Q-state Potts Models Research Articles

Generalized mutual information of quantum critical chains

We study the R\'enyi mutual information $\tilde{I}_n$ of the ground state of different critical quantum chains. The R\'enyi mutual information definition that we use is based on the well established concept of the R\'enyi divergence. We calculate this quantity numerically for several distinct quantum chains having either discrete $Z(Q)$ symmetries (Q-state Potts model with $Q=2,3,4$ and $Z(Q)$ parafermionic models with $Q=5,6,7,8$ and also Ashkin-Teller model with different anisotropies) or the $U(1)$ continuous symmetries(Klein-Gordon field theory, XXZ and spin-1 Fateev-Zamolodchikov quantum chains with different anisotropies). For the spin chains these calculations were done by expressing the ground-state wavefunctions in two special basis. Our results indicate some general behavior for particular ranges of values of the parameter $n$ that defines $\tilde{I}_n$. For a system, with total size $L$ and subsystem sizes $\ell$ and $L-\ell$, the$\tilde{I}_n$ has a logarithmic leading behavior given by $\frac{\tilde{c}_n}{4}\log(\frac{L}{\pi}\sin(\frac{\pi \ell}{L}))$ where the coefficient $\tilde{c}_n$ is linearly dependent on the central charge $c$ of the underlying conformal field theory (CFT) describing the system's critical properties.

Non compact continuum limit of two coupled Potts models

We study two Q-state Potts models coupled by the product of their energy operators, in the regime 2 < Q ⩽ 4 where the coupling is relevant. A particular choice of weights for the square lattice is shown to be equivalent to the integrable vertex model. It corresponds to a selfdual system of two antiferromagnetic Potts models, coupled ferromagnetically. We derive the Bethe ansatz equations and study them numerically for two arbitrary twist angles. The continuum limit is shown to involve two compact bosons and one non compact boson, with discrete states emerging from the continuum at appropriate twists. The non compact boson entails strong logarithmic corrections to the finite-size behaviour of the scaling levels, an understanding of which allows us to correct an earlier proposal for some of the critical exponents. In particular, we infer the full set of magnetic scaling dimensions (watermelon operators) of the Potts model.

Superstable cycles for antiferromagnetic Q-state Potts and three-site interaction Ising models on recursive lattices

We consider the superstable cycles of the Q-state Potts (QSP) and the three-site interaction antiferromagnetic Ising (TSAI) models on recursive lattices. The rational mappings describing the models’ statistical properties are obtained via the recurrence relation technique. We provide analytical solutions for the superstable cycles of the second order for both models. A particular attention is devoted to the period three window. Here we present an exact result for the third order superstable orbit for the QSP and a numerical solution for the TSAI model. Additionally, we point out a non-trivial connection between bifurcations and superstability: in some regions of parameters a superstable cycle is not followed by a doubling bifurcation. Furthermore, we use symbolic dynamics to understand the changes taking place at points of superstability and to distinguish areas between two consecutive superstable orbits.

CUDA programs for the GPU computing of the Swendsen–Wang multi-cluster spin flip algorithm: 2D and 3D Ising, Potts, and XY models

We present sample CUDA programs for the GPU computing of the Swendsen–Wang multi-cluster spin flip algorithm. We deal with the classical spin models; the Ising model, the q-state Potts model, and the classical XY model. As for the lattice, both the 2D (square) lattice and the 3D (simple cubic) lattice are treated. We already reported the idea of the GPU implementation for 2D models (Komura and Okabe, 2012). We here explain the details of sample programs, and discuss the performance of the present GPU implementation for the 3D Ising and XY models. We also show the calculated results of the moment ratio for these models, and discuss phase transitions. Program summaryProgram title: SWspinCatalogue identifier: AERM_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AERM_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 5632No. of bytes in distributed program, including test data, etc.: 14688Distribution format: tar.gzProgramming language: C, CUDA.Computer: System with an NVIDIA CUDA enabled GPU.Operating system: System with an NVIDIA CUDA enabled GPU.Classification: 23.External routines: NVIDIA CUDA Toolkit 3.0 or newerNature of problem:Monte Carlo simulation of classical spin systems. Ising, q-state Potts model, and the classical XY model are treated for both two-dimensional and three-dimensional lattices.Solution method:GPU-based Swendsen–Wang multi-cluster spin flip Monte Carlo method. The CUDA implementation for the cluster-labeling is based on the work by Hawick et al. [1] and that by Kalentev et al. [2].Restrictions:The system size is limited depending on the memory of a GPU.Running time:For the parameters used in the sample programs, it takes about a minute for each program. Of course, it depends on the system size, the number of Monte Carlo steps, etc.

Phase transitions and critical phenomena in a three-dimensional site-diluted Potts model

The phase transitions and critical phenomena in the three-dimensional (3D) site-diluted q-state Potts models on a simple cubic lattice are explored. We systematically study the phase transitions of the models for q=3 and q=4 on the basis of Wolff high-effective algorithm by the Monte–Carlo (MC) method. The calculations are carried out for systems with periodic boundary conditions and spin concentrations p=1.00–0.65. It is shown that introducing of weak disorder (p∼0.95) into the system is sufficient to change the first order phase transition into a second order one for the 3D 3-state Potts model, while for the 3D 4-state Potts model, such a phase transformation occurs when introducing strong disorder (p∼0.65). Results for 3D pure 3-state and 4-state Potts models (p=1.00) agree with conclusions of mean field theory. The static critical exponents of the specific heat α, susceptibility γ, magnetization β, and the exponent of the correlation radius ν are calculated for the samples on the basis of finite-size scaling theory.

Cluster densities at 2D critical points in rectangular geometries

Making use of the complete calculation [1] of the chiral six-point correlation functionwith the four ϕ1, 2 operators at the corners of an arbitrary rectangle and the point z = x + i y in the interior, for arbitrary central charge (equivalently, SLE parameter κ > 0), we calculate various quantities of interest for percolation (κ = 6) and many other two-dimensional critical points. In particular, we use C to specify the density at z of critical clusters conditioned to touch either or both vertical sides of the rectangle, with these sides ‘wired’, i.e. constrained to be in a single cluster, and the horizontal sides free. These quantities probe the structure of various cluster configurations, including those that contribute to the crossing probability. We first examine the effects of boundary conditions on C for the critical O(n) loop models in both high- and low-density phases and for both Fortuin–Kasteleyn (FK) and spin clusters in the critical Q-state Potts models. A Coulomb gas analysis then allows us to calculate the cluster densities with various conditionings in terms of the conformal blocks calculated in []. Explicit formulas generalizing Cardy’s horizontal crossing probability to these models (using previously known results) are also presented. These solutions are employed to generalize previous results demonstrating factorization of higher order correlation functions to the critical systems mentioned. An explicit formula for the density of critical percolation clusters that cross a rectangle horizontally with free boundary conditions is also given. Simplifications of the hypergeometric functions in our solutions for various models are presented. High-precision simulations verify these predictions for percolation and for the Q = 2- and 3-state Potts models, including both FK and spin clusters. Our formula for the density of crossing clusters in percolation with free boundary conditions is also verified.

ARNOLD TONGUES AND FEIGENBAUM EXPONENTS OF THE RATIONAL MAPPING FOR Q-STATE POTTS MODEL ON RECURSIVE LATTICE: Q &lt; 2

We considered Q-state Potts model on Bethe lattice in presence of external magnetic field for Q &lt; 2 by means of recursion relation technique. This allows to study the phase transition mechanism in terms of the obtained one dimensional rational mapping. The convergence of Feigenabaum α and δ exponents for the aforementioned mapping is investigated for the period doubling and three cyclic window. We regarded the Lyapunov exponent as an order parameter for the characterization of the model and discussed its dependence on temperature and magnetic field. Arnold tongues analogs with winding numbers w = 1/2, w = 2/4 and w = 1/3 (in the three cyclic window) are constructed for Q &lt; 2. The critical temperatures of the model are discussed and their dependence on Q is investigated. We also proposed an approximate method for constructing Arnold tongues via Feigenbaum δ exponent.

Geometric exponents, SLE and logarithmic minimal models

In statistical mechanics, observables are usually related to local degrees of freedom such as theQ<4 distinctstates of the Q-state Potts models or the heights of the restricted solid-on-solid models. In the continuumscaling limit, these models are described by rational conformal field theories, namely theminimal models for suitable p,p′. More generally, as in stochastic Loewner evolution(SLEκ), one can consider observables related to non-local degrees of freedom such as paths orboundaries of clusters. This leads to fractal dimensions or geometric exponents related tovalues of conformal dimensions not found among the finite sets of values allowed by therational minimal models. Working in the context of a loop gas with loop fugacityβ = −2cos(4π/κ), we use Monte Carlo simulations to measure the fractal dimensions of variousgeometric objects such as paths and the generalizations of cluster mass, cluster hull,external perimeter and red bonds. Specializing to the case where the SLE parameterκ = (4p′/p) is rationalwith p<p′, we argue that the geometric exponents are related to conformal dimensions found in theinfinitely extended Kac tables of the logarithmic minimal models . These theories describe lattice systems with non-local degrees of freedom. We presentresults for critical dense polymers , critical percolation , the logarithmic Ising model , the logarithmic tricritical Ising model as well as . Our results are compared with rigorous results fromSLEκ, with predictions from theoretical physics and with other numericalexperiments. Throughout, we emphasize the relationships betweenSLEκ, geometric exponents and the conformal dimensions of the underlying CFTs.

Strong Spatial Mixing and Rapid Mixing with Five Colours for the Kagome Lattice

We consider proper 5-colourings of the kagome lattice. Proper q-colourings correspond to configurations in the zero-temperature q-state anti-ferromagnetic Potts model. Salas and Sokal have given a computer assisted proof of strong spatial mixing on the kagome lattice for q ≥ 6 under any temperature, including zero temperature. It is believed that there is strong spatial mixing for q ≥ 4. Here we give a computer assisted proof of strong spatial mixing for q = 5 and zero temperature. It is commonly known that strong spatial mixing implies that there is a unique infinite-volume Gibbs measure and that the Glauber dynamics is rapidly mixing. We give a proof of rapid mixing of the Glauber dynamics on any finite subset of the vertices of the kagome lattice, provided that the boundary is free (not coloured). The Glauber dynamics is not necessarily irreducible if the boundary is chosen arbitrarily for q = 5 colours. The Glauber dynamics can be used to uniformly sample proper 5-colourings. Thus, a consequence of rapidly mixing Glauber dynamics is that there is fully polynomial randomised approximation scheme for counting the number of proper 5-colourings.

Critical points in coupled Potts models and critical phases in coupled loop models

We show how to couple two critical Q-state Potts models to yield a new self-dual critical point. We also present strong evidence of a dense critical phase near this critical point when the Potts models are defined in their completely packed loop representations. In the continuum limit, the new critical point is described by an SU(2) coset conformal field theory, while in this limit of the critical phase, the two loop models decouple. Using a combination of exact results and numerics, we also obtain the phase diagram in the presence of vacancies. We generalize these results to coupling two Potts models at different Q.

STATISTICAL PHYSICS APPROACH TO POLITICAL DISTRICTING PROBLEM

The Political Districting Problem is to partition a zone into several electoral districts subject to some constraints such as contiguity, population equality, etc. In this paper, we apply statistical physics methods to Political Districting Problem. This political problem is mapped to a q-state Potts model system, and the political constraints are written in the form of an energy function with the interactions between sites or external fields acting on the system. Districting into q voter districts is equivalent to finding the ground state of this q-state Potts model. We illustrate this problem by districting Taipei city and compare it to a computer-generated artificial system.

Thermodynamic formalism and large deviation functions in continuous time Markov dynamics

The thermodynamic formalism, which was first developed for dynamical systems and then applied to discrete Markov processes, turns out to be well suited for continuous time Markov processes as well, provided the definitions are interpreted in an appropriate way. Moreover, it can be reformulated in terms of the generating function of an observable, and then extended to other observables. In particular, the simple observable K giving the number of events occurring over a given time interval turns out to contain already the signature of dynamical phase transitions. For mean-field models in equilibrium, and in the limit of large systems, the formalism is rather simple to apply and shows how thermodynamic phase transitions may modify the dynamical properties of the systems. This is exemplified with the q-state mean-field Potts model, for which the Ising limit q = 2 is found to be qualitatively different from the other cases. To cite this article: V. Lecomte et al., C. R. Physique 8 (2007).

ADE and SLE

We point out that the probability law of a single domain wall separating clusters in ADE lattice models in a simply connected domain is identical to that of corresponding chordal curves in the lattice O(n) and Q-state Potts models, for suitable n or Q. They are conjectured to be described in the scaling limit by chordal SLEκ with κ rational and >2. However in a multiply connected domain the laws can differ from those for the corresponding O(n) or Potts model. The correspondence also sheds light on the scaling limit of multiple curves.

Interfacial wetting in the 4-state Potts model: A cluster approach

We formulate an inhomogeneous generalization of the bond-cluster approach to deal with interfacial phenomena in q-state Potts models. We apply this formalism to the wetting by the disordered phase of the interface between two bulk phases. For the case q = 4 we obtain the full interfacial profiles, provide an estimate for the wetting temperature and verify the predicted logarithmic divergence of the wetting layer.

Ferromagnetic Potts model under an external magnetic field: An exact renormalization group approach

The q-state ferromagnetic Potts model under a non-zero magnetic field coupled with the 0^th Potts state was investigated by an exact real-space renormalization group approach. The model was defined on a family of diamond hierarchical lattices of several fractal dimensions d_F. On these lattices, the renormalization group transformations became exact for such a model when a correlation coupling that singles out the 0^th Potts state was included in the Hamiltonian. The rich criticality presented by the model with q=3 and d_F=2 was fully analyzed. Apart from the Potts criticality for the zero field, an Ising-like phase transition was found whenever the system was submitted to a strong reverse magnetic field. Unusual characteristics such as cusps and dimensional reduction were observed on the critical surface.

Complex-temperature phase diagram of Potts and RSOS models

We study the phase diagram of Q-state Potts models, for Q = 4 cos 2 ( π / p ) a Beraha number ( p > 2 integer), in the complex-temperature plane. The models are defined on L × N strips of the square or triangular lattice, with boundary conditions on the Potts spins that are periodic in the longitudinal ( N) direction and free or fixed in the transverse ( L) direction. The relevant partition functions can then be computed as sums over partition functions of an A p − 1 type RSOS model, thus making contact with the theory of quantum groups. We compute the accumulation sets, as N → ∞ , of partition function zeros for p = 4 , 5 , 6 , ∞ and L = 2 , 3 , 4 and study selected features for p > 6 and/or L > 4 . This information enables us to formulate several conjectures about the thermodynamic limit, L → ∞ , of these accumulation sets. The resulting phase diagrams are quite different from those of the generic case (irrational p). For free transverse boundary conditions, the partition function zeros are found to be dense in large parts of the complex plane, even for the Ising model ( p = 4 ). We show how this feature is modified by taking fixed transverse boundary conditions.

Relations between Potts and RSOS models on a torus

We study the relationship between Q-state Potts models and staggered RSOS models of the A p − 1 type on a torus, with Q 1 / 2 = 2 cos ( π / p ) . In general the partition functions of these models differ due to clusters of non-trivial topology. However we find exact identities, valid for any temperature and any finite size of the torus, between various modified partition functions in the two pictures. The field theoretic interpretation of these modified partition function is discussed.

Calculation of Partition Functions by Measuring Component Distributions

A new algorithm is presented, which allows us to calculate numerically the partition function Z for systems, which can be described by arbitrary interaction graphs and lattices, e.g., Ising models or Potts models (for arbitrary values q>0), including random or diluted models. The new approach is suitable for large systems. The basic idea is to measure the distribution of the number of connected components in the corresponding Fortuin-Kasteleyn representation and to compare with the case of zero degrees of freedom, where the exact result Z=1 is known. As an application, d=2 and d=3 dimensional ferromagnetic Potts models are studied, and the critical values qc, where the transition changes from second to first order, are determined. Large systems of sizes N=1000(2) and N=100(3) are treated. The critical value qc(d=2)=4 is confirmed and qc(d=3)=2.35(5) is found.

High-temperature series expansions for random Potts models

We discuss recently generated high-temperature series expansions for the free energy and the susceptibility of random-bond q-state Potts models on hypercubic lattices. Using the star-graph expansion technique quenched disorder averages can be calculated exactly for arbitrary uncorrelated coupling distributions while keeping the disorder strength p as well as the dimension d as symbolic parameters. We present analyses of the new series for the susceptibility of the Ising (q=2) and 4-state Potts model in three dimensions up to order 19 and 18, respectively, and compare our findings with results from field-theoretical renormalization group studies and Monte Carlo simulations.

Growth model on (1 + 1) dimensions with local relaxation and discrete number of orientations

We introduced in this work a simple model for studying the texture formation during the electrodeposition process. Monte Carlo simulations were used to describe the formation of the deposits, and scaling concepts were also employed to characterize their growth and roughness properties. Particles are randomly deposited on a substrate, and their main axis can be aligned in a discrete set of possible directions. The final orientation of the deposited particle is determined by the interaction energy with its first neighboring particles and substrate temperature. Particle interactions are chosen according to the q-state ferromagnetic Potts model hamiltonian. Simulations were performed on (1 + 1) dimensions, for different values of temperature and substrate size. We have found different behaviors at low and high temperatures. Only at zero temperature the system reaches an absorbing state with all the layers occupied by particles oriented in the same direction. At this temperature we found the dynamic, roughness and growth exponents of the model, which satisfies the well known Family–Vicsek scaling relation for the self-affine interfaces. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)