Showing papers on "Potts model published in 1986"
TL;DR: In this paper, the authors studied the effects at a temperature-driven first-order transition by analyzing various moments of the energy distribution and the rounding of the singularities and the shifts in the location of the specific heat maximum.
Abstract: We study the finite-size effects at a temperature-driven first-order transition by analyzing various moments of the energy distribution. The distribution function for the energy is approximated by the superposition of two weighted Gaussian functions yielding quantitative estimates for various quantities and scaling form for the specific heat. The rounding of the singularities and the shifts in the location of the specific-heat maximum are analyzed and the characteristic features of a first-order transition are identified. The predictions are tested on the ten-state Potts model in two dimensions by carrying out extensive Monte Carlo calculations. The results are found to be in good agreement with theory. Comparison is made with the second-order transitions in the two- and three-state Potts models.
529 citations
TL;DR: In this article, the spectrum of the transfer matrix for finite width strips and a variety of boundary conditions were derived for the Ising model and for the three-state Potts model, demonstrating how the internal symmetries of these theories are tied in with their conformal properties.
488 citations
TL;DR: In this paper, it was shown that the two-dimensional free fermion model is equivalent to a checkboard Ising model, which is a special case of the general Z -invariant Ising Model.
Abstract: It is shown that the two-dimensional free fermion model is equivalent to a checkerboard Ising model, which is a special case of the general ‘ Z -invariant’ Ising model Expressions are given for the partition function and local correlations in terms of those of the regular square lattice Ising model Corresponding results are given for the self-dual Potts model, and the application of the methods to the three-dimensional Zamolodchikov model is discussed The paper ends with a discussion of the critical and disorder surfaces of the checkerboard Potts model
127 citations
TL;DR: In this article, the Bethe ansatz method was used to solve the critical O(n) model on the honeycomb lattice, which is equivalent to the zero-temperature antiferromagnetic Potts model of the triangular lattice.
Abstract: Nienhuis (1982,4) has shown that the critical O(n) model, on the honeycomb lattice is equivalent to a zero-temperature antiferromagnetic Potts model on the triangular lattice, i.e. to the chromatic polynomial of the triangular lattice. Here the critical O(n) model is solved by the Bethe ansatz method, thereby giving the large-lattice limit of the chromatic polynomial.
127 citations
TL;DR: In this article, the corrections to the finite-size scaling behavior of the eigenvalues of the transfer matrix of a critical theory defined on an infinitely long strip of finite width, which occur when the Hamiltonian contains a marginal operator, are computed using conformal invariance.
Abstract: The corrections to the finite-size scaling behaviour of the eigenvalues of the transfer matrix of a critical theory defined on an infinitely long strip of finite width, which occur when the Hamiltonian contains a marginal operator, are computed using conformal invariance. They show a calculable universal logarthmic character. For the four-state Potts model they agree with numerical data.
124 citations
TL;DR: In this paper, the authors showed that the Potts model exhibits a first order phase transition at some inverse temperature βt between ordered and disordered phases for q large as proved in [1].
Abstract: Theq states Potts model exhibits a first order phase transition at some inverse temperature βt between “ordered” and “disordered” phases forq large as proved in [1] In space dimension 2 we use theduality transformation as aninternal symmetry of the partition function at βt to derive an estimate on the probability of a contour This enables us to prove the preceding result and the following new results:
(i)
The discontinuity of the mass gap at βt
(ii)
The existence of astrictly positive surface tension between two ordered phases up to βt
(iii)
The existence of a non-zero surface tension between an “ordered” and the “disordered” phase at βt
77 citations
TL;DR: In this paper, the existence of a first-order phase transition in an infinite volume was shown to be possible in the Potts model with respect to local minima of energy.
Abstract: CONTENTS § 1. Introduction and statement of the problem § 2. Gibbs states and reflection positivity § 3. The existence of Gibbs states in an infinite volume § 4. A strategy for proving the existence of a first-order phase transition § 5. First applications § 6. Phase transitions associated with local minima of energy § 7. The Potts model § 8. The Potts gauge model References
49 citations
TL;DR: In this paper, the lowest excitations of the three-state Potts quantum chain with periodic and twisted boundary conditions were computed in terms of eighteen irreducible representations of the Virasoro algebra with c=4/5.
Abstract: The authors compute the lowest excitations of the spectra of the three-state Potts quantum chain with periodic and twisted boundary conditions. These spectra can be understood in terms of eighteen irreducible representations of the Virasoro algebra with c=4/5.
44 citations
TL;DR: The set of all translation invariant Gibbs states in theq-state Potts model for the case ofq large enough and the other parameters to be arbitrary is described.
Abstract: We describe the set of all translation invariant Gibbs states in theq-state Potts model for the case ofq large enough and the other parameters to be arbitrary.
39 citations
TL;DR: In this paper, an infinite set of conserved charges for some Z(N) symmetric quantum spin models by constructing their Lax pairs was derived, which correspond to the Potts model, Ashkin-Teller model and the particular set of self-dual Z (N) models solved by Fateev and Zamolodchikov.
35 citations
TL;DR: In this article, it was shown that the partition function for the A-state Potts model with pair interactions is related to the expected number of integer mod-A flows in a percolation model.
Abstract: It is shown that the partition function for the A-state Potts model with pair interactions is related to the expected number of integer mod-A flows in a percolation model. The relation is generalised to the pair correlation function. The resulting high- temperature expansion coefficients are shown to be the flow polynomials of graph theory. We also prove an observation of Tsallis and Levy concerning the equivalent transmissivity of a cluster.
TL;DR: In this article, the exact equivalence of the partition function of a q-state nonintersecting string (NIS) model on an arbitrary planar, even-valenced, lattice with the q2-state Potts model on a related lattice was established.
Abstract: Using a graphical method we establish the exact equivalence of the partition function of aq-state nonintersecting string (NIS) model on an arbitrary planar, even-valenced, lattice with that of a q2-state Potts model on a related lattice. The NIS model considered in this paper is one in which the vertex weights are expressible as sums of those of basic vertex types, and the resulting Potts model generally has multispin interactions. For the square and Kagome lattices this leads to the equivalence of a staggered NIS model with Potts models with anisotropic pair interactions, indicating that these NIS models have a first-order transition forq > 2. For the triangular lattice the NIS model turns out to be the five-vertex model of Wu and Lin and it relates to a Potts model with two- and three-site interactions. The most general model we discuss is an oriented NIS model which contains the six-vertex model and the NIS models of Stroganov and Schultz as special cases.
TL;DR: In this paper, a general q-state nonintersecting string model for q≧2 is studied, and solved, using a graphical approach, and the result is applied to obtain exact solutions for specific string models.
Abstract: A general q-state nonintersecting string model for q≧2 is studied, and solved, using a graphical approach. This model, of which special cases were first introduced by Strogonov and Schultz, has recently drawn further attention because of its relation to Potts models. In its simplest version, a uniform, separable nonintersecting string model on a quadratic lattice, it corresponds to a q2-state Potts model with Potts-spins on every other lattice face. We first formulate the related star-triangle equation which we solve in terms of a line-variable parametrization. Such a parametrization of a solution of the star-triangle equation leads to a simple and direct graphical derivation of an inversion relation for the partition function. Our graphical analysis also shows that the inversion relation holds if certain boundary effects can be neglected, as we shall give a special finite lattice from which the inversion relation can be read off immediately. This relation is next solved under appropriate analyticity assumptions, again using a simple and direct approach, and the result is applied to obtain exact solutions for specific string models.
TL;DR: In this article, the authors obtained the zeros of the partition function for the isotropic triangular lattice three-state Potts model in a finite lattice and identified a symmetry approximately relating all six transition points.
Abstract: The authors obtain the zeros of the partition function for the isotropic triangular lattice three-state Potts model in a finite lattice. The distribution exhibits new points on the real axis which are well fitted by an algebraic equation deduced from the inversion relation and the symmetries of the anisotropic model. They identify a symmetry approximately relating all six transition points.
TL;DR: In this article, the relationship between zeros of the partition function, analyticity and degeneracy of absolute magnitude of eigenvalues of the transfer matrix for statistical mechanical models is discussed.
Abstract: The author explains with examples, the relationship between zeros of the partition function, analyticity and degeneracy of absolute magnitude of eigenvalues of the transfer matrix for statistical mechanical models. It is shown how to write down a polynomial representation of the partition function for any model with a two-sheet largest eigenvalue. The staggered ice model representation of the q-state Potts model, on a sequence of semi-infinite strips, is solved and it is shown that for 0
TL;DR: In this article, a graph theoretic analysis is made of the m-spin correlation functions of the lambda-state Potts model, and the correlation functions for m=2 were expressed in terms of rooted mod- lambda flow polynomials.
Abstract: For pt.I see ibid. vol.19 p.411 (1985). A graph theoretic analysis is made of the m-spin correlation functions of the lambda -state Potts model. In paper I, the correlation functions for m=2 were expressed in terms of rooted mod- lambda flow polynomials. The authors introduce a more general type of polynomial, the partitioned m-rooted flow polynomial, which plays a fundamental role in the calculation of the multispin correlation functions. The m-rooted equivalent transmissivities of Tsallis and Levy (1981) are interpreted in terms of percolation theory and are expressed as linear combinations of the above correlation functions.
TL;DR: In this article, the authors performed extensive Monte Carlo simulations on the two-dimensional Ising models with n-spin interactions described by Debierre and Turban, and obtained an improved estimate of the temperature exponent close to 3 2, in agreement with 4-state Potts universality.
Abstract: We have performed extensive Monte Carlo simulations on the two-dimensional Ising models with n -spin interactions described recently by Debierre and Turban. Results for n = 3 models with sizes up to 128 × 128 are analyzed by means of finite-size scaling. This yields a value of the magnetic exponent y h close to 15 8 . Direct estimates of the temperature exponent y T do not converge convincingly. However, assuming the presence of logarithmic corrections such as in the 4-state Potts model, we obtain an improved estimate of the temperature exponent close to 3 2 , in agreement with 4-state Potts universality. This result is further supported by an exact mapping between the n = 3 model and the 4-state Potts model in an anisotropic limit. For the n = 4 model, we confirm that the phase transition is first order, and we estimate the discontinuities in the energy and the magnetization.
TL;DR: A Monte Carlo renormalization-group analysis of the dynamics of the eight-state Potts model with nearest- and next-nearest-neighbor interactions after a quench below its phase-transition point finds the existence of at least two distinct fixed points at zero temperature.
Abstract: We present a Monte Carlo renormalization-group analysis of the dynamics of the eight-state Potts model with nearest- and next-nearest-neighbor interactions after a quench below its phase-transition point. Our results sugest the existence of at least two distinct fixed points at zero temperature. One is associated with a freezing behavior while the second is associated with equilibration. Our conclusion is that the freezing fixed point is only attractive for quenches to zero temperature, while the equilibration fixed point is attractive for quenches to finite temperature.
TL;DR: In this paper, it was shown that aZ(N2)-spin model proposed by A. B. Zamolodchikov and M. I. Monastyrskii can be conveniently described by two interacting N-state Potts models.
Abstract: We show that aZ(N2)-spin model proposed by A. B. Zamolodchikov and M. I. Monastyrskii can be conveniently described by two interactingN-state Potts models. We study its properties, especially by using a dual invariant quantity of the model. A partial duality performed on one set of Potts spins yields a staggeredZ(N)-symmetric vertex model, which turns out to be a generalization of theN-state “nonintersecting string model” of C. L. Schultz and J. H. H. Perk. We describe its properties and elaborate on its (pseudo) “weak-graph symmetry” As by-products we find alternative representations of the N2-state andN-state Potts models by staggered Schultz-Perk vertex models, as compared to the usual representation by staggered six-vertex models.
TL;DR: In this paper, an interface between two ordered phases of a q-state Potts model below its bulk first-order transition temperature is considered and the mean field equations are solved analytically for small q-2 and the solution shows that the interface is wetted by the disordered phase as the transition is approached.
Abstract: The authors consider an interface between two ordered phases of a q-state Potts model below its bulk first-order transition temperature. The mean-field equations are solved analytically for small q-2 and the solution shows that the interface is wetted by the disordered phase as the transition is approached. The excess absorption and surface entropy diverge logarithmically. Numerical calculations indicate this wetting also occurs at larger q.
TL;DR: A detailed proof of Vdovichenko's method is given in this article, giving a exact expression of the free energy of the Ising model on a two-dimensional lattice.
Abstract: A detailed proof is presented of Vdovichenko's method (1965) giving a exact expression of the free energy of the Ising model on a two-dimensional lattice.
TL;DR: In this paper, the authors consider a general kinetic model for a chain of three-state Potts spins and derive points in two-dimensional Potts systems where certain spin correlations have one-dimensional character and the model is exactly solvable.
Abstract: We consider a general kinetic model for a chain of three-state Potts spins. From the time-evolution operator we infer points in two-dimensional Potts systems where certain spin correlations have one-dimensional character and the model is exactly solvable. This occurs in square lattice models with different kinds of competing interactions.
TL;DR: In this paper, the critical behavior of the ferromagnetic Potts Model on families of fractal lattices called Sierpinski Carpets and Pastry Shells was studied.
Abstract: We study the critical behaviour of the ferromagnetic Potts Model on families of fractal lattices called Sierpinski Carpets and Sierpinski Pastry Shells. We find the influence of geometrical parameters on critical temperature and thermal exponents, which confirms lacunarity as a relevant geometrical parameter in the definition of universality classes. We distinguish the inner surface structure from the bulk and study the influence of both structures independently. The phase diagram for the Pastry Shell family exhibit a crossover between bulk and surface behaviour which shows the increasing importance of the surface bonds on the full fractal geometry as the fractal dimension or the lacunarity is lowered.
TL;DR: In this article, simple physical arguments about the movement of domain walls are used to determine the dependence of the dynamical critical exponent z on the transition rates for the one-dimensional q-state Potts model.
Abstract: Simple physical arguments about the movement of domain walls are used to determine the dependence of the dynamical critical exponent z on the transition rates for the one-dimensional q-state Potts model.
TL;DR: In this paper, a method for studying the high-order behavior of perturbation expansion in theories in which the number of field components, n, is taken to zero is presented.
Abstract: The author develops a method for studying the high-order behaviour of the perturbation expansion in theories in which the number of field components, n, is taken to zero. This procedure is illustrated on the field theory formulation of the percolation problem, which can be considered as the n=0 limit of the (n+1)-state Potts model. The saddle points controlling the asymptotic behaviour are labelled by an integer r=1, 2, . . ., n for positive integer n, but after continuation to n=0 the dominant contribution effectively comes from the saddle point with r= infinity . The perturbation expansion is found to be oscillatory at large orders and its behaviour is calculated.
TL;DR: The random-cluster expansion of the n-state Potts model can be interpreted as an interacting percolation problem as discussed by the authors, and the phase transition in the Potts models can be viewed as a per-colation transition.
Abstract: The random-cluster expansion of the n-state Potts model can be understood as an interacting percolation problem: 'cluster-weighted bond percolation'. This interpretation explains the phase transition in the Potts model as a percolation transition, and it also suggests several generalisation. The author employs a position space renormalisation group to calculate the transition temperature and the critical exponents.
TL;DR: In this article, the authors discuss two recent papers on the dynamics of the one-dimensional Potts model which yield different values of the dynamical critical exponent and show that in fact there are choices of the transition rates which yield arbitrarily large non-universal exponents.
Abstract: The authors discuss two recent papers on the dynamics of the one-dimensional Potts model which yield different values of the dynamical critical exponent. They show that in fact there are choices of the transition rates which yield arbitrarily large non-universal exponents.
TL;DR: In this paper, a new method is proposed for asymptotic analysis of power series, based on the relation that an n-term power series of a singular quantity with critical exponent beta behaves asympticically like n- beta at the phase transition point.
Abstract: A new method is proposed for asymptotic analysis of power series. In the method the author uses the relation that an n-term power series of a singular quantity with critical exponent beta behaves asymptotically like n- beta at the phase transition point. The method is tested on known series with satisfactory results and used to analyse the power series of the two-dimensional three-state Potts model of Enting, in good agreement with den Nijs' conjecture.