Four-state Potts Model Research Articles

Weakening and disappearance of the jaming behavior in systems of self-propelled particles

In this study, we utilize the Eigen Microstate Approach (EMA) to investigate the active four-state Potts model, with a focus on the impact of volume repulsion. Our findings reveal that the introduction of long-range interactions significantly weakens or completely eliminates the jamming state, as indicated by changes in the probability curve of the second-largest eigen microstate. The emergence of sub-band structures in the system correlates with the appearance of a two-peak structure in the probability curve. We construct a phase diagram to explore the complex interplay between long-range interactions and emergent system behaviors. Additionally, we conjecture that the active four-state Potts model with long-range interactions undergoes a transition higher than second order.

Phase transitions of the four-state Potts model on the square and the honeycomb lattices

Abstract It is widely believed that the phase transition for the four-state ferromagnetic Potts model on the square lattice is of the pseudo-first order. Specifically, it is expected that first-order phase transition behavior is found on small lattices and that the true nature of second-order phase transition only emerges with large system sizes. It is also intuitively expected that for other geometries, the types of the associated phase transitions should be identical to that of the square lattice. However, after simulating more than 16 million spins for the four-state Pott model, we observe that a feature of first-order phase transition persists on the square lattice. Additionally, a characteristic of second-order phase transition already appears on a small honeycomb lattice. Indications of a pseudo-first-order phase transition were not found in our investigation. This suggests that a thorough analytic calculation may be required to develop a better understanding of the presented results.

Two-dimensional dilute Baxter-Wu model: Transition order and universality.

We investigate the critical behavior of the two-dimensional spin-1 Baxter-Wu model in the presence of a crystal-field coupling Δ with the goal of determining the universality class of transitions along the second-order part of the transition line as one approaches the putative location of the multicritical point. We employ extensive Monte Carlo simulations using two different methodologies: (i) a study of the zeros of the energy probability distribution, closely related to the Fisher zeros of the partition function, and (ii) the well-established multicanonical approach employed to study the probability distribution of the crystal-field energy. A detailed finite-size scaling analysis in the regime of second-order phase transitions in the (Δ,T) phase diagram supports previous claims that the transition belongs to the universality class of the four-state Potts model. For positive values of Δ, we observe the presence of strong finite-size effects, indicative of crossover effects due to the proximity of the first-order part of the transition line. Finally, we demonstrate how a combination of cluster and heat-bath updates allows one to equilibrate larger systems, and we demonstrate the potential of this approach for resolving the ambiguities observed in the regime of Δ≳0.

Spectral signatures of symmetry-breaking dynamical phase transitions.

Large deviation theory provides the framework to study the probability of rare fluctuations of time-averaged observables, opening new avenues of research in nonequilibrium physics. Some of the most appealing results within this context are dynamical phase transitions (DPTs), which might occur at the level of trajectories in order to maximize the probability of sustaining a rare event. While macroscopic fluctuation theory has underpinned much recent progress on the understanding of symmetry-breaking DPTs in driven diffusive systems, their microscopic characterization is still challenging. In this work we shed light on the general spectral mechanism giving rise to continuous DPTs not only for driven diffusive systems, but for any jump process in which a discrete Z_{n} symmetry is broken. By means of a symmetry-aided spectral analysis of the Doob-transformed dynamics, we provide the conditions whereby symmetry-breaking DPTs might emerge and how the different dynamical phases arise from the specific structure of the degenerate eigenvectors. In particular, we show explicitly how all symmetry-breaking features are encoded in the subleading eigenvectors of the degenerate subspace. Moreover, by partitioning configuration space into equivalence classes according to a proper order parameter, we achieve a substantial dimensional reduction which allows for the quantitative characterization of the spectral fingerprints of DPTs. We illustrate our predictions in several paradigmatic many-body systems, including (1) the one-dimensional boundary-driven weakly asymmetric exclusion process (WASEP), which exhibits a particle-hole symmetry-breaking DPT for current fluctuations, (2) the three- and four-state Potts model for spin dynamics, which displays discrete rotational symmetry-breaking DPTs for energy fluctuations, and (3) the closed WASEP which presents a continuous symmetry-breaking DPT into a time-crystal phase characterized by a rotating condensate.

Phase transitions in the four-state Potts model with competing exchange interactions: Application of the Wang-Landau algorithm

Phase transitions in the four-state Potts model with competing exchange interactions: Application of the Wang-Landau algorithm

Critical properties of quantum three- and four-state Potts models with boundaries polarized along the transverse field

By computing the low-lying energy excitation spectra with the density matrix renormalization group algorithm we show that boundaries polarized in the direction of the transverse field lead to scale-invariant conformal towers of states at the critical point of the quantum four-state Potts model - a special symmetric case of the Ashkin-Teller model. Furthermore, by direct comparison of the excitation spectra we phenomenologically establish the duality between the transverse-polarized and three-state-mixed boundary conditions at the four-state Potts critical point. Finally, for completeness, we verify that in the quantum three-state Potts model the "new" boundary conditions dual to the mixed ones can be realized by polarizing edge spins along the transverse field.

Universality in the two-dimensional dilute Baxter-Wu model.

We study the question of universality in the two-dimensional spin-1 Baxter-Wu model in the presence of a crystal field Δ. We employ extensive numerical simulations of two types, providing us with complementary results: Wang-Landau sampling at fixed values of Δ and a parallelized variant of the multicanonical approach performed at constant temperature T. A detailed finite-size scaling analysis in the regime of second-order phase transitions in the (Δ,T) phase diagram indicates that the transition belongs to the universality class of the four-state Potts model. Previous controversies with respect to the nature of the transition are discussed and attributed to the presence of strong finite-size effects, especially as one approaches the pentacritical point of the model.

Numerical study on a crossing probability for the four-state Potts model: Logarithmic correction to the finite-size scaling

Abstract A crossing probability for the critical four-state Potts model on an $L\times M$ rectangle on a square lattice is numerically studied. The crossing probability here denotes the probability that spin clusters cross from one side of the boundary to the other. First, by employing a Monte Carlo method, we calculate the fractal dimension of a spin cluster interface with a fluctuating boundary condition. By comparison of the fractal dimension with that of the Schramm–Loewner evolution (SLE), we numerically confirm that the interface can be described by the SLE with $\kappa=4$, as predicted in the scaling limit. Then, we compute the crossing probability of this spin cluster interface for various system sizes and aspect ratios. Furthermore, comparing with the analytical results for the scaling limit, which have been previously obtained by a combination of the SLE and conformal field theory, we numerically find that the crossing probability exhibits a logarithmic correction ${\sim} 1/\log(L M)$ to the finite-size scaling.

Paramagnetic phases of four-state Potts model on a Cayley tree in the presence of competing interactions

ABSTRACTWe consider a Potts model on a Cayley tree with competing nearest-neighbor and next-nearest-neighbor interactions restricted to spins belonging to the same branch of the tree with four spins and investigate the problem of phase transition for paramagnetic phases of this model.

Ferroic multipolar order and disorder in cyanoelpasolite molecular perovskites.

We use a combination of variable-temperature high-resolution synchrotron X-ray powder diffraction measurements and Monte Carlo simulations to characterize the evolution of two different types of ferroic multipolar order in a series of cyanoelpasolite molecular perovskites. We show that ferroquadrupolar order in [C3N2H5]2Rb[Co(CN)6] is a first-order process that is well described by a four-state Potts model on the simple cubic lattice. Likewise, ferrooctupolar order in [NMe4]2B[Co(CN)6] (B = K, Rb, Cs) also emerges via a first-order transition that now corresponds to a six-state Potts model. Hence, for these particular cases, the dominant symmetry breaking mechanisms are well understood in terms of simple statistical mechanical models. By varying composition, we find that the effective coupling between multipolar degrees of freedom-and hence the temperature at which ferromultipolar order emerges-can be tuned in a chemically sensible manner. This article is part of the theme issue 'Mineralomimesis: natural and synthetic frameworks in science and technology'.

A tunable magnetic metamaterial based on the dipolar four-state Potts model

Metamaterials, tunable artificial materials, are useful playgrounds to investigate magnetic systems. So far, artificial Ising spin systems have revealed features such as emergent magnetic monopoles1,2 and charge fragmentation3. Here we present a metasystem composed of a lattice of dipolarly coupled nanomagnets. The magnetic spin of each nanomagnet is constrained to lie along a body diagonal, which yields four possible spin states. We show that the magnetic ordering of this metasystem (antiferromagnetic, ferromagnetic or spin ice like) is determined by the spin states orientation relative to the underlying lattice. The dipolar four-state Potts model explains our experimental observations and sheds light on the role of symmetry, as well as short- and long-range dipolar magnetic interactions, in such non-Ising spin systems.

Scaling in the vicinity of the four-state Potts fixed point

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.

Extended canonical Monte Carlo methods: Improving accuracy of microcanonical calculations using a reweighting technique

Velazquez and Curilef [J. Stat. Mech. (2010); J. Stat. Mech. (2010)] have proposed a methodology to extend Monte Carlo algorithms that are based on canonical ensemble. According to our previous study, their proposal allows us to overcome slow sampling problems in systems that undergo any type of temperature-driven phase transition. After a comprehensive review about ideas and connections of this framework, we discuss the application of a reweighting technique to improve the accuracy of microcanonical calculations, specifically, the well-known multihistograms method of Ferrenberg and Swendsen [Phys. Rev. Lett. 63, 1195 (1989)]. As an example of application, we reconsider the study of the four-state Potts model on the square lattice L×L with periodic boundary conditions. This analysis allows us to detect the existence of a very small latent heat per site qL during the occurrence of temperature-driven phase transition of this model, whose size dependence seems to follow a power law qL(L)∝(1/L)z with exponent z≃0.26±0.02. Discussed is the compatibility of these results with the continuous character of temperature-driven phase transition when L→+∞.

The entanglement entropy for quantum system in one spatial dimension

Ground-state entanglement entropies were investigated for the one-dimension quantum two- and three-spin interaction models, the four-state Potts model, and the XXZ model with uniaxial single-ion-type anisotropy, which were obtained on an infinite-size lattice in one spatial dimension. Thus we show that the entanglement, a key concept of quantum information science, is quantified by the ground-state entanglement entropy. The relationships between ground-state entanglement entropy and quantum phase transition was analyzed. These results were obtained using the infinite matrix product state algorithm which works in the thermodynamical limit.

Frustrated electrons on a spatially anisotropic triangular lattice: Emergent competition of charge orders and exotic disorders due to thermal fluctuations

We study the interplay of correlation and thermal fluctuation in a system consisting of two species of classical particles with up and down spin on a geometrically frustrated anisotropic triangular lattice, described by an extended four-state Potts model. The model corresponds to the strong coupling limit of the extended Hubbard model at quarter-filling, which is known to host several competing charge ordered phases as well as an exotic quantum state called pinball liquid. The frustrated intersite Coulomb interactions together with the on-site Coulomb interaction generate macroscopically degenerate manifolds of low-energy states. They compete entropically at finite temperature and two characteristic states emerge; a threefold periodic charge ordered state and a quasi-one-dimensionally disordered state called ``good defect state'' characterized by the systematic generation of ferroelectric bonds. The two states show good correspondence with the threefold charge order and the pinball liquid in the extended Hubbard model, and are separated by the partial Mott transition taking place on one of the three sublattices of the triangular lattice.

Four-state standard Potts model

We investigated magnetic and thermal properties of a one-dimensional 4-state Potts model by the method of Kramers–Wannier transfer-matrix. Not only magnetic field but also the interactions between nearest J and second J′ neighbors were taken into account. It was shown that in the case of a single crystal, this magnet could have up to four frustration fields at which magnetization has jumps and entropy tends to non-zero values at T→0. We found the exact analytical expressions for these frustration fields. When an external field crosses over the frustration fields cardinal changes of magnetic structures take place.

Effects of Oxygen Vacancy on Ferroelectric Hysteresis under External Electric and Stress Fields

Effects of oxygen vacancies on properties of ferroelectric materials were investigated through their hysteresis characteristics using Monte Carlo simulation. The purpose of this work is to study the roles of oxygen vacancy in ferroelectrics. The 2D four-state Potts model was considered for dipole switching. Under the application of external electric and stress fields, for both static and periodic field conditions, the electrical polarization and strain were numerically calculated with varying stress field and oxygen-vacancy probability (or concentration). It was found that, with or without an applied stress, the coercivity, remnant and saturation of the ferroelectric and elastic properties decrease with increasing oxygen vacancies. Moreover, under an applied periodic stress, the shapes of strain-electric-field butterfly loops were found similar to those obtained from applying the combination of two static stresses.

Nonequilibrium scaling explorations on a two-dimensional Z(5)-symmetric model.

We have investigated the dynamic critical behavior of the two-dimensional Z(5)-symmetric spin model by using short-time Monte Carlo (MC) simulations. We have obtained estimates of some critical points in its rich phase diagram and included, among the usual critical lines the study of first-order (weak) transition by looking into the order-disorder phase transition. In addition, we also investigated the soft-disorder phase transition by considering empiric methods. A study of the behavior of β/νz along the self-dual critical line has been performed and special attention has been devoted to the critical bifurcation point, or Fateev-Zamolodchikov (FZ) point. First, by using a refinement method and taking into account simulations out of equilibrium, we were able to localize parameters of this point. In a second part of our study, we turned our attention to the behavior of the model at the early stage of its time evolution in order to find the dynamic critical exponent z as well as the static critical exponents β and ν of the FZ point on square lattices. The values of the static critical exponents and parameters are in good agreement with the exact results, and the dynamic critical exponent z≈2.28 very close to the four-state Potts model (z≈2.29).

Baxter-Wu model in a transverse magnetic field.

We investigate the low-energy properties as well as quantum and thermal phase transitions of the Baxter-Wu model in a transverse magnetic field. Our study relies on stochastic series expansion quantum Monte Carlo and on series expansions about the low- and high-field limits at zero temperature using the quantum finite-lattice method on the triangular lattice. The phase boundary consists of a second-order critical line in the four-state Potts model universality class starting from the pure Baxter-Wu limit meeting a first-order line connected to the zero-temperature transition point (h≈2.4, T=0). Both lines merge at a tricritical point approximately located at (h≈2.3J, T≈J).

Improving the efficiency of Monte Carlo simulations of systems that undergo temperature-driven phase transitions

Recently, Velazquez and Curilef proposed a methodology to extend Monte Carlo algorithms based on a canonical ensemble which aims to overcome slow sampling problems associated with temperature-driven discontinuous phase transitions. We show in this work that Monte Carlo algorithms extended with this methodology also exhibit a remarkable efficiency near a critical point. Our study is performed for the particular case of a two-dimensional four-state Potts model on a square lattice with periodic boundary conditions. This analysis reveals that the extended version of Metropolis importance sampling is more efficient than the usual Swendsen-Wang and Wolff cluster algorithms. These results demonstrate the effectiveness of this methodology to improve the efficiency of MC simulations of systems that undergo any type of temperature-driven phase transition.