Showing papers on "Potts model published in 1989"

Superintegrable Chiral Potts Model: Thermodynamic Properties, an "Inverse" Model, and a Simple Associated Hamiltonian

Abstract: The partition function of theN-state Superintegrable chiral Potts model is obtained exactly and explicitly (if not completely rigorously) for a finite lattice with particular boundary conditions. This yields the bulk and surface free energies, and horizontal and vertical correlation lengths and interfacial tensions. The critical exponents are α=1−2/N,μhor=υhor=2/N, and μvert=υvert=1, and the finite-size corrections are obtained at criticality. The eigenvalue spectrum of the column-to-column transfer matrix is that of a direct product ofN byN matrices. Inverting this matrix gives a related solvable model which is a generalization of the free-fermion model. The associated Hamiltobian has a very simple form, suggesting there may be a more direct algebraic method (perhaps a generalized Clifford algebra) for obtaining its eigenvalues.

Fractal structure of Ising and Potts clusters: Exact results.

Abstract: It is shown that previously defined clusters, which give a geometrical description of the fluctuations in the q-state Potts model, at criticality have a fractal structure made of links and blobs as in percolation. Using the mapping from the Potts model to the Coulomb gas it is found that the fractal dimension of the links or red bonds ${D}_{R}$ is given by 5/4, 3/4, 13/24, 7/20, 0, while the fractal dimension of the external hull ${\mathit{D}}_{\mathit{h}}$ is given by 2, 7/4, 5/3, 8/5, 3/2, for q=0,1,2,3,4. A model originally introduced by Mandelbrot and Given for percolation clusters is found to correctly describe the fractal structure of the Potts clusters.

On a certain value of the Kauffman polynomial

Abstract: IfF L (a, x) is the Kauffman polynomial of a linkL we show thatF L (1, 2 cos 2π/5) is determind up to a sign by the rank of the homology of the 2-fold cover of the complement ofL. This value corresponds to a certain Wenzl subfactor defined by the Birman-Wenzl algebra, which we describe in simple terms. There also corresponds a “solvable” model in statistical mechanics similar to the 5-state Potts model. It is the 5-state case of a general model of Fateev and Zamolodchikov.

Antiferromagnetic Potts models

Abstract: We present a study of the antiferromagnetic Potts model in two and three dimensions, using a new method of Monte Carlo simulations, which enables us to perform simulations with greatly improved efficiency. Illustrating the method for the three-state model, we have obtained new results for the entropy and critical exponents in two and three dimensions. The low-temperature phase in three dimensions is shown to have long-range order with a finite-size dependence similar to that of the XY model

Rigorous lower bound on the dynamic critical exponents of the Swendsen-Wang algorithm

Abstract: We prove the rigorous lower bound z sw ≥α/v for the dynamic critical exponent of the Swendsen-Wang algorithm For two-dimensional q-state Potts models with q=2, 3, 4, this implies z sw ≥0, 2/5, 1 We present numerical data indicating that z sw =055±003, 089±005 for q=3, 4 (95% confidence limits, statistical errors only) The discrepancy for q=4 appears to be caused by multiplicative logarithmic corrections

Solving Models in Statistical Mechanics

Abstract: Publisher Summary This chapter discusses solving models in statistical mechanics. One of the main aims of statistical mechanics is to calculate the partition function Z. This can be done for a certain class of two-dimensional lattice models. They are by definition solvable and most of them can also be related to one-dimensional integrable Hamiltonians. The Ising, Potts, chiral Potts, and Fateev-Zamolodchikov models are all of this edge-interaction type. There are many relations between these models, for example, the Ising model is a special case of both the 8-vertex and chiral Potts models. The 8-vertex model is equivalent to the 8-vertex solidon-solid (SOS) model, in the sense that they both have the same partition function, even though they are formulated differently and have different order parameters. The hard hexagon model is a special case of the 8-vertex SOS model.

Monte Carlo study of finite-size effects at a weakly first-order phase transition.

Abstract: A new kind of finite-size phenomenon in the five-state Potts model in two dimensions, a ``false'' ground state, was recently reported by Katznelson and Lauwers based on Monte Carlo studies of large lattices. We reexamine the behavior of this weakly first-order system by performing extensive Monte Carlo simulations, and demonstrate that the ``novel'' states previously observed in internal-energy-distribution histograms tend to disappear when runs of sufficient length are used. We present evidence that very strong critical slowing down and a pseudodivergent correlation length are responsible for the observed effects. Our additional analysis of finite-size effects in the ten-state Potts model on small lattices (L\ensuremath{\le}10) shows a crossover of scaling from ${L}^{\mathrm{\ensuremath{-}}d}$ behavior (observed earlier on larger lattices) to a nonanalytic form. This result is used to discuss the observed nonanalytic scaling of thermodynamic quantities in the five-state Potts model.

Simulations without critical slowing down: Ising and three-state Potts models.

Abstract: We have developed a novel simulation method that combines a multigrid technique with a stochastic blocking procedure. Our algorithm eliminates critical slowing down completely, as we demonstrated previously by simulating the two-dimensional Ising model at criticality. Here we describe results of much more extensive simulations of the two-dimensional Ising and three-state Potts models and provide a heuristic argument to explain why our method works.

Behavior in large dimensions of the potts and heisenberg models

Abstract: We extend to the Potts and Heisenberg models some of the results proven in [6] for the Ising model. For both these models we prove that if the interaction is properly normalized, then as the space dimensionality goes to ∞, the spontaneous magnetization converges to the value given by the corresponding Curie-Weiss model (except possibly at the Curie-Weiss transition point, in the case of the Potts model.) For the Potts model we prove also that the ordered phases approach product measures. The proofs are based on the convergence of the free energy to the Curie-Weiss value and on infrared bounds. A consequence of our result for the q-state Potts model is an asymptotic upper bound for the transition temperature, which for q > 2 is better than the one obtained by the conventional use of infrared bounds, or comparison inequalities between different Potts models.

Mean-field theory and fluctuations in Potts spin glasses. I

Abstract: A short-range Potts spin glass is studied. A stable, non-marginal, mean-field theory is found with one level of replica symmetry breaking and a discontinuous transition for p>4. A complete stability analysis is provided, and two different correlation lengths are found above eight dimensions. The fluctuations in the ordered phase around this solution are incorporated in a renormalisation group approach and it is found that for a small range of the parameters they restore scaling close to the upper critical dimension. For the three-state case it appears that fluctuations destroy the stability of the solution for d

Correlation Length Of The Three State Potts Model In Three-Dimensions.

Abstract: A Monte Carlo study is carried out for the three-state Potts model in three dimensions. The correlation length is shown to remain finite and to be discontinuous at the transition point. It is also shown that the transition exhibits genuine first-order characteristics in all aspects

Inversion relations, phase transitions and transfer matrix excitations for special spin models in two dimensions

Abstract: For a special class of “Interaction-Round-Faces’ models (SIRF) which include the eightvertex model and the standard Potts model in a decoupling limit we derive inversion relations both for the partition function and the transfer matrix. These relations lead to some new exact results. In the general case the location of phase transitions is determined which reduces to known results for the Potts model. For the (solvable) self-dual Potts model we calculate all eigenvalues of the transfer matrix, i.e. the excitations of the model, from which the exact correlation length is derived.

Surface free energy of the critical six-vertex model with free boundaries

Abstract: The Bethe ansatz equations are derived for the six-vertex model with general boundary weights on a lattice in a diagonal orientation. These are solved in the thermodynamic limit. Finite-size corrections to the free energy (for a restricted class of boundary weights, including those corresponding to a Potts model with free boundaries) are calculated in the critical region. The first-order term gives the surface free energy of the model. The second-order term is found to be - pi okT tan( pi nu /2 mu )c/48N'2 where c=(1-6 mu 2/( pi 2- pi mu )) is the conformal anomaly. This can be compared to - pi kT sin( pi nu / mu )c/6N'2 for a calculation on the lattice in the standard orientation and with periodic boundary conditions. This difference can be explained geometrically using conformal invariance.

Yang-Lee zeros, Julia sets, and their singularity spectra.

Abstract: We have studied the global scaling properties of the Julia sets of the Yang-Lee zeros of the s-state Potts model on the diamond hierarchical lattice. The singularity spectrum f(\ensuremath{\alpha}) and the generalized dimension ${D}_{q}$ are calculated for different s values. General observations are made on their variations.

Percolation on a fractal with the statistics of planar feynman graphs: exact solution

Abstract: A new bond-percolation problem on a graph (fractal) randomly chosen from all planar Feynman graphs of zero-dimensional φ3 (or φ4) theory in the thermodynamical limit (infinite order of graphs) is solved exactly. At the percolation transition point pc the mean number of clusters per volume unit has the singularity (pc−p)4log (pc−p) which corresponds to the critical exponent α=−2. This model is a particular example of Potts models on dynamical planar lattice1 and the result agrees with the formulae obtained in Ref. 2 by conformal field theory approach for Potts spins interacting with 2D quantum gravity.

Bulk, surface and hull fractal dimension of critical Ising clusters in d=2

Abstract: The authors present accurate numerical calculations of the fractal dimension d and surface dimension ds of the critical Ising cluster, in d=2. The results clearly support the values d=187/96, ds=5/6 which are consistent with Ising clusters being described by tricritical q=1 Potts model exponents. From this, the hull dimension dH of critical Ising clusters is found to be dH=11/8, consistent with numerical work of other authors.

Interface Approach to Phase Transitions and Ordering by Monte Carlo Simulations and Its Applications to Three-Dimensional Antiferromagnetic Potts Models

Abstract: We develop an interface approach to bulk phase transitions and ordered states by using Monte Garlo simulations, and apply it to antiferromagnetic q -state Potts models with q =3∼6 on the simple cubic lattice. A stiffness exponent a defined by Δ F ∼ L a , where Δ F is the interface free energy for a system of size L , is introduced as a measure of the stiffness of the ordered phase against an external stress. Applying finite-size scaling to Δ F and to the squared order parameter enable one to determine each of T c , ν, β and γ in order where T c is the critical temperature and the others are critical exponents. This approach also provides a means to study properties of the ordered state through a and interface profiles. In the q =3 and 4 models the estimated exponents ν, β, γ and a indicate new universality classes. The q =5 model shows a phase transition, while the q =6 model does not. The q =3∼5 models have non-integer values of a at low temperatures. The phase transitions in these models are confirmed t...

Monte Carlo study of the Potts model on the square and the simple cubic lattices.

Abstract: On utilise la methode de simulation de Monte Carlo de Swendsen et Wang, et la technique d'etiquetage multiple de Hashen et Kopelman, pour calculer le modele de Pott du modele de percolation des liaisons correlees a l'etat q, sur le reseau carre et cubique simple. Les calculs donnent les points critiques exacts du modele de Pott

Finite-Size Real-Space Renormalization Group

Abstract: We propose a method for calculating the critical indices of statistical systems based on the analysis of the flow of coupling constants renormalized at a scale equal to half of the total lattice size. We have tested the method in the two and three-state Potts model in two dimensions.

The continuum Potts model and continuum percolation

Abstract: Recently, integral equations have been developed for continuum percolation models by relating them to the one-state limit of a continuum Potts model (CPM). In this paper, we use the CPM formalism to develop series expansions for the main quantities of continuum percolation theory. This provides necessary background for the work with integral equations and also sheds light on the basic structure of percolation theory. We develop virial series for the mean number of clusters 〈nc〉 and the mean cluster size S( ρ ). A basic relation between these series is developed using a correlation function new to percolation theory which we denote h′(x). We develop the virial theorem for continuum percolation, which relates the mean number of clusters to the two-point blocking function. A geometric interpretation of this theorem is offered in the case of random sphere percolation. For a general, correlated percolation model, the virial theorem involves our new correlation function h′(x). We use the CPM formalism to give computationally efficient expressions for the general coefficient in the virial expansion of all of these quantities. The combinatorial weighting factors that appear in these expansions are discussed in detail. For completeness, we prove the equivalence of our formulas to the others that have appeared in the literature. We do this in the hope of providing a useful reference for further work in continuum percolation.

Semi-infinite Potts model and percolation at surfaces

Abstract: The effects of surfaces on percolation are investigated near the bulk percolation threshold ind=6−e dimensions. Using field-theoretic methods, this is done within the framework of a semi-infinite continuousq-state Potts model withq→1. Renormalization-group equations are obtained which imply that the usual scaling laws for surface and bulk exponents are valid to all orders in e, and the surface exponents at the ordinary and special transition are computed to order e. Our result for β1ord is in conformity with the one by Carton.

Discontinuity of the Wilson string tension in the $4$-dimensional lattice pure gauge Potts model

Abstract: We consider the 4-dimensionalq-state pure gauge Potts model. Forq large enough, we give a new proof of the existence of a unique coupling constant β t , where a first order phase transition occurs. Moreover we prove the following new results: The string tension is discontinuous at β t , the Wilson parameter exhibits at β t a direct transition from an area law decay (quark confinement) to a perimeter law decay (quark deconfinement).