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arXiv:hep-lat/9312083 · Journal of Physics A Mathematical and General

Series studies of the Potts model: III. The 3-state model on the simple cubic lattice

Anthony John Guttmann, Ian Graham Enting

Published Sep 7, 1994• 33 citations
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Abstract

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.

DOI
10.1088/0305-4470/27/17/014

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Journal of Physics A Mathematical and General•Sep 7, 1994•0 citations
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 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.
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 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.
A linked-cluster type of series expansion of the free energy with a variational parameter is developed to the eighth order for the q-state Potts model. The order parameter and the internal energy are then derived. By analyzing these series expansions, critical temperatures Tc{\mathit{T}}_{\mathit{c}}Tc​, latent heats, and discontinuities of the order parameters at Tc{\mathit{T}}_{\mathit{c}}Tc​ are determined for the cubic lattices.
Both 'high-' and 'low-temperature' perturbation series are used to locate and characterize the first-order transition in the three-state Potts model in (3+1) dimensions on the simple-cubic, body-centred cubic and face-centred cubic lattices. Estimates are presented for the vacuum energy, 'latent heat', magnetization, susceptibility and mass gap at the transition. The results show a remarkable degree of universality between the different lattices, and a 'law of corresponding states' is followed very closely.