Exact eigenvalues of the Ising Hamiltonian in one-, two- and three-dimensions in the absence of a magnetic field

The Hamiltonian of the Ising model in one-, two- and three-dimensions has b een analysed using unitary transformations and combinatorics. We have been able to obtain closed formulas for the eigenvalues of the Ising Hamiltonian for an arbitrary number of dimensions and sites. Although the solution pro vided assumes the absence of external magnetic fields an extension to inclu de a magnetic field along the z-axis is readily extracted. Furthermore, gen eralisations to a higher number of spin components on each site are possibl e within this method. We made numerical comparisons with the partition func tion from the earlier analytical expressions known in the literature for on e- and two-dimensional cases. We find complete agreement with these studies . (C) 2001 Elsevier Science B.V. All rights reserved.