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arXiv:cond-mat/9704148 · Journal of Physics A Mathematical and General

Low-temperature series for the square lattice Potts model using the improved finite-lattice method

H. Arisue, Kanehito Tabata

Published May 21, 1997• 6 citations
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Abstract

The low-temperature series are calculated for the free energy, magnetization and susceptibility in the Q-state Potts model on the square lattice, using the improved algorithm of the finite lattice method. The series are obtained to the order of for each of Q = 5 - 50, and the result of their Padé-type analysis is compared with those of the large-Q expansion and the Monte Carlo simulations.

DOI
10.1088/0305-4470/30/10/011

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The low-temperature series are calculated for the free energy, magnetization and susceptibility in the Q-state Potts model on the square lattice, using the improved algorithm of the finite lattice method. The series are obtained to the order of for each of Q = 5 - 50, and the result of their Padé-type analysis is compared with those of the large-Q expansion and the Monte Carlo simulations.
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 q-state Potts model to order z 56 (i.e. u 28 ), z 47 , z 43 , z 39 , z 39 , z 39 , z 35 , z 31 and z 31 for q=2, 3, 4, ..., 9 and 10 respectively. These series are used to test techniques designed to distinguish first-order transitions from continuous transitions. New numerical values are also obtained for the q-state Potts model with q>4.
The finite lattice method is used to derive low-temperature series expansions for surface magnetisations for the 3-state Potts model on the square lattice, for both fixed and free boundary conditions. Combinatorial weights required for surface corrections to the free energy have been generalised to include cases with explicit surface interactions. Analysis of series for the surface correction to the bulk magnetisation is consistent with the generic scaling prediction, assuming a slow convergence comparable to that found in analysis of Potts model series for bulk properties. For free boundaries, analysis of the ordering of the surface layer confirms the specific Potts model result derived from conformal invariance.
The finite-lattice method of series expansion is generalized 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 the authors have extended the low-temperature series for the partition functions, magnetization and zero-field susceptibility to u 26 from u 20 . The high-temperature series for the zero-field partition function is extended from nu 18 to nu 22 . Subsequent analysis gives critical exponents in agreement with those from field theory.
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.