A close correspondence is demonstrated between the quanatum Hamiltonian for an Ising model in d space dimensions and continuous time with the one-spin transfer matrix for a (d+1)-dimensional Ising model. The analysis is done analytically, showing the relation in detail of the eigenvectors and eigenvalues of the two problems.
The Hamiltonian of the Ising model in d dimensions has been analysed using unitary transformations and combinatorics. We have been able to obtain closed formulas for the eigenvalues of the Ising Hamiltonian in an arbitrary number of dimensions. Furthermore, generalisations to a higher number of spin components on each site are possible within this method. We have made numerical comparisons with the partition function from the earlier analytical expressions known in the literature for one and two-dimensional cases and found complete agreement. (C) 2001 Elsevier Science B.V. All rights reserved.
By relating a transfer matrix associated with a classical system to the Hamiltonian of its corresponding quantum mechanical system, it is proved that the two-dimensional Ising model in the absence of a magnetic field is equivalent to the ground state of the linear XY-model in the presence of a magnetic field, under appropriate relations among coupling parameters appearing in the two Hamiltonians. Simple relations are established for spin correlations in both systems. Consequently, the Ising system and the ground state of the XY-model are shown to exhibit similar singularities with respect to temperature and a magnetic field, respectively. It is also proved that the two-dimensional dimer problem is equivalent to the ground state of a generalized XY-model, which is solved exactly in general.
Abstract The connection matrix of the Ising model on a d-dimensional hypercube is investigated. In addition to the interactions between the nearest neighbors, the interactions between the next-nearest neighbors are taken into account. For such a matrix, the exact relations for the eigenvalues and eigenvectors, which are reasonably simply expressed through the corresponding characteristics of the one-dimensional Ising model, are obtained. Both periodic and free boundary conditions are considered.
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.