Exact Potts Model Partition Functions for Strips of the Honeycomb Lattice

We present exact calculations of the Potts model partition function Z(G,q,v) for arbitrary q and temperature-like variable v on strip graphs G of the honeycomb lattice for a variety of transverse widths equal to L _y vertices and for arbitrarily great length, with free longitudinal boundary conditions and free and periodic transverse boundary conditions. These partition functions have the form , where m denotes the number of repeated subgraphs in the longitudinal direction. We give general formulas for N _Z,G,j for arbitrary L _y . We also present plots of zeros of the partition function in the q plane for various values of v and in the v plane for various values of q. Plots of specific heat for infinite-length strips are also presented, and, in particular, the behavior of the Potts antiferromagnet at is investigated.     

Exact Potts Model Partition Functions for Strips of the Triangular Lattice

We present exact calculations of the Potts model partition function Z(G,q,v) for arbitrary q and temperature-like variable v on n-vertex strip graphs G of the triangular lattice for a variety of transverse widths equal to L vertices and for arbitrarily great length equal to m vertices, with free longitudinal boundary conditions and free and periodic transverse boundary conditions. These partition functions have the form Z(G,q,v)= $\sum _{j = 1}^{N_{Z,G,\lambda } }$ c _z,G,j (λ _z,G,j ) ^m-1. We give general formulas for N _Z,G,j and its specialization to v=?1 for arbitrary L. The free energy is calculated exactly for the infinite-length limit of the graphs, and the thermodynamics is discussed. It is shown how the internal energy calculated for the case of cylindrical boundary conditions is connected with critical quantities for the Potts model on the infinite triangular lattice. Considering the full generalization to arbitrary complex q and v, we determine the singular locus ${\mathcal{B}}$ , arising as the accumulation set of partition function zeros as m→∞, in the q plane for fixed v and in the v plane for fixed q. Explicit results for partition functions are given in the text for L=3 (free) and L=3, 4 (cylindrical), and plots of partition function zeros and their asymptotic accumulation sets are given for L up to 5. A new estimate for the phase transition temperature of the q=3 Potts antiferromagnet on the 2D triangular lattice is given.     

An Analytical Study of Three-State Potts Model on Lattice

We investigate the phase structure of the three-state Potts model by the variational cumulant expansion approach. It is shown that there is a weak first-order phase transition in three and four dimensions. The critical coupling given by this method is in good agreement with MC data.     

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 J/kT according to Ae ² (1+be ^– +···), 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.     

Exact results for a dilute Potts model

Exact results are obtained for the annealed, dilute,q-component Potts model on the decorated square lattice. The phase diagram is found to consist of a high-temperature region, a low-temperature region, and a two-phase region in between which arises only forq>4: exact expressions for the phase boundary and the critical probability are derived. At the critical point the specific heat is generally finite and has a cusp; the slope of the cusp is finite forq=4 and infinite (vertical) forq=2 and 3.Work supported in part by NSF Grant No. DMR 78-18808.     

Exact results for the Potts model in two dimensions

By considering the zeros of the partition function, we establish the following results for the Potts model on the square, triangular, and honeycomb lattices: (i) We show that there exists only one phase transition; (ii) we give an exact determination of the critical point; (iii) we prove the exponential decay of the correlation functions, in one direction at least, for all temperatures above the critical point. The results are established forq 4, whereq is the number of components.Work supported by the Fond. National Suisse de la Recherche Scientifique.Work supported in part by NSF Grant No. DMR 76-20643.     

Potts Model on Maple Leaf Lattice with Pure Three-Site Interaction

We use Monte Carlo method to study three-state Potts model on maple leaf lattice with pure three-site interaction. The critical behavior of both ferromagnetic and antiferromagnetic cases is studied. Our results confirm that the critical behavior of the ferromagnetic model is independent of the lattice details and lies in the universality class of the three-state ferromagnetic Potts model. For the antiferromagnetic case the transition is of the first order. We have calculated the energy jump and critical temperature in this area. We find there is a tricritical point separating the first order and second order phases for this system.     

Structure of the Partition Function and Transfer Matrices for the Potts Model in a Magnetic Field on?Lattice Strips

We determine the general structure of the partition function of the q-state Potts model in an external magnetic field, Z(G,q,v,w) for arbitrary q, temperature variable v, and magnetic field variable w, on cyclic, M?bius, and free strip graphs G of the square (sq), triangular (tri), and honeycomb (hc) lattices with width L _y and arbitrarily great length L _x . For the cyclic case we prove that the partition function has the form Z(L,L_y×L_x,q,v,w)=?_d=0^L_y[(c)\tilde]^(d)Tr[(T_Z,L,Ly,d)^m]Z(\Lambda,L_{y}\times L_{x},q,v,w)=\sum_{d=0}^{L_{y}}\tilde{c}^{(d)}\mathrm{Tr}[(T_{Z,\Lambda,L_{y},d})^{m}] , where Λ denotes the lattice type, [(c)\tilde]^(d)\tilde{c}^{(d)} are specified polynomials of degree d in q, T_Z,L,Ly,dT_{Z,\Lambda,L_{y},d} is the corresponding transfer matrix, and m=L _x (L _x /2) for Λ=sq,tri (hc), respectively. An analogous formula is given for M?bius strips, while only T_Z,L,Ly,d=0T_{Z,\Lambda,L_{y},d=0} appears for free strips. We exhibit a method for calculating T_Z,L,Ly,dT_{Z,\Lambda,L_{y},d} for arbitrary L _y and give illustrative examples. Explicit results for arbitrary L _y are presented for T_Z,L,Ly,dT_{Z,\Lambda,L_{y},d} with d=L _y and d=L _y −1. We find very simple formulas for the determinant det(T_Z,L,Ly,d)\mathrm{det}(T_{Z,\Lambda,L_{y},d}) . We also give results for self-dual cyclic strips of the square lattice.     

Two-Dimensional Potts Model and Annular Partitions

Using the random cluster expansion, correlations of the Potts model on a graph can be expressed as sums over partitions of the vertices where the spins are fixed. For a planar graph, only certain partitions can occur in these sums. For example, when all fixed spins lie on the boundary of one face, only noncrossing partitions contribute. In this paper we examine the partitions which occur when fixed spins lie on the boundaries of two disjoint faces. We call these the annular partitions, and we establish some of their basic properties. In particular we demonstrate a partial duality for these partitions, and show some implications for correlations of the Potts model.     

Interface sharpness in the Potts model

A simple proof is given for the existence of a sharp interface between two ordered phases for the three-dimensional 2 ⁿ -state Potts model (n integer).     

New Correlation Duality Relations for the Planar Potts Model

We introduce a new method to generate duality relations for correlation functions of the Potts model on a planar graph. The method extends previously known results, by allowing the consideration of the correlation function for arbitrarily placed vertices on the graph. We show that generally it is linear combinations of correlation functions, not the individual correlations, that are related by dualities. The method is illustrated in several non-trivial cases, and the relation to earlier results is explained. A graph-theoretical formulation of our results in terms of rooted dichromatic, or Tutte, polynomials is also given.     

Finite-Size Effects for the Potts Model with Weak Boundary Conditions

Using the Pirogov–Sinai theory, we study finite-size effects for the ferromagnetic q-state Potts model in a cube with boundary conditions that interpolate between free and constant boundary conditions. If the surface coupling is about half of the bulk coupling and q is sufficiently large, we show that only small perturbations of the ordered and disordered ground states are dominant contributions to the partition function in a finite but large volume. This allows a rigorous control of the finite-size effects for these weak boundary conditions. In particular, we give explicit formulæ for the rounding of the infinite-volume jumps of the internal energy and magnetization, as well as the position of the maximum of the finite-volume specific heat. While the width of the rounding window is of order L ^–d , the same as for periodic boundary conditions, the shift is much larger, of order L ^–1. For strong boundary conditions—the surface coupling is either close to zero or close to the bulk coupling—the finite size effects at the transition point are shown to be dominated by either the disordered or the ordered phase, respectively. In particular, it means that sufficiently small boundary fields lead to the disordered, and not to the ordered Gibbs state. This gives an explicit proof of A. van Enter's result that the phase transition in the Potts model is not robust.     

Yang-Lee zeros of the Potts model

The Yang-Lee zeros of the three-component ferromagnetic Potts model in one dimension in the complex plane of an applied field are determined. The phase diagram consists of a triple point where three phases coexist. Emerging from the triple point are three lines on which two phases coexist and which terminate at critical points (Yang-Lee edge singularity). The zeros do not all lie on the imaginary axis but along the three two-phase lines. The model can be generalized to give rise to a tricritical point which is a new type of Yang-Lee edge singularity. Gibbs phase rule is generalized to apply to coexisting phases in the complex plane.Supported in part by the National Science Foundation under Grant No. DMR-81-06151.     

A twist on chiral Potts

We show that the chiral Potts model may be formulated so that the rapidity lines carry a second integer variable-an increment or twist in each bond crossing it. This modification does not affect those properties of the chiral Potts model which lead to integrability, since it is equivalent to one of the automorphisms allowed for in the theory. In particular, transfer matrices still form commuting families and still satisfy hierarchies of functional equations. Surprisingly, the superintegrable case with twists retains the special algebraic properties which lead to its Ising-like spectra. The formalism should be useful for considering systems with twisted boundary conditions or embedded interfaces.     

Antiferromagnetic Potts Models on the Square Lattice: A High-Precision Monte Carlo Study

We study the antiferromagnetic q-state Potts model on the square lattice for q=3 and q=4, using the Wang–Swendsen–Kotecký (WSK) Monte Carlo algorithm and a powerful finite-size-scaling extrapolation method. For q=3 we obtain good control up to correlation length 5000; the data are consistent with ()=Ae ^{2 p} (1+a ₁ e ^– + ...) as , with p1. The staggered susceptibility behaves as _stagg ^5/3. For q=4 the model is disordered (2) even at zero temperature. In appendices we prove a correlation inequality for Potts antiferromagnets on a bipartite lattice, and we prove ergodicity of the WSK algorithm at zero temperature for Potts antiferromagnets on a bipartite lattice.     

Equivalence of the Two Results for the Free Energy of the Chiral Potts Model

The free energy of the chiral Potts model has been obtained in two ways. The first used only the star-triangle relation, symmetries, and invariances, and led to a system of equations that implicitly define the free energy, and from which the critical behavior can be obtained The second used the functional relations derived by Bazhanov and Stroganov, solving them to obtain the free energy explicitly as a double integral. Here we obtain, for the first time, a direct verification that the two results are identical at all temperatures.     

Functional Relations for the Order Parameters of the Chiral Potts Model

Following the method of Jimbo, Miwa, and others, we obtain functional relations for the order parameters of the chiral Potts model. We have not yet solved these relations. Here we discuss their properties and show how one should beware of spurious solutions.     

Exact Results for the Universal Area Distribution of Clusters in Percolation, Ising, and Potts Models

At the critical point in two dimensions, the number of percolation clusters of enclosed area greater than A is proportional to A ^–1, with a proportionality constant C that is universal. We show theoretically (based upon Coulomb gas methods), and verify numerically to high precision, that . We also derive, and verify to varying precision, the corresponding constant for Ising spin clusters, and for Fortuin–Kasteleyn clusters of the Q = 2, 3 and 4-state Potts models.     

Onsager's reaction field for the Potts model from the path integral

A new path integral formulation for theq-state Potts model is proposed. This formulation reproduces known results for the Ising model (q=2) and naturally extends these results for arbitraryq. The mean field results for both the Ising and the Potts models are obtained as a leading saddle point contribution to the corresponding functional integrals, while the systematic computation of corrections to the saddle point contribution produces the Onsager reaction field terms, which forq=2 coincide with results already known for the Ising model.