good approximation only for large coordination number of the lattice.

Ordered phase and scaling inZnmodels and the three-state antiferromagnetic Potts model in three dimensions

Based on a renormalization-group picture of ${Z}_{n}$ symmetric models in three dimensions, we derive a scaling law for the ${Z}_{n}$ order parameter in the ordered phase. An existing Monte Carlo calculation on the three-state antiferromagnetic Potts model, which has the effective ${Z}_{6}$ symmetry, is shown to be consistent with the proposed scaling law. It strongly supports the renormalization-group picture that there is a single massive ordered phase, although an apparently rotationally symmetric region in the intermediate temperature was observed numerically.

Vortex dynamics and entropic forces in antiferromagnets and antiferromagnetic Potts models.

It is well known in models with an interface representation, such as the dimer model, the triangular Ising antiferromagnet, the six-vertex ice model, and the three-state antiferromagnetic Potts model on the square lattice, that topological defects of opposite charge are attracted with an entropically-driven Coulomb force. We examine the Potts model in detail, show explicitly how this force is felt through local fields, and calculate the defects' mobility. We then take two approaches to measuring this force numerically. First, we quench a random initial state to zero temperature and measure the density of defects rho(t) as a function of time. While this gives some evidence for a local force, we compare it with a free diffusion experiment, and show that the asymptotic decay of rho(t) depends on the initial distribution of defects rather than the forces between them. Second, we set up initial conditions with a single pair of vortices, and measure the force between them as a function of distance. This gives reasonable agreement with theory, although finite-size effects and a lack of ergodicity play a significant role.

Magnetic properties of the Blume-Emery-Griffiths model in the vicinity of an antiferromagnetic Potts line

The Blume-Emery-Griffiths model is studied for the range of parameters in a close neighborhood of the antiferromagnetic three-state Potts model. The cluster variational method in pair approximation is adopted for the studies. Particular attention is paid to the properties of ferrimagnetic and staggered quadrupolar phases which occur on a bipartite lattice. In particular, the isothermal susceptibility is calculated, revealing its splitting into two branches, each with different temperature behavior. The reentrant ferrimagnetism phenomenon is also discussed.

Phase transitions in the three-state antiferromagnetic Potts model for different lattices

The ordered phases in the three-state antiferromagnetic Potts model for different lattices are investigated using the cluster-variation method in the pair approximation. There are two types of low-temperature ordered phases. The intermediate-temperature ordered phases are also analyzed.

Intermediate-temperature ordering in a three-state antiferromagnetic Potts model

We investigate the ordering in a three-state antiferromagnetic Potts model on a simple cubic lattice by using an efficient cluster-flipping Monte Carlo algorithm. By considering the degeneracy due to the sublattices in two different ways, we arrive at two entirely different phases [rotationally symmetric (RS) and permutationally symmetric sublattice (PSS)] for the intermediate temperature region. Each of these phases has been claimed to be the correct phase by different authors in a recent controversy over ordering in this system. We show that the system moves from a broken sublattice symmetry phase at low temperature to the disordered phase by going through the RS phase at intermediate temperature, and that the PSS phase results from taking the sublattice degeneracy into account in an incorrect fashion.

Two Types of Ordering in the Three-State Antiferromagnetic Potts Model on 3D Lattices: Monte Carlo Evidence

Two Types of Ordering in the Three-State Antiferromagnetic Potts Model on 3D Lattices: Monte Carlo Evidence

Mean-field analysis of antiferromagnetic three-state Potts model with next-nearest-neighbor interaction

The three-state Potts model with antiferromagnetic nearest-neighbor (NN) and ferromagnetic next-nearest-neighbor (NNN) interaction is investigated within a mean-field theory. We find that the phase diagram contains two kinds of ordered phases, the so-called broken-sublattice-symmetry and permutationally-symmetric-sublattice phases, separated by a discontinuous phase transition line. The order-disorder transition is continuous for weak NNN interactions and becomes a discontinuous transition, via a tricritical point, when the strength of the NNN interaction is increased.

Low-temperature phase of the three-state antiferromagnetic Potts model on the simple-cubic lattice.

The three-state antiferromagnetic Potts model on the simple-cubic lattice is investigated using the cluster variation method in the cube and the star-cube approximations. The broken-sublattice-symmetry phase is found to be stable in the whole low-temperature region, contrary to previous results obtained using a modified cluster variation method. The tiny free-energy difference between the broken-sublattice-symmetry and the permutationally-symmetric sublattice phases is calculated in the two approximations and turns out to be smaller in the (more accurate) star-cube approximation than in the cube one.

MULTI-SPIN CODING OF THE MONTE CARLO SIMULATION OF THE THREE-STATE RANDOM POTTS MODEL AND THE BLOCK-SPIN TRANSFORMATION

The multi-spin coding of the Monte Carlo simulation of the three-state Potts model on the simple cubic lattice is presented. The ferromagnetic (F) model, the antiferromagnetic (AF) model, and the random mixture of the F and AF couplings are treated. The multispin coding technique is also applied to the block-spin transformation. The block-spin transformation of the F Potts model is simply realized by the majority rule, whereas the AF three-state Potts model is transformed to the block spin having a six-fold symmetry.

Rotationally symmetric ordered phase in the three-state antiferromagnetic Potts model.

We investigate the three-state antiferromagnetic Potts model on a simple cubic lattice with a cluster flipping Monte Carlo simulation algorithm in the temperature region below the transition into disorder at ${\mathit{T}}_{\mathit{c}1}$. We find both the well-established broken-sublattice-symmetry (BSS) phase at low temperature and a rotationally symmetric phase at higher temperature, but below ${\mathit{T}}_{\mathit{c}1}$. The properties of the second phase and the transition temperature to the BSS phase are in disagreement with recent cluster-variation and Monte Carlo simulation results, but in agreement with simulations by Kolesik and Suzuki. \textcopyright{} 1996 The American Physical Society.