Two-dimensional Potts Model Research Articles

Evaporation-condensation transition of the two-dimensional Potts model in the microcanonical ensemble

The evaporation-condensation transition of the Potts model on a square lattice is numerically investigated by the Wang-Landau sampling method. An intrinsically system-size-dependent discrete transition between supersaturation state and phase-separation state is observed in the microcanonical ensemble by changing constrained internal energy. We calculate the microcanonical temperature, as a derivative of microcanonical entropy, and condensation ratio, and perform a finite-size scaling of them to indicate the clear tendency of numerical data to converge to the infinite-size limit predicted by phenomenological theory for the isotherm lattice gas model.

Monte Carlo simulation of the two-dimensional Potts model using nonextensive statistics

The standard Potts model is investigated in the framework of nonextensive statistical mechanics. We performed Monte Carlo simulations on two-dimensional lattices with linear sizes ranging from 16 to 64 using the Metropolis algorithm, where the classical Boltzmann–Gibbs transition probabilities were modified for the nonextensive case. We found that the Potts model undergoes a phase transition in the nonextensive scenario. We established the order of the phase transition and we computed the critical temperature for different values of the Tsallis entropic index.

Influence of aperiodic modulations on first-order transitions: Numerical study of the two-dimensional Potts model

We study the Potts model on a rectangular lattice with aperiodic modulations in its interactions along one direction. Numerical results are obtained using the Wolff algorithm and for many lattice sizes, allowing for a finite-size scaling analyses to be carried out. Three different self-dual aperiodic sequences are employed, which leads to more precise results, since the exact critical temperature is known. We analyze two models, with 6 and 15 number of states: both present first-order transitions on their uniform versions. We show that the Harris-Luck criterion, originally introduced in the study of continuous transitions, is obeyed also for first-order ones. Also, we show that the new universality class that emerges for relevant aperiodic modulations depends on the number of states of the Potts model, as obtained elsewhere for random disorder, and on the aperiodic sequence. We determine the occurrence of log-periodic behavior, as expected for models with aperiodic modulated interactions.

Evidences of the Instability Fixed Points of First-Order Phase Transitions

We study the dynamics of the first-order phase transitions in the two-dimensional Potts model driven by a linearly varying temperature using a finite-time scaling with extended dynamic Monte Carlo renormalization group method. It is found that, for sufficiently large lattice sizes, the flows of apparent exponents of different temperature sweep rates upon renormalization show characteristics that are markedly distinct from those of continuous transitions and are argued to result from the instability fixed points involved and provide a method for estimating the associated instability exponents.

Bulk and boundary critical behaviour of thin and thick domain walls in the two-dimensionalPotts model

The geometrical critical behaviour of the two-dimensionalQ-state Potts model is usually studied in terms of the Fortuin–Kasteleyn (FK) clusters, ortheir surrounding loops. In this paper we study a quite different geometrical object: thespin clusters, defined as connected domains where the spin takes a constant value. Unlikethe usual loops, the domain walls separating different spin clusters can cross and branch.Moreover, they come in two versions, ‘thin’ or ‘thick’, depending on whether they separatespin clusters of different or identical colours. For these reasons their critical behaviour isdifferent from, and richer than, those of FK clusters. We develop a transfer matrixtechnique enabling the formulation and numerical study of spin clusters even whenQ is not an integer. We further identify geometrically the crossing events which give rise toconformal correlation functions. We study the critical behaviour both in the bulk,and at a boundary with free, fixed, or mixed boundary conditions. This leads toinfinite series of fundamental critical exponents, in the bulk and at the boundary, valid for 0 ≤ Q ≤ 4, that describe the insertion of thin and thick domain walls. We argue that these exponents imply that the domain walls are ‘thin’and ‘thick’ also in the continuum limit. A special case of the bulk exponents is derivedanalytically from a massless scattering approach.

Duality and Fisher zeros in the two-dimensional Potts model on a square lattice

A phenomenological approach to the ferromagnetic two-dimensional (2D) Potts model on square lattice is proposed. Our goal is to present a simple functional form that obeys the known properties possessed by the free energy of the q-state Potts model. The duality symmetry of the 2D Potts model together with the known results on its critical exponent α allows us to fix consistently the details of the proposed expression for the free energy. The agreement of the analytic ansatz with numerical data in the q=3 case is very good at high and low temperatures as well as at the critical point. It is shown that the q>4 cases naturally fit into the same scheme and that one should also expect a good agreement with numerical data. The limiting q=4 case is shortly discussed.

Finite-time scaling via linear driving: Application to the two-dimensional Potts model

We apply finite-time scaling to the q-state Potts model with q=3 and 4 on two-dimensional lattices to determine its critical properties. This consists in applying to the model a linearly varying external field that couples to one of its q states to manipulate its dynamics in the vicinity of its criticality and that drives the system out of equilibrium and thus produces hysteresis and in defining an order parameter other than the usual one and a nonequilibrium susceptibility to extract coercive fields. From the finite-time scaling of the order parameter, the coercivity, and the hysteresis area and its derivative, we are able to determine systematically both static and dynamic critical exponents as well as the critical temperature. The static critical exponents obtained in general and the magnetic exponent delta in particular agree reasonably with the conjectured ones. The dynamic critical exponents obtained appear to confirm the proposed dynamic weak universality but unlikely to agree with recent short-time dynamic results for q=4. Our results also suggest an alternative way to characterize the weak universality.

Curvature-driven coarsening in the two-dimensional Potts model

We study the geometric properties of polymixtures after a sudden quench in temperature. We mimic these systems with the q -states Potts model on a square lattice with and without weak quenched disorder, and their evolution with Monte Carlo simulations with nonconserved order parameter. We analyze the distribution of hull-enclosed areas for different initial conditions and compare our results with recent exact and numerical findings for q=2 (Ising) case. Our results demonstrate the memory of the presence or absence of long-range correlations in the initial state during the coarsening regime and exhibit superuniversality properties.

Optimized broad-histogram simulations for strong first-order phase transitions: droplet transitions in thelarge-Q Potts model

The numerical simulation of strongly first-order phase transitions has remained anotoriously difficult problem even for classical systems due to the exponentially suppressed(thermal) equilibration in the vicinity of such a transition. In the absence of efficientupdate techniques, a common approach for improving equilibration in Monte Carlosimulations is broadening the sampled statistical ensemble beyond the bimodal distributionof the canonical ensemble. Here we show how a recently developed feedback algorithm cansystematically optimize such broad-histogram ensembles and significantly speed upequilibration in comparison with other extended ensemble techniques such asflat-histogram, multicanonical and Wang–Landau sampling. We simulate, as a prototypicalexample of a strong first-order transition, the two-dimensional Potts model with up toQ = 250 different states in large systems. The optimized histogram develops a distinct multi-peakstructure, thereby resolving entropic barriers and their associated phase transitions in thephase coexistence region—such as droplet nucleation and annihilation, and droplet–striptransitions for systems with periodic boundary conditions. We characterize theefficiency of the optimized histogram sampling by measuring round-trip timesτ(N, Q) across the phase transition for samples comprised ofN spins. While we findpower-law scaling of τ versus N for small and , we observe a crossover to exponential scaling for largerQ. These results demonstrate that despite the ensemble optimization, broad-histogramsimulations cannot fully eliminate the supercritical slowing down at strongly first-ordertransitions.

Crossing bonds in the random-cluster model

We derive the scaling dimension associated with crossing bonds in the random-cluster representation of the two-dimensional Potts model by means of a mapping on the Coulomb gas. The scaling field associated with crossing bonds appears to be irrelevant on the critical as well as on the tricritical branch. The latter result stands in a remarkable contrast with the existing result for the tricritical O(n) model that crossing bonds are relevant. Although the O(1) model is equivalent with the q=2 random-cluster model, the crossing-bond exponents obtained for these two models appear to be different. We provide an explanation of this peculiar observation. In order to obtain an independent confirmation of the Coulomb gas result for the crossing-bond exponent, we perform a finite-size-scaling analysis based on numerical transfer-matrix calculations.

Critical dynamics of the two-dimensional random-bond Potts model with nonequilibrium Monte Carlo simulations

We study two-dimensional q -state random-bond Potts models for both q=8 and q=5 with a linearly varying temperature. By applying a successive Monte Carlo renormalization group procedure, both the static and dynamic critical exponents are obtained for randomness amplitudes (the strong to weak coupling ratio) of r_{0}=3 , 10, 15, and 20. The correlation length exponent nu increases with disorder from less than to larger than unity and this variation is justified by the good collapse of the specific heat near the critical region. The specific heat exponent is obtained by the usual hyperscaling relation alpha=2-dnu and thus indicates no possibility of the activated dynamic scaling. Both r_{0} and q have effects on the critical dynamics of the disordered systems, which can be seen from variations of the rate exponent, the hysteresis exponent, and the dynamic critical exponent. Implications of these results are discussed.

Multifractal statistics of the local order parameter at random critical points: Application to wetting transitions with disorder

Disordered systems present multifractal properties at criticality. In particular, as discovered by Ludwig [A.W.W. Ludwig, Nucl. Phys. B 330, 639 (1990)] in the case of a diluted two-dimensional Potts model, the moments rho(q) (r) of the local order parameter rho(r) scale with a set x(q) of nontrivial exponents x(q) not = qx(1). We reexamine these ideas to incorporate more recent findings: (i) whenever a multifractal measure w(r) normalized over space sum(r) w(r) = 1 occurs in a random system, it is crucial to distinguish between the typical values and the disorder-averaged values of the generalized moments Y(q) = sum(r) w(q) (r), since they may scale with different generalized dimensions D(q) and D(q), and (ii), as discovered by Wiseman and Domany [S. Wiseman and E. Domany, Phys. Rev. E 52, 3469 (1995)], the presence of an infinite correlation length induces a lack of self-averaging at critical points for thermodynamic observables, in particular for the order parameter. After this general discussion, valid for any random critical point, we apply these ideas to random polymer models that can be studied numerically for large sizes and good statistics over the samples. We study the bidimensional wetting or the Poland-Scheraga DNA model with loop exponents c = 1.5 (marginal disorder) and c = 1.75 (relevant disorder). Finally, we argue that the presence of finite Griffiths-ordered clusters at criticality determines the asymptotic value x(q-->infinity) = d and the minimal value alpha(min) = D(q-->infinity) = d - x(1) of the typical multifractal spectrum f(alpha).

Invaded cluster algorithm for a tricritical point in a diluted Potts model

The invaded cluster approach is extended to the two-dimensional Potts model with annealed vacancies by using the random-cluster representation. Geometrical arguments are used to propose the algorithm which converges to the tricritical point in the two-dimensional parameter space spanned by temperature and the chemical potential of the vacancies. The tricritical point is identified as a simultaneous onset of the percolation of a Fortuin-Kasteleyn cluster and of a percolation of the "geometrical disorder cluster." The location of the tricritical point and the concentration of vacancies for q=1,2,3 are found to be in good agreement with the best known results. Scaling properties of the percolating scaling cluster and related critical exponents are also presented.

Liquid Crystals in Two Dimensions: First-Order Phase Transitions and Nonuniversal Critical Behavior

Liquid crystals in two dimensions undergo a first-order isotropic-to-quasi-nematic transition, provided the particle interactions are sufficiently "sharp and narrow." This implies phase coexistence between isotropic and quasi-nematic domains, separated by interfaces. The corresponding line tension is determined and shown to be very small, giving rise to strong interface fluctuations. When the interactions are no longer "sharp and narrow," the transition becomes continuous, with nonuniversal critical behavior obeying hyperscaling and approximately resembling the two-dimensional Potts model.

Critical behavior of repulsive lineark-mers on triangular lattices

Monte Carlo (MC) simulations and finite-size scaling analysis have been carried out to study the critical behavior in a submonolayer two-dimensional gas of repulsive linear $k$-mers on a triangular lattice at coverage $k∕(2k+1)$. A low-temperature ordered phase, characterized by a repetition of alternating files of adsorbed $k$-mers separated by $k+1$ adjacent empty sites, is separated from the disordered state by an order-disorder phase transition occurring at a finite critical temperature, ${T}_{c}$. The MC technique was combined with the recently reported free-energy minimization criterion approach (FEMCA) [F. Rom\'a et al., Phys. Rev. B 68, 205407 (2003)] to predict the dependence of the critical temperature of the order-disorder transformation. The dependence on $k$ of the transition temperature, ${T}_{c}(k)$, observed in MC is in qualitative agreement with FEMCA. In addition, an accurate determination of the critical exponents has been obtained for adsorbate sizes ranging between $k=1$ and 3. For $k>1$, the results reveal that the system does not belong to the universality class of the two-dimensional Potts model with $q=3$ ($k=1$, monomers). Based on symmetry concepts, we suggested that the behavior observed for $k=1$, 2, and 3 could be generalized to include larger particle sizes $(k\ensuremath{\geqslant}2)$.

Two-dimensional critical Potts and its tricritical shadow

These notes give examples of how suitably defined geometrical objects encode in their fractal structure thermal critical behavior. The emphasis is on the two-dimensional Potts model for which two types of spin clusters can be defined. Whereas the Fortuin-Kasteleyn clusters describe the standard critical behavior, the geometrical clusters describe the tricritical behavior that arises when including vacant sites in the pure Potts model. Other phase transitions that allow for a geometrical description discussed in these notes include the superfluid phase transition and Bose-Einstein condensation.

Targeted manipulation of grain boundaries and triple junctions on thin films using lasers: A Potts model simulation

A two-dimensional Potts model is used to simulate the motion of a laser spot on a polycrystalline thin film. The model is used to demonstrate the feasibility of reshaping individual grain boundaries by either dragging grain boundaries or by repositioning triple junctions.

Realizing non-Abelian statistics in time-reversal-invariant systems

We construct a series of 2+1-dimensional models whose quasiparticles obey non-Abelian statistics. The adiabatic transport of quasiparticles is described by using a correspondence between the braid matrix of the particles and the scattering matrix of 1+1-dimensional field theories. We discuss in depth lattice and continuum models whose braiding is that of SO(3) Chern-Simons gauge theory, including the simplest type of non-Abelian statistics, involving just one type of quasiparticle. The ground-state wave function of an SO(3) model is related to a loop description of the classical two-dimensional Potts model. We discuss the transition from a topological phase to a conventionally-ordered phase, showing in some cases there is a quantum critical point.

Simulations of viscous shape relaxation in shuffled foam clusters

We simulate the shape relaxation of foam clusters and compare them with the time exponential expected for Newtonian fluid. Using two-dimensional Potts Model simulations, we artificially create holes in a foam cluster and shuffle it by applying shear strain cycles. We reproduce the experimentally observed time exponential relaxation of cavity shapes in the foam as a function of the number of strain steps. The cavity rounding up results from local rearrangement of bubbles, due to the conjunction of both a large applied strain and local bubble wall fluctuations.

Spontaneous edge order and geometric aspects of two-dimensional Potts models

Using suitable Monte Carlo methods and finite-size scaling, we investigate critical and tricritical surface phenomena of two-dimensional Potts models. For the critical two- and three-state models, we determine a surface scaling dimension describing percolation properties of the so-called Potts clusters near the edges. On this basis, we propose an exact expression describing this exponent for the whole critical branch. For tricritical Potts models we find that varying the surface coupling constant or the surface magnetic field can induce a continuous phase transition. At bulk tricriticality and sufficiently strong surface couplings, spontaneous one-dimensional order occurs on the edges. We determine several critical exponents describing these edge transitions. On the basis of these results and conformal field theory, we conjecture exact expressions for these exponents.