Two-dimensional Three-state Potts Model Research Articles

Form factors of the tricritical three-state Potts model in its scaling limit

We compute the form factors of the order and disorder operators, together with those of the stress–energy tensor, of a two-dimensional three-state Potts model with vacancies along its thermal deformation at the critical point. At criticality, the model is described by the non-diagonal partition function of the unitary minimal model M6,7 of conformal field theories and is accompanied by an internal S 3 symmetry. The off-critical thermal deformation is an integrable massive theory that is still invariant under S 3. The presence of infinitely many conserved quantities, whose spin spectrum is related to the exceptional Lie algebra E 6, allows us to determine the analytic S-matrix, the exact mass spectrum and the matrix elements of local operators of this model in an exact non-perturbative way. We use the spectral representation series of the correlators and the fast convergence of these series to compute several universal ratios of the renormalization group.

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.

Tangent space to the manifold of critical classical Hamiltonians representable by tensor networks.

We introduce a scheme to perform Monte Carlo renormalization group with the coupling constants of the system Hamiltonian encoded in a tensor network. With this scheme we compute the tangent space to the manifold of the critical Hamiltonians representable by a tensor network at the nearest-neighbor critical coupling for three models: the two- and three-dimensional Ising models and the two-dimensional three-state Potts model.

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.

Universality class of the structural phase transition in the normal phase of cuprate superconductors.

The tetragonal-orthorhombic structural phase transition of oxygen atoms in the basal plane of YBa_{2}Cu_{3}O_{6+δ} high-T_{C} cuprate superconductors is studied numerically. By mapping the system onto the asymmetric next-nearest-neighbor Ising model, we characterize this phase transition. Results indicate the degrees of critical behavior. We show that this phase transition occurs at the temperature T_{C}≃0.148eV in the thermodynamic limit. By analyzing the critical exponents, it is found that this universality class displays some common features, with the two-dimensional three-state Potts model universality class, although the possibility of other universality classes cannot be ruled out. Conformal invariance at T=T_{c} is investigated using the Schramm-Loewner evolution (SLE) technique, and it is found that the SLE diffusivity parameter for this system is 3.34±0.01.

Critical behavior of a triangular lattice Ising AF/FM bilayer

We study a bilayer Ising spin system consisting of antiferromagnetic (AF) and ferromagnetic (FM) triangular planes, coupled by ferromagnetic exchange interaction, by standard Monte Carlo and parallel tempering methods. The AF/FM bilayer is found to display the critical behavior completely different from both the single FM and AF constituents as well as the FM/FM and AF/AF bilayers. Namely, by finite-size scaling (FSS) analysis we identify at the same temperature a standard Ising transition from the paramagnetic to FM state in the FM plane that induces a ferrimagnetic state with a finite net magnetic moment in the AF plane. At lower temperatures there is another phase transition, that takes place only in the AF plane, to different ferrimagnetic state with spins on two sublattices pointing parallel and on one sublattice antiparallel to the spins on the FM plane. FSS indicates that the corresponding critical exponents are close to the two-dimensional three-state ferromagnetic Potts model values.

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.

Block-Spin Distribution Functions at Criticality

The block-spin distribution functions are studied for the two-dimensional Ising model, the three-dimensional Ising model and the two-dimensional three-state Potts model using the Monte Carlo technique. The nearest-neighbour block-spin correlation function for the two-dimensional Ising model is also calculated. These functions which reflect the local properties of the systems are proved to be good probes for the critical phenomena.

Application of simulated tempering and magnetizing to a two-dimensional Potts model

We apply the simulated tempering and magnetizing (STM) method to the two-dimensional three-state Potts model in an external magnetic field in order to investigate STM’s applicability further. The temperature and the external field are treated as dynamical variables updated during the STM simulations. On the basis of adequate information obtained by STM for several lattice sizes L × L (up to 160×160), we also perform a number of conventional canonical simulations of larger lattices in order to illustrate the crossover behavior of the Potts model in an external field with increasing L. The temperature and external field for the larger lattice size simulations are chosen by extrapolation of the detailed information obtained by STM. We present a careful analysis of the crossover-scaling behavior at the phase transitions with respect to the lattice size as well as the temperature and external field. The crossover behavior is clearly observed in the simulations in agreement with theoretical predictions.

Crossover scaling in the two-dimensional three-state Potts model

We apply simulated tempering and magnetizing (STM) Monte Carlo simulations to the two-dimensional three-state Potts model in an external magnetic field in order to investigate the crossover scaling behaviour in the temperature-field plane at the Potts critical point and towards the Ising universality class for negative magnetic fields. Our data set has been generated by STM simulations of several square lattices with sizes up to 160 x 160 spins, supplemented by conventional canonical simulations of larger lattices at selected simulation points. We present careful scaling and finite-size scaling analyses of the crossover behaviour with respect to temperature, magnetic field and lattice size.

Determination of the dynamic and static critical exponents of the two-dimensional three-state Potts model using linearly varying temperature

We employ a successive Monte Carlo renormalization group procedure in the presence of a linearly varying temperature to study the three-state (q=3) Potts model on square lattices. By matching correlation functions of different lattice sizes at different renormalization levels, a rate exponent r associated with the temperature sweep rate R and the correlation length exponent nu are obtained. The dynamic critical exponent z is then obtained by the scaling law z=r-1/nu derived in our method. The results of z=2.171(62) for q=3 in this work and the previously obtained z=2.15(13) for q=2 seem to support the extension of the "weak universality" hypothesis to dynamic critical behavior. With these calculated exponents, the dynamic scaling forms of both specific heat and order parameter at various R are presented and all the other static critical exponents are determined, which verify our method. Discussions are made on how to improve the accuracy of the estimation of the dynamic critical exponent.

Potts correlators and the static three-quark potential

We discuss the two- and three-point correlators in the two-dimensional three-state Pottsmodel in the high temperature phase of the model. By using the form factor approach andperturbed conformal field theory methods we are able to describe both the large distanceand the short distance behaviours of the correlators. We compare our predictions with a setof high precision Monte Carlo simulations (performed on the triangular lattice realizationof the model) finding a complete agreement in both regimes. In particular we use thetwo-point correlators to fix the various non-universal constants involved in thecomparison (whose determination is one of the results of our analysis) and then usethese constants to compare numerical results and theoretical predictions for thethree-point correlator with no free parameter. Our results can be used to shed somelight on the behaviour of the three-quark correlator in the confining phase of the(2 + 1)-dimensional SU(3) lattice gauge theory which is related by dimensional reduction to the three-spin correlatorin the high temperature phase of the three-state Potts model. The picture which emerges isthat of a smooth crossover between a Δ-type law at short distances and aY-type law at large distances.

Sweeny and Gliozzi dynamics for simulations of Potts models in the Fortuin-Kasteleyn representation.

We compare the correlation times of the Sweeny and Gliozzi dynamics for two-dimensional Ising and three-state Potts models, and the three-dimensional Ising model for the simulations in the percolation representation. The results are also compared with Swendsen-Wang and Wolff cluster dynamics. It is found that Sweeny and Gliozzi dynamics have essentially the same dynamical critical behavior. Contrary to Gliozzi's claim [Phys. Rev. E 66, 016115 (2002)], the Gliozzi dynamics has critical slowing down comparable to that of other cluster methods. For the two-dimensional Ising model, both Sweeny and Gliozzi dynamics give good fits to logarithmic size dependences for the correlation times; for two-dimensional three-state Potts model, their dynamical critical exponent z is 0.49+/-0.01; the three-dimensional Ising model has z=0.37+/-0.02.

Susceptibility amplitude ratio in the two-dimensional three-state Potts model

We analyze Monte Carlo simulation and series-expansion data for the susceptibility of the three-state Potts model in the critical region. The amplitudes of the susceptibility on the high- and the low-temperature sides of the critical point as extracted from the Monte Carlo data are in good agreement with those obtained from the series expansions and their (universal) ratio compares quite well with a recent quantum field theory prediction by Delfino and Cardy.

Zeros of the singular partition function of the fully frustrated superconducting array in the complex temperature plane

We have analyzed complex zeros of the singular partition function of the fully frustrated square superconducting array. Zero contours of the singular partition function for different lattice sizes collapse into single scaling curves, providing an evidence that the predetermined exponent $\ensuremath{\nu}=0.813$ is correct. By analyzing zero-contour lines we have estimated the universal specific-heat amplitude ratio ${A}_{+}{/A}_{\ensuremath{-}}=1.04(4)$ and checked the Abe-Itzykson-Pearson-Zuber relation. By comparing several critical parameters, we have suggested possibilities that the fully frustrated square superconducting array belongs to the same universality class with the two-dimensional three-state Potts model.

Dynamic critical behavior of the Swendsen-Wang algorithm: The two-dimensional three-state Potts model revisited

We have performed a high-precision Monte Carlo study of the dynamic critical behavior of the Swendsen-Wang algorithm for the two-dimensional 3-state Potts model. We find that the Li-Sokal bound ($\tau_{int,E} \geq const \times C_H$) is almost but not quite sharp. The ratio $\tau_{int,E} / C_H$ seems to diverge either as a small power ($\approx 0.08$) or as a logarithm.

Estimation of the central charge by Monte Carlo simulations

Using an idea put forward by J.L. Cardy, we demonstrate how to obtain a good estimate for the central charge c of a conformally invariant field theory by Monte Carlo simulations of the corresponding statistical model away from critically, but in the scaling region; the two-dimensional three-state Potts model in a magnetic field is used as a test case. Determining c in this way has become feasible because of a big increase in the availability of computational power and because of the invention of stochastic cluster algorithms, which allow an efficient simulation of statistical models near criticality. This method provides a reliable way to determine c for models, for which the other known methods have failed up to now.

Critical relaxation of the two-dimensional three-state Potts model with conserved dynamics

The dynamical critical exponent z is obtained using the finite-size scaling method for the two-dimensional three-state Potts model with conserved dynamics. It is observed that there is a change in the dynamical behaviour of the system at the size n=15. For finite lattices of sizes n or=15, the asymptotic behaviour is observed and z is calculated as z=2.78+or-0.20.

Dynamics of q-state Potts models at the percolation threshold

The critical dynamics of the q-state Potts model at the percolation threshold is studied using some of the techniques that have been applied recently to diluted Ising systems. Various formulations of single-spin-flip dynamics are considered and shown to lead to a breakdown of conventional dynamic scaling. Furthermore, the predictions are checked by performing Monte Carlo simulations of the two-dimensional three-state Potts model at the bond percolation threshold.

A scaling method for asymptotic analysis of power series

A new method is proposed for asymptotic analysis of power series. In the method the author uses the relation that an n-term power series of a singular quantity with critical exponent beta behaves asymptotically like n- beta at the phase transition point. The method is tested on known series with satisfactory results and used to analyse the power series of the two-dimensional three-state Potts model of Enting, in good agreement with den Nijs' conjecture.