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Journal of Physics A Mathematical and General

Series studies of the four-state Potts model

R.V. Ditzian, Leo P. Kadanoff

Published Sep 1, 1979• 24 citations
Publisher

Abstract

The four-state Potts model is studied via series expansions. Free energy and susceptibility high-temperature series are calculated for hypercubic lattices of general dimensionality. Estimates of critical temperatures and critical indices gamma and alpha are obtained as functions of dimensionality. The free energy, magnetisation, and susceptibility low-temperature series are calculated for d=4. The high-and low-temperature series together indicate that the phase transition in this dimensionality is first order. From this result the authors suggest that the apparent second-order transitions shown by the susceptibility for d>or=4 might be explained as spinodal pseudotransitions.

DOI
10.1088/0305-4470/12/9/002

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High-field series expansions and critical properties for the three-state Potts model

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Series studies of the Potts model: I. the simple cubic Ising model

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The finite lattice method of series expansion is generalised to the $q$-state Potts model on the simple cubic lattice. It is found that the computational effort grows exponentially with the square of the number of series terms obtained, unlike two-dimensional lattices where the computational requirements grow exponentially with the number of terms. For the Ising ($q=2$) case we have extended low-temperature series for the partition functions, magnetisation and zero-field susceptibility to $u^{26}$ from $u^{20}$. The high-temperature series for the zero-field partition function is extended from $v^{18}$ to $v^{22}$. Subsequent analysis gives critical exponents in agreement with those from field theory.

•Jan 1, 1993•31 citations
The four-state Potts model is studied via series expansions. Free energy and susceptibility high-temperature series are calculated for hypercubic lattices of general dimensionality. Estimates of critical temperatures and critical indices gamma and alpha are obtained as functions of dimensionality. The free energy, magnetisation, and susceptibility low-temperature series are calculated for d=4. The high-and low-temperature series together indicate that the phase transition in this dimensionality is first order. From this result the authors suggest that the apparent second-order transitions shown by the susceptibility for d>or=4 might be explained as spinodal pseudotransitions.
High temperature series expansions of the staggered susceptibility of the q -state antiferromagnetic Potts model (AFPM) on a d -dimensional hypercubic lattice are calculated. For arbitrary values of q and d , the transition temperatures and the critical exponents γ are determined by use of the ratio method and the Padé approximation. It is shown that the 3- and 4-state AFPM exhibit a second-order phase transition when d ≥3, and the 5- and 6-state AFPM do when d ≥4, etc. In the limit of infinite d , the transition temperature T c is obtained rigorously as k T c / J ={log [2 d /(2 d - q )]} -1 ( k : the Boltzmann constant, J : the interaction parameter), and the ordered phase exists when d > q /2.
The three-state antiferromagnetic Potts model on d-dimensional hypercubic lattices is studied, where d is 2, 3, 4, 5 and 6. The Hamiltonian is expressed by the spin Hamiltonian with spin 1. The staggered susceptibility is calculated by the high temperature series expansion. By use of the ratio method and the Padé approximation, the critical temperature and the critical exponent γ are obtained. It is concluded that the three-state antiferromagnetic Potts model exhibits the second-order transition when d ≥3.
We develop high-temperature expansions for the Edwards-Anderson susceptibility of the \ifmmode\pm\else\textpm\fi{}J three-state Potts spin glasses on d-dimensional hypercubic lattices. For d\ensuremath{\ge}6, d log Pad\'e approximant analysis yields a second-order phase transition with an exponent \ensuremath{\gamma} consistent with the mean-field value of unity. In lower dimensions the extrapolations are not convergent enough to extract the critical behavior. In d=3, the series are analyzed assuming various finite- and zero-temperature transitions and their consistency with available Monte Carlo data is studied.
For paper II, see ibid, vol. 27, p.1503 (1994). The finite-lattice method of series expansion has been used to extend low-temperature series for the partition function, order parameter and susceptibility of the 3-state Potts model on the simple cubic lattice to order z 43 and the high-temperature expansion of the partition function to order v 21 . We use the numerical data to show that the transition is first-order, and estimate the latent heat, the discontinuity in the magnetization, and a number of other critical parameters.
The finite lattice method of series expansion is generalised to the qqq-state Potts model on the simple cubic lattice. It is found that the computational effort grows exponentially with the square of the number of series terms obtained, unlike two-dimensional lattices where the computational requirements grow exponentially with the number of terms. For the Ising (q=2q=2q=2) case we have extended low-temperature series for the partition functions, magnetisation and zero-field susceptibility to u26u^{26}u26 from u20u^{20}u20. The high-temperature series for the zero-field partition function is extended from v18v^{18}v18 to v22v^{22}v22. Subsequent analysis gives critical exponents in agreement with those from field theory.