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Physical review. B, Condensed matter

New effective-field theory for the Potts model

R. Honmura, Enaldo F. Sarmento, Constantino Tsallis, I. P. Fittipaldi

Published Mar 1, 1984• 17 citations
Publisher

Abstract

A new type of effective-field theory that has recently been used with success for many applications concerning the Ising model is, through the generalization of Callen identities, herein extended to the $q$-state Potts model. Although mathematically simple, it yields results quite superior to those currently obtained within the molecular-field approximation. In order to test its reliability we check the following properties: (a) The critical temperature ${T}_{c}$ associated with a linear chain vanishes for all $q$, (b) the value of ${T}_{c}$ associated with a $z$-fold-coordinated lattice ($z>2$) exhibits a qualitatively (and, to a certain extent, quantitatively) satisfactory $q$ dependence, (c) the $z\ensuremath{\rightarrow}\ensuremath{\infty}$ limit reproduces the exact value for ${T}_{c}$, and (d) for fixed $z$, all values of $q>2$ provide first-order phase transitions, which are exact for $d>4$. Furthermore, the procedures for obtaining, as functions of temperature and for any values of $q$ and $z$, the order parameter, internal energy, and specific heat are outlined, and some typical illustrations are presented.

DOI
10.1103/physrevb.29.2761

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Physical review. B, Condensed matter•Jul 1, 1984•42 citations
A new type of effective-field theory that has recently been used with success for many applications concerning the Ising model is, through the generalization of Callen identities, herein extended to the qqq-state Potts model. Although mathematically simple, it yields results quite superior to those currently obtained within the molecular-field approximation. In order to test its reliability we check the following properties: (a) The critical temperature Tc{T}_{c}Tc​ associated with a linear chain vanishes for all qqq, (b) the value of Tc{T}_{c}Tc​ associated with a zzz-fold-coordinated lattice (z>2z>2z>2) exhibits a qualitatively (and, to a certain extent, quantitatively) satisfactory qqq dependence, (c) the z\ensuremath→\ensuremath∞z\ensuremath{\rightarrow}\ensuremath{\infty}z\ensuremath→\ensuremath∞ limit reproduces the exact value for Tc{T}_{c}Tc​, and (d) for fixed zzz, all values of q>2q>2q>2 provide first-order phase transitions, which are exact for d>4d>4d>4. Furthermore, the procedures for obtaining, as functions of temperature and for any values of qqq and zzz, the order parameter, internal energy, and specific heat are outlined, and some typical illustrations are presented.
A new type of effective-field theory of the Ising model is presented. The differential-operator method is introduced into the exact spin correlation function identity obtained by Callen. The Curie temperatures are evaluated by using two different types of effective Hamiltonians. It is also shown how the Zernike and the Bethe-Peierls equations can be reproduced within one framework depending on the choice of effective fields in an effective Hamiltonian. The spin correlation function and the specific heat are presented.
Critical properties of the QQQ-state Potts model for dimensions $3\ensuremath{\le}d\ensuremath{\le}6$ are calculated by means of Wilson's exact momentum-space renormalization-group equation. The scaling-field method of Golner and Riedel is used to approximate the functional differential equation by a set of 11 ordinary coupled differential equations. For $d=4\ensuremath{-}\ensuremath{\epsilon}$, lines of critical and tricritical Potts fixed points are found as functions of QQQ that annihilate as QQQ approaches a critical value ${Q}_{c}=\frac{2+{\ensuremath{\epsilon}}^{2}}{a+O({\ensuremath{\epsilon}}^{3})}$. For Q>QcQ>{Q}_{c}Q>Qc​, the Potts transition is first order. Along these fixed lines the critical and tricritical exponents (upper and lower sign, respectively) are to leading order: $\frac{1}{\ensuremath{\nu}}=2\ensuremath{-}\frac{1}{6}[\ensuremath{\epsilon}\ifmmode\pm\else\textpm\fi{}{({\ensuremath{\epsilon}}^{2}\ensuremath{-}a\ensuremath{\delta})}^{\frac{1}{2}}]$, $\frac{\ensuremath{\varphi}}{\ensuremath{\nu}}=\ensuremath{\mp}{({\ensuremath{\epsilon}}^{2}\ensuremath{-}a\ensuremath{\delta})}^{\frac{1}{2}}$, and $\ensuremath{\eta}=\frac{{[\ensuremath{\epsilon}\ifmmode\pm\else\textpm\fi{}{({\ensuremath{\epsilon}}^{2}\ensuremath{-}a\ensuremath{\delta})}^{\frac{1}{2}}]}^{2}}{216}+b\ensuremath{\delta}$, where $\ensuremath{\epsilon}=4\ensuremath{-}d$, $\ensuremath{\delta}=Q\ensuremath{-}2$, and $\ensuremath{\delta}\ensuremath{\le}{\ensuremath{\delta}}_{c}=\frac{{\ensuremath{\epsilon}}^{2}}{a}+O({\ensuremath{\epsilon}}^{3})$. While the form of the $\ensuremath{\epsilon}$ and $\ensuremath{\delta}$ dependences is exact, the coefficients aaa and bbb cannot be obtained systematically by $\ensuremath{\epsilon}$ expansion, since the upper critical dimensionality of the Potts model is six when $Q\ensuremath{\ne}2$. In our truncation, a=6.52a=6.52a=6.52 and b=0.065b=0.065b=0.065. The results have been extended to dimensions $3.4\ensuremath{\le}d\ensuremath{\lesssim}4$ by solving the renormalization-group equations numerically. The percolation limit of the Potts model, Q=1Q=1Q=1, is also investigated and the critical exponents ${\ensuremath{\nu}}^{P},{\ensuremath{\varphi}}^{P}$, and ${\ensuremath{\eta}}^{P}$ determined as functions of dimension for $3\ensuremath{\le}d\ensuremath{\le}6$.
We present a large- q expansion of the 2d q -states Potts model free energies up to order 9 in 1/√ q . Its analysis leads us to an ansatz which, in the first-order region, incorporates properties inferred from the known critical regime at q = 4, and predicts, for q > 4, the n -th energy cumulant scales as the power (3 n /2 - 2) of the correlation length. The parameter-free energy distributions reproduce accurately, without reference to any interface effect, the numerical data obtained in a simulation for q = 10 with lattices of linear dimensions up to L = 50. The pure-phase specific heats are predicted to be much larger, at q ⩽ 10, than the values extracted from the current finite-size scaling analysis of the extrema. Implications for safe numerical determinations of interface tensions are discussed.
Earlier studies of the simple Ising model have been extended and the techniques applied to a variety of multistate and multispin models. Briefly presented is the authors' research results in which they have developed methods for the derivation of series developments for the Potts model, for which one of the most interesting questions is the order of the transition, for Ising models with four-spin interactions (including the eight-vertex model) and the three-spin triangular Ising model. For the three-spin triangular Ising model it has been found possible to conjecture the exact form of the spontaneous magnetisation. They have also continued to study the general cluster expansion for the high temperature specific heat of the Ising model of a ferromagnet. High field expansions have been derived for the Ising model on the hydrogen peroxide lattice. The exploration of the size and shape of polymer chains using insight gained from critical phenomena has continued.
A new type effective field approach is used to derive the thermodynamical properties of the transverse Ising model. The method is based on an exact formal identity for the two-state transverse Ising model and utilizes an exponential operator technique. The lines of critical points in the Ω-T plane are derived and the critical transverse fields ΩC are obtained analytically, for lattices with coordination numbers z = 2, 4 and 6. This simple method yields results which represent a remarkable improvement on the usual mean field approximation. Our results are also compared with those from series expansion method.
Within a new type of correlated-effective-field theory we calculate the effective-field parameters (including their response to external field), critical temperatures, susceptibility, short-range order parameters, and specific heat associated with an anisotropic, square-lattice, spin-\textonehalf{} Ising ferromagnet. The present formalism, which is shown to recover the Bethe-Peierls thermodynamics for the pure ZZZ-coordinated-lattice, spin-\textonehalf{} Ising model, yields satisfactory results; in particular, the crossover from a pure Ising ferromagnet on a square lattice to that on a linear chain is reasonably exhibited in the behavior of the susceptibility and the specific heat.