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Journal of Physics Conference Series

Surface magnetisation of the 3-state Potts model with free and fixed boundaries on the square lattice

Ian Graham Enting

Published Jun 1, 2006• 2 citations
Publisher

Abstract

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.

DOI
10.1088/1742-6596/42/1/009

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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.
We obtain long series expansions for the bulk, surface and corner free energies for several two-dimensional statistical models, by combining Enting's finite lattice method (FLM) with exact transfer matrix enumerations. The models encompass all integrable curves of the Q -state Potts model on the square and triangular lattices, including the antiferromagnetic transition curves and the Ising model ( Q = 2) at temperature T , as well as a fully packed O( n ) type loop model on the square lattice. The expansions are around the trivial fixed points at infinite Q , n or 1/ T . By using a carefully chosen expansion parameter, q ≪ 1, all expansions turn out to be of the form , where the coefficients α k and β k are periodic functions of k . Thanks to this periodicity property, we can conjecture the form of the expansions to all orders (except in a few cases where the periodicity is too large). These expressions are then valid for all 0 ⩽ q < 1. We analyse in detail the q → 1 − limit in which the models become critical. In this limit the divergence of the corner free energy defines a universal term which can be compared with the conformal field theory (CFT) predictions of Cardy and Peschel. This allows us to deduce the asymptotic expressions for the correlation length in several cases. Finally we work out the FLM formulae for the case where some of the system's boundaries are endowed with particular (non-free) boundary conditions. We apply this in particular to the square-lattice Potts model with Jacobsen–Saleur boundary conditions, conjecturing the expansions of the surface and corner free energies to arbitrary order for any integer value of the boundary interaction parameter r . These results are in turn compared with CFT predictions.
The analysis of renormalisation group equations for the q-state Potts model in two dimensions is generalised to include the presence of a surface field H s besides the bulk one, H B . Combined with conformal invariance results, this allows prediction of a surface susceptibility behaviour delta 2/f delta H s delta H B approximately L -1/8 (1-C'lnL) 15/16 for a q=4 system of size L at the critical temperature. This prediction is checked by a Monte Carlo based finite-size scaling analysis, which also nicely reproduces the exact and conjectured magnetic surface exponents for q=2 and 3, respectively. For the Baxter-Wu model similar methods give results consistent with the q=4 behaviour without logarithmic corrections (c'=0).
We consider the anisotropic Ising model on the triangular lattice with finite boundaries, and use Kaufman's spinor method to calculate low-temperature series expansions for the partition function to high order. From these we can obtain 108-term series expansions for the bulk, surface and corner free energies. We extrapolate these to all terms and thereby conjecture the exact results for each. Our results agree with the exactly known bulk free energy. For the isotropic case, they also agree with Vernier and Jacobsen's conjecture for the 60∘60^{\circ}60∘ corners, and with Cardy and Peschel's conformal invariance predictions for the dominant behaviour at criticality.
We consider the anisotropic Ising model on the triangular lattice with finite boundaries, and use Kaufman’s spinor method to calculate low-temperature series expansions for the partition function to high order. From these, we can obtain 108-term series expansions for the bulk, surface and corner free energies. We extrapolate these to all terms and thereby conjecture the exact results for each. Our results agree with the exactly known bulk-free energy and with Cardy and Peschel’s conformal invariance predictions for the dominant behaviour at criticality. For the isotropic case, they also agree with Vernier and Jacobsen’s conjecture for the 60 ° corners.
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.