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Physica A Statistical Mechanics and its Applications

On solving the Ising model functional relation without elliptic functions

R. J. Baxter

Published Sep 1, 1991• 7 citations
Publisher
DOI
10.1016/0378-4371(91)90140-8

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Abstract In this chapter we study the solution of the two-dimensional Ising model by means of the transfer matrix. Unlike the methods discussed in the previous chapter, the transfer matrix approach has greater generality and can used to solve exactly other two-dimensional models. Even if the general ideas behind this approach have been explained in Chapter 2 by means of the one-dimensional case, their application to the two-dimensional cases requires more powerful and sophisticated mathematical tools: for instance, the study the eigenvalues of the transfer matrix in the Ising model for T = Tcneeds to employ elliptic functions. The same is also true for other models. In order to present in the simplest possible way the main lines of this method, in the following we focus attention only on the solution of the model at T = Tcbecause this case can be analyzed in terms of simple trigonometric functions.

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The quantum annealing devices, which encode the solution to a computational problem in the ground state of a quantum Hamiltonian, are implemented in D-Wave systems with more than 2,000 qubits. However, quantum annealing can solve only a classical combinatorial optimization problem such as an Ising model, or equivalently, a quadratic unconstrained binary optimization (QUBO) problem. In this paper, we formulate the QUBO model to solve elliptic problems with Dirichlet and Neumann boundary conditions using the finite element method. In this formulation, we develop the objective function of quadratic binary variables represented by qubits and the system finds the binary string combination minimizing the objective function globally. Based on the QUBO formulation, we introduce an iterative algorithm to solve the elliptic problems. We discuss the validation of the modeling on the D-Wave quantum annealing system.
Correlation functions of the two-dimensional Ising model on the periodic lattice can be expressed in terms of form factors — matrix elements of the spin operator in the basis of common eigenstates of the transfer matrix and translation operator. Free-fermion structure of the model implies that any multiparticle form factor is given by the pfaffian of a matrix constructed from the two-particle ones. Crossed two-particle form factors can be obtained by inverting a block of the matrix of linear transformation induced on fermions by the spin conjugation. We show that the corresponding matrix is of elliptic Cauchy type and use this observation to solve the inversion problem explicitly. Non-crossed two-particle form factors are then obtained using theta functional interpolation formulas. This gives a new simple proof of the factorized formulas for periodic Ising form factors, conjectured by A. Bugrij and one of the authors.
Abstract In this chapter we study the solution of the two-dimensional Ising model by means of the transfer matrix. Unlike the methods discussed in the previous chapter, the transfer matrix approach has greater generality and can used to solve exactly other two-dimensional models. Even if the general ideas behind this approach have been explained in Chapter 2 by means of the one-dimensional case, their application to the two-dimensional cases requires more powerful and sophisticated mathematical tools: for instance, the study the eigenvalues of the transfer matrix in the Ising model for T = Tcneeds to employ elliptic functions. The same is also true for other models. In order to present in the simplest possible way the main lines of this method, in the following we focus attention only on the solution of the model at T = Tcbecause this case can be analyzed in terms of simple trigonometric functions.