Showing papers on "Potts model published in 1990"

Chiral Potts model as a descendant of the six-vertex model

Abstract: We observe that theN-state integrable chiral Potts model can be considered as a part of some new algebraic structure related to the six-vertex model. As a result, we obtain a functional equation which is supposed to determine all the eigenvalues of the chiral Potts model transfer matrix.

Rounding effects of quenched randomness on first-order phase transitions

Abstract: Frozen-in disorder in an otherwise homogeneous system, is modeled by interaction terms with random coefficients, given by independent random variables with a translation-invariant distribution. For such systems, it is proven that ind=2 dimensions there can be no first-order phase transition associated with discontinuities in the thermal average of a quantity coupled to the randomized parameter. Discontinuities which would amount to a continuous symmetry breaking, in systems which are (stochastically) invariant under the action of a continuous subgroup ofO(N), are suppressed by the randomness in dimensionsd≦4. Specific implications are found in the Random-Field Ising Model, for which we conclude that ind=2 dimensions at all (β,h) the Gibbs state is unique for almost all field configurations, and in the Random-Bond Potts Model where the general phenomenon is manifested in the vanishing of the latent heat at the transition point. The results are explained by the argument of Imry and Ma [1]. The proofs involve the analysis of fluctuations of free energy differences, which are shown (using martingale techniques) to be Gaussian on the suitable scale.

New numerical method to study phase transitions.

Abstract: A powerful method of detecting first order transitions by numerical simulations of finite systems is presented. The method relies on simulations and the finite size scaling properties of free energy barriers between coexisting states. It is demonstrated that the first order transitions in d = 2, q = 5 and d = q = 3 Potts models are easily seen with modest computing time. The method can also be used to obtain quite accurate estimates of critical exponents by studying the barriers in the vicinity of a critical point. Some new results on exponents and conformal charge in frustrated XY models and a related coupled XY-Ising model in d = 2 are presented. These show that the transitions in these models are in new universality classes and that the conformal charge varies with a parameter.

Functional relations for transfer matrices of the chiral potts model

Abstract: It has recently been shown that the solvable N-state chiral Potts model is related to a vertex model with N-state spins on vertical edges, two-state spins on horizontal edges. Here we generalize this to a “j-state by N-state” model and establish three sets of functional relations between the various transfer matrices. The significance of the “super-integrable” case of the chiral Potts model is discussed, and results reported for its finite-size corrections at criticality.

Coarsening in the two-dimensional soap froth and the large-Q Potts model: A detailed comparison

Abstract: A detailed comparison between the experimental evolution of a two-dimensional soap froth and a next-nearest-neighbour Q = ∞ Potts model on a square lattice starting from identical initial conditions is presented. We compare the pattern evolution, dynamics, distribution functions and correlations of the two systems. We also examine in detail the relation between number or sides and area (Lewis' law) and two measures of pattern disorder. Overall agreement is found between the model and simulation, with a few systematic deviations which suggest that side redistribution is affected by subtle anisotropy and equilibration effects.

Zeroes of chromatic polynomials: a new approach to Beraha conjecture using quantum groups

Abstract: The number of colourings of a graphG withQ or fewer colors is a polynomial inQ known as the chromatic polynomialPG(Q). It coincides with the partition functionFG of theQ state Potts model onG at zero temperature and in the antiferromagnetic regimeeK=0. In the planar case, the Beraha conjecture particularizes the numbers\(B_n = 4\cos ^2 \frac{\pi }{n}\) as possible accumulation points of real zeroes ofPG in the infinite graph limit. We suggest in this work an approach based on recent developments of quantum groups to handle this conjecture. For the square, triangular and honeycomb lattices and systems wrapped on a cylinderl×t, we first exhibit in the (Q, eK) Potts parameter space a critical line, whereFG(Q,eK) has real zeroes converging to and only to theBn's asl, t→∞. The analysis is based on the vertex representation of theQ state Potts model, quantum algebraUqSl(2) properties forq a root of unity, and conformal invariance.UqSl(2) symmetry is present for anyeK, including the chromatic polynomial caseeK=0. Using an additional hypothesis on the eigenvalues structure and knowledge of the Potts parameter space, we then argue that forPG(Q), real zeros occur and converge toBn's, 2≦n≦n0 only, wheren0 depends on the lattice. Extensions to other kinds of graphs and size dependence of the zeros are discussed. Finally physical applications are also mentioned.

Helix-Coil transition in polypeptides: A microscopical approach

Abstract: Analogous with the Potts model that describes the helix-coil transition in the isolated polypeptide chain (a Hamiltonian model allowing for the energy U of hydrogen bond formation) the number Q of conformational states of a repeating unit of the chain and the topology of Δ = 3 hydrogen bond formation (the hydrogen bond fixing three pairs of φψ chain rotations) has been constructed and the corresponding transfer-matrix has been obtained. In the thermodynamical limit, the partition function is expressed through the principal root of the cubic equation. The degree of helicity, the transition point and range, the correlation length, the number of junctions between the helical and coiling sections as well as the mean length of helical and coiling sections are calculated. Empirically introduced parameters of the Zimm–Bragg theory, constants of hydrogen bond formation s, and the cooperativity parameter σ as functions of microscopic parameters U, Q, and Δ are obtained by direct calculations. The behavior of this model was investigated at other topologies of the hydrogen-bond closing Δ = 2 and Δ = 4, and it was suggested that the actual polypeptide chain (Δ = 3) provides the optimum correlation of helical structure of the order of globule dimensions. An expression was obtained for the maximum correlation length of the order ξ ∼ Q(Δ-1)/2. For a System with solvent competing for the formation of hydrogen bonds with peptide groups a Hamiltonian model was constructed that took account of the energy E of the formation of hydrogen bond with the solvent and the number q of orientations of a solvent molecule about the peptide groups. It is shown that by the redefinition of the temperature parameter, the model with solvent reduces to the model of an isolated chain. Aside from the definition relationship that exists between the parameters of the theory U < 2E < Uq and the ordinary helix-coil transitions (“melting”), the model also describes the transition from the coiling state to the helical one (“arrangement”) under heating. The change in temperature and transition range with solvent parameters was discussed and it was shown, that despite the difference in ΔT for the given polypeptide chain (Q = constant) with different solvent parameters, at “melting” and “arrangement,” the transition occurred at the same correlation length (the same cooperativity).

Exact results on the dimerisation transition in SU(n) antiferromagnetic chains

Abstract: The author shows that SU(n) antiferromagnetic chains (or equivalently SU(2) spin-s-chains with Hamiltonians which project out singlet states) are exactly equivalent to the n2-state quantum Potts chain (obtained from the transfer matrix for the 2D classical Potts model), for arbitrary n (or s with n=2s+1). This implies that the models are spontaneously dimerised with a finite gap for n<2 but have a unique ground state and vanishing gap for n

Quantum groups and generalized statistics in integrable models

Abstract: The paper deals with the integrable massive models of quantum field theory. It is shown that generalized statistics of physical particles is closely connected with the invariance under quantum groups. This invariance provides the possibility to construct quasi-local operators (parafermions) possessing generalized statistics which interpolates the physical particles. For the particular examples of SG, RSG models and scaling 3-state Potts model the parafermions are described completely (all their matrix elements in the space of states are presented).

A numerical method to compute exactly the partition function with application toZ(n) theories in two dimensions

Abstract: I present a new method to exactly compute the partition function of a class of discrete models in arbitrary dimensions. The time for the computation for ann-state model on anL d lattice scales like\(n^{L^{d - 1} } nL^d \). I show examples of the use of this method by computing the partition function of the 2D Ising and 3-state Potts models for maximum lattice sizes 10×10 and 8×8, respectively. The critical exponentsv andα and the critical temperature one obtains from these are very near the exactly known values. The distribution of zeros of the partition function of the Potts model leads to the conjecture that the ratio of the amplitudes of the specific heat below and above the critical temperature is unity.

Polynomial-time approximation algorithms for the Ising model

Abstract: This paper is concerned with computational solutions to some classical combinatorial problems in statistical physics These problems stem from the Ising model, which has been the focus of much attention in the physics and mathematics communities since it was first introduced by Lenz [14] and Ising [7] in the early 1920s We will not present a detailed historical account here: a very readable survey is given by Cipra [2], while Welsh [18] sets the Ising model in the context of other combinatorial problems in statistical physics

Some comments on the solvable chiral potts model

Abstract: We present a simple proof of the conjecture produced by Baxter, Perk and Au-Yang on the structure of the normalization factorR(p, q, r) corresponding to their new solution of the star-triangle equation related with the generalized Fermat curve. Some important properties of the underlying curvex N y N+x N+y N+1/k 2=0 for theN=3 state case are also established. Particularly, we calculate exactly its matrix of theb-periods for some normalized basis of holomorphic differentials. We also show that associated four-dimensional theta function may be decomposed into a sum containing 12 terms, each term being the product of four one-dimensional theta functions. We also derive Picard-Fuchs equations for the periods of holomorphic differentials of the same curve. The remarkable appearance of the hypergeometric functions in our answers seems to be closely related with an expression for the groundstate energy per site, obtained for the superintegrable case by Albertini, Perk, and McCoy and independently by Baxter, although for a moment the connection is not clear.

The q-state Potts model in the standard Pirogov-Sinai theory: Surface tensions and Wilson loops

Abstract: Theq-state Potts model (both scalar and gauge versions) is rewritten, with the help of the duality transformation, into a form of the Pirogov-Sinai theory with noninteracting contours that can be controlled by cluster expansions onceq is large enough. This is then used in a new proof of the existence of a unique transition (inverse) temperatureβ t , where the mean internal energy is discontinuous. Moreover, we prove for the scalar model (again forq large enough) that there are discontinuities atβ t of the magnetization and of the mass gap, with the magnetization vanishing belowβ t and the mass gap vanishing aboveβ t . We also show that the surface tensions between ordered stable phases are strictly positive up toβ t , and the surface tension between an ordered phase and the disordered one is strictly positive atβ t . For the three-dimensional gauge model, the Wilson parameter exhibits a direct transition from an area law decay (quark confinement) to a perimeter law decay (deconfinement).

Random surface dynamics for Z2 gauge theory.

Abstract: A new Monte Carlo dynamics is proposed for {ital Z}{sub 2} lattice gauge theory that reduces the dynamical exponent for critical slowing down in three dimensions to {ital z}=0.61{plus minus}0.05 in comparison with a heat-bath exponent of {ital z}=2.1{plus minus}0.2. The dynamics is based on plaquette percolation and a nonlocal Monte Carlo update rule analogous to the Swendsen-Wang algorithm for the Potts model. However, the acceleration mechanism for the gauge theory, unlike the spin models, is driven by the percolation and topology of extended surfaces, not the percolation of connected clusters.

Disordering of the (3×1) reconstruction on Si(113) and the chiral three-state Potts model

Abstract: We have used high-resolution low-energy electron diffraction to study the disordering; The reconstruction disorders via a continuous transition at approximately 850 K. Above the transition temperature the positions of the superlattice diffraction beams shift away from their commensurate positions. The shift is proportional to the broadening of the beams, as expected from basic scaling arguments for a system in the chiral three-state Potts-model universality class

Damage spreading in the Potts and Ashkin-Teller models: exact results

Abstract: Following the approach introduced by Coniglio et al. (1989) for the Ising model, the author studies the relation between damage spreading and thermal properties in the q-state Potts model as well as in the Ashkin-Teller model. The author shows that by selecting appropriate combinations of different types of damage (q(q-1) in the Potts model and 12 in the Ashkin-Teller model), it is possible to find exact relations between damage and thermodynamic functions, like the magnetisation and the pair correlation function.

Hypercube stacking: A potts-spin model for surface growth

Abstract: We present a deposition and evaporation model for surface growth under a solid-on-solid constraint. We generalize the Ising-spin representation of a two-dimensional surface by Blote and Hilhorst to ad-dimensional surface of a (d+1)-dimensional hypercubic lattice. The allowed surface configurations correspond to the (degenerate) ground states of a chirald-state Potts model. We describe a vectorized multisite-coding implementation for the corresponding kinetic Potts-spin model ford=2 andd=3. For thed=2 equilibrium surface our simulation results show excellent agreement with an exact analysis.

Finite-size scaling of the three-state Potts model on a simple cubic lattice

Abstract: Finite-size scaling is studied for the three-state Potts model on a simple cubic lattice. We show that the specific heat and the magnetic susceptibility scale accurately as the volume. The correlation length exhibits behaviors expected for a genuine first-order transition; the one extracted from the unsubtracted correlation function shows a characteristic finite-size behavior, whereas the physical correlation length that characterizes the first excited state stays at a finite value and is discontinuous at the transition point.

Surface energy of integrable quantum spin chains

Abstract: The surface energy of the antiferromagnetic spin-1/2 XXZ Heisenberg chain is derived in the region Delta (-1 from the known Bethe ansatz solution for free boundaries with surface fields. The result gives the surface energy of related models satisfying the Temperley-Lieb algebra. The models discussed are the quantum Q-state Potts chain and a family of isotropic spin-s chains including the spin-1 biquadratic model.

Resolving the order of phase transitions in Monte Carlo simulations

Abstract: The authors made a Monte Carlo simulation of the two-dimensional Potts model with q=3, 4 and 5 to examine the question whether numerical methods can distinguish the order of a phase transition for the subtle cases that this model exemplifies. They found that the finite-size scaling test for susceptibilities has sufficient power for this purpose, whereas the simple method of detecting metastability signals often fails.

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

Abstract: For pt.I, see ibid., vol.22, p.4971 (1989). The fluctuations in the ordered phase around the mean-field solution are incorporated in a renormalization 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 3-state case it appears that fluctuations destroy the stability of the solution and cause the system to undergo a first-order phase transition. Non-universal fluctuation corrections to the equation of state above six dimensions are obtained.

Exchange algebra and exotic supersymmetry in the chiral potts model

Abstract: We obtain an exchange algebra for the Chiral Potts model, the elements of which are linear in the parameters defining the rapidity curve. This enables us to connect the Chiral Potts model to a Uq(GL(2)) algebra. On the other hand, looking at the model from the S-matrix point of view relates it to a ZZN generalisation of the supersymmetric algebra.