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The Pfaffian solution of the Ising model

Barry M. McCoy

Published Dec 1, 2009• 0 citations
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Abstract

Abstract This chapter presents the computation of the partition function of the two-dimensional Ising model in terms of a sum of Pfaffians using a combinatorial approach. This method is extended to derive the determinental expressions for the correlation functions.
DOI
10.1093/acprof:oso/9780199556632.003.0011

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The combinatorial approach to the triangular Ising model is shown to be considerably simplified at a given temperature TD. This enables the partition function and spin-correlation functions to be evaluated without the use of pfaffians, thus providing a simple derivation of a few of the results obtained previously by Stephenson. We confirm the results of Stephenson, that at the temperature TD, the correlation functions have the same structure as the correlations of a one-dimensional Ising lattice. The only new result is a general expression for some of the spin–spin correlation functions when the spins do not lie on the axes of the lattice.
The partition function for the Ising problem for a general class of plane lattices is evaluated exactly using the Pfaffian method. A classification of plane lattices in terms of basic lattices with vertices decorated by sublattices is given, and the most general class which can be solved by the Pfaffian method described. The solutions given in this paper are for the special case when neighbouring lattice points of a basic rectangular lattice are connected by single bonds only. All of the exact solutions known so far are obtained as special cases. The critical points are defined as singularities of an analytic function and equations to determine their location are found. The derivatives of the partition function are evaluated in terms of elliptic integrals.
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We give a self-contained exposition of the combinatorial solution of quantum mechanical systems of coupled spins on a one-dimensional lattice. Using Trotter formula, we write the partition function as a generating function of a spanning subgraph of a two-dimensional lattice and solve the combinatorial problem by the method of Pfaffians provided the weights satisfy the so-called free fermion condition. The free energy and the ground state energy as a function of the inverse temperature, couplings J and magnetic fields h, for the XY model in a transverse field with period p=1 and 2, is then obtained.
In the algebraic formulation of the Ising model, the partition function is expressed as the trace of a power VM of the transfer operator V, or equivalently, as the sum of Mth powers of the eigenvalues of V. In the derivations of Kaufman, Onsager, and more recently of Schultz, Mattis, and Lieb (SML), the transfer operator is first reduced to a more amenable form for computation and then, in principle at least, diagonalized. For the infinite lattice, only the largest eigenvalue of V is needed, and this is all Onsager and SML compute. Kaufman finds all the eigenvalues and is thus able to write down the partition function for the finite lattice. In the present work we give an alternative derivation of the SML form for V, and show how the Kaufman result can be obtained from this form without actual diagonalization. Instead of diagonalizing V, the evaluation of the trace is done directly after assigning a simple representation to V.