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

A method for calculating complete secular polynomials is discussed that is based on the evaluation of matrix elements of a specific Hamiltonian. Sever al Hamiltonians are presented and described in detail as well as their phys ical significance. It is shown that they can be transformed into an equival ent form in terms of raising and lowering operators, and the third componen t 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 polyno mials for an anisotropic Hamiltonian with one anisotropy parameter and a Ha miltonian with two anisotropy parameters for several values of N are report ed. These polynomials that have not appeared before are given in terms of b oth the energy and anisotropy variables.