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Journal of Statistical Mechanics Theory and Experiment

Inclusion–exclusion relations for series expansions of boundary effects using the finite lattice method

Ian Graham Enting

Published Dec 14, 2005• 3 citations
Publisher

Abstract

A number of new results are derived for the weights applied to finite lattice partition functions in the finite lattice method of series expansions on the square lattice. New weights are given for a truncation appropriate for calculating low-temperature surface series for the case of free boundaries, allowing a 50% increase in the length of series obtained for a given size of transfer matrix.

DOI
10.1088/1742-5468/2005/12/p12007

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A number of new results are derived for the weights applied to finite lattice partition functions in the finite lattice method of series expansions on the square lattice. New weights are given for a truncation appropriate for calculating low-temperature surface series for the case of free boundaries, allowing a 50% increase in the length of series obtained for a given size of transfer matrix.
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 has been used to extend low-temperature series for the partition function, order parameter and susceptibility of the spin-1 Ising model on the square lattice. A new formalism is described which uses two distinct transfer-matrix approaches in order to significantly reduce computer memory requirements and which permits the derivation of the series to 79th order. Subsequent analysis of the series clearly confirms that the spin-1 model has the same dominant critical exponents as the spin- 1/2 Ising model. Accurate estimates for both the critical temperature and non-physical singularities are obtained. In addition, evidence for a non-analytic confluent correction with exponent Delta 1 =1.1+or-0.1 is found.
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.
Abstract The partition function of the square lattice Ising model on the rectangle with open boundary conditions in both directions is calculated exactly for arbitrary system size L × M and temperature. We start with the dimer method of Kasteleyn, McCoy and Wu, construct a highly symmetric block transfer matrix and derive a factorization of the involved determinant, effectively decomposing the free energy of the system into two parts, F ( L , M ) = F s t r i p ( L , M ) + F s t r i p r e s ( L , M ) , where the residual part F s t r i p r e s ( L , M ) contains the nontrivial finite- L contributions for fixed M . It is given by the determinant of a M / 2 × M / 2 matrix and can be mapped onto an effective spin model with M Ising spins and long-range interactions. While F s t r i p r e s ( L , M )
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.