Antiferromagnetic Potts Model Research Articles

Improved Mixing Bounds for the Anti-Ferromagnetic Potts Model on Z2

AbstractThe authors of this paper consider the anti-ferromagnetic Potts model on the the integer lattice Z2. The model has two parameters: q, the number of spins, and λ = exp(−β), where β 4 is ‘inverse temperature’. It is known that the model has strong spatial mixing if q > 7, or if q = 7 and λ = 0 or λ > 1/8, or if q = 6 and λ = 0 or λ > 1/4. The λ = 0 case corresponds to the model in which configurations are proper q-colourings of Z2. It is shown that the system has strong spatial mixing for q ≥ 6 and any λ. This implies that Glauber dynamics is rapidly mixing (so there is a fully-polynomial randomised approximation scheme for the partition function), and also that there is a unique infinite-volume Gibbs state. We also show that strong spatial mixing occurs for a larger range of λ than was previously known for q = 3, q = 4 and q = 5.

Glauber Dynamics on Trees: Boundary Conditions and Mixing Time

We give the first comprehensive analysis of the effect of boundary conditions on the mixing time of the Glauber dynamics in the so-called Bethe approximation. Specifically, we show that spectral gap and the log-Sobolev constant of the Glauber dynamics for the Ising model on an n-vertex regular tree with plus-boundary are bounded below by a constant independent of n at all temperatures and all external fields. This implies that the mixing time is O(log n) (in contrast to the free boundary case, where it is not bounded by any fixed polynomial at low temperatures). In addition, our methods yield simpler proofs and stronger results for the spectral gap and log-Sobolev constant in the regime where there are multiple phases but the mixing time is insensitive to the boundary condition. Our techniques also apply to a much wider class of models, including those with hard-core constraints like the antiferromagnetic Potts model at zero temperature (proper colorings) and the hard--core lattice gas (independent sets).

Extremal optimization at the phase transition of the three-coloring problem.

We investigate the phase transition in vertex coloring on random graphs, using the extremal optimization heuristic. Three-coloring is among the hardest combinatorial optimization problems and is equivalent to a 3-state anti-ferromagnetic Potts model. Like many other such optimization problems, it has been shown to exhibit a phase transition in its ground state behavior under variation of a system parameter: the graph's mean vertex degree. This phase transition is often associated with the instances of highest complexity. We use extremal optimization to measure the ground state cost and the "backbone," an order parameter related to ground state overlap, averaged over a large number of instances near the transition for random graphs of size n up to 512. For these graphs, benchmarks show that extremal optimization reaches ground states and explores a sufficient number of them to give the correct backbone value after about O (n(3.5)) update steps. Finite size scaling yields a critical mean degree value alpha(c) =4.703 (28). Furthermore, the exploration of the degenerate ground states indicates that the backbone order parameter, measuring the constrainedness of the problem, exhibits a first-order phase transition.

Perfect sampling using bounding chains

Bounding chains are a technique that offers three benefits to Markov chain practitioners: a theoretical bound on the mixing time of the chain under restricted conditions, experimental bounds on the mixing time of the chain that are provably accurate and construction of perfect sampling algorithms when used in conjunction with protocols such as coupling from the past. Perfect sampling algorithms generate variates exactly from the target distribution without the need to know the mixing time of a Markov chain at all. We present here the basic theory and use of bounding chains for several chains from the literature, analyzing the running time when possible. We present bounding chains for the transposition chain on permutations, the hard core gas model, proper colorings of a graph, the antiferromagnetic Potts model and sink free orientations of a graph.

Phase transition in multiprocessor scheduling.

An "easy-hard" phase transition is shown to characterize the multiprocessor scheduling problem in which one has to distribute the workload on a parallel computer such as to minimize the overall run time. The transition can be analyzed in detail by mapping it on a mean-field antiferromagnetic Potts model. The static phase transition, characterized by a vanishing ground state entropy, corresponds to a transition in the performance of practical scheduling algorithms.

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.

Yang-Lee zeros of the Q-state Potts model on recursive lattices.

The Yang-Lee zeros of the Q-state Potts model on recursive lattices are studied for noninteger values of Q. Considering one-dimensional (1D) lattice as a Bethe lattice with coordination number equal to 2, the location of Yang-Lee zeros of 1D ferromagnetic and antiferromagnetic Potts models is completely analyzed in terms of neutral periodical points. Three different regimes for Yang-Lee zeros are found for Q>1 and 0<Q<1. An exact analytical formula for the equation of phase transition points is derived for the 1D case. It is shown that Yang-Lee zeros of the Q-state Potts model on a Bethe lattice are located on arcs of circles with the radius depending on Q and temperature for Q>1. Complex magnetic field metastability regions are studied for the Q>1 and 0<Q<1 cases. The Yang-Lee edge singularity exponents are calculated for both 1D and Bethe lattice Potts models. The dynamics of metastability regions for different values of Q is studied numerically.

Potts Model on Infinite Graphs and the Limit of Chromatic Polynomials

Given an infinite graph $\GI$ quasi-transitive and amenable with maximum degree $\D$, we show that reduced ground state degeneracy per site $W_r(\GI,q)$ of the q-state antiferromagnetic Potts model at zero temperature on $\GI$ is analytic in the variable 1/q, whenever $|2\D e^3/q|< 1$. This result proves, in an even stronger formulation, a conjecture originally sketched in [KE] (Kim and Enting, 1979) and explicitly formulated in [ST] (Shrock and Tsai,1997), based on which a sufficient condition for $W_r(\GI,q)$ to be analytic at 1/q=0 is that $\GI$ is a regular lattice.

ON THE FREE ENERGY STRUCTURE OF ANTIFERROMAGNETIC POTTS MODEL

Ordered phase for the antiferromagnetic Potts model is a controversial problem. For the three state Potts model, whether the phase just below the antiferromagnetic transition temperature is rotationally symmetric or not has not been settled. This comes from subtle free energy structure near the phase transition. Numerical simulations can be hampered by the subtlety because of finite size effects. Necessary size of systems for simulations to distinguish possible ordered states is estimated by solving self-consistent equations within the pair approximation to obtain free energies of ordered states and saddle points between them.

Unusual Corrections to Scaling in the 3-State Potts Antiferromagnet on a Square Lattice

At zero temperature, the 3-state antiferromagnetic Potts model on a square lattice maps exactly onto a point of the 6-vertex model whose long-distance behavior is equivalent to that of a free scalar boson. We point out that at nonzero temperature there are two distinct types of excitation: vortices, which are relevant with renormalization-group eigenvalue 1/2; and non-vortex unsatisfied bonds, which are strictly marginal and serve only to renormalize the stiffness coefficient of the underlying free boson. Together these excitations lead to an unusual form for the corrections to scaling: for example, the correlation length diverges as \beta \equiv J/kT \to \infty according to \xi \sim A e^{2\beta} (1 + b\beta e^{-\beta} + ...), where b is a nonuniversal constant that may nevertheless be determined independently. A similar result holds for the staggered susceptibility. These results are shown to be consistent with the anomalous behavior found in the Monte Carlo simulations of Ferreira and Sokal.

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.

Height Representation, Critical Exponents, and Ergodicity in the Four-State Triangular Potts Antiferromagnet

We study the four-state antiferromagnetic Potts model on the triangular lattice. We show that the model has six types of defects which diffuse and annihilate according to certain conservation laws consistent with their having a vector-valued topological charge. Using the properties of these defects, we deduce a (2+2)-dimensional height representation for the model and hence show that the model is equivalent to the three-state Potts antiferromagnet on the Kagome lattice and to bond-coloring models on the triangular and honeycomb lattices. We also calculate critical exponents for the ground-state ensemble of the model. We find that the exponents governing the spin–spin correlation function and spin fluctuations violate the Fisher scaling law because of constraints on path length which increase the effective wavelength of the spin operator on the height lattice. We confirm our predictions by extensive Monte Carlo simulations of the model using the Wang–Swendsen–Kotecký cluster algorithm. Although this algorithm is not ergodic on lattices with toroidal boundary conditions, we prove that it is ergodic on lattices whose topology has no noncontractible loops of infinite order, such as the projective plane. To guard against biases introduced by lack of ergodicity, we perform our simulations on both the torus and the projective plane.

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.

Monte Carlo simulation of the antiferromagnetic four-state Potts model on simple cubic and body-centered-cubic lattices

The phase transition of four-state antiferromagnetic Potts model on simple cubic and body-centered-cubic lattice is studied by cluster Monte Carlo simulations of finite systems up to ${96}^{3}$ lattice sizes. The result indicates that the size ${96}^{3}$ is still sufficient to extract asymptotic behavior. However, if one assumes that both models are well described by the usual Ginzburg-Landau Hamiltonian, the result is consistent with the prediction of the field-theoretical analysis that the phase transition is first-order one.

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

Upper and lower bounds for the ground state entropy of antiferromagnetic Potts models

We derive rigorous upper and lower bounds for the ground state entropy of the $q$-state Potts antiferromagnet on the honeycomb and triangular lattices. These bounds are quite restrictive, especially for large $q$.

Description of periodic extreme gibbs measures of some lattice models on the Cayley tree

The uniqueness of the translation-invariant extreme Gibbs measure for the antiferromagnetic Potts model with an external field and the existence of an uncountable number of extreme Gibbs measures for the Ising model with an external field on the Cayley tree are proved. The classes of normal subgroups of finite index of the Cayley tree group representation are constructed. The periodic extreme Gibbs measures, which are invariant with respect to subgroups of index 2, are constructed for the Ising model with zero external field. From these measures, the existence of an uncountable number of nonperiodic extreme Gibbs measures for the antiferromagnatic Ising model follows.