Secular polynomials in energy and coupling for lattice spin models calculated using Hamiltonian matrix elements

Abstract

A method for calculating complete secular polynomials is discussed that is based on the evaluation of matrix elements of a specific Hamiltonian. Several Hamiltonians are presented and described in detail as well as their physical significance. It is shown that they can be transformed into an equivalent form in terms of raising and lowering operators, and the third component of the spin operator. A basis set is defined and the action of a specific Hamiltonian on the basis set is described in detail. Several Hamiltonians are given explicitly and in matrix form. Results in terms of secular polynomials for an anisotropic Hamiltonian with one anisotropy parameter and a Hamiltonian with two anisotropy parameters for several values of are reported. These polynomials that have not appeared before are given in terms of both the energy and anisotropy variables.