Standard-Basis Operator Method in the Green's-Function Technique of Many-Body Systems with an Application to Ferromagnetism

Abstract

A new formalism, using standard-basis matrix operators, is presented for the study of the collective excitations and the thermodynamic properties of an ensemble of identical interacting quantum-mechanical systems subsequently called ions, each ion having discrete energy levels. A model Hamiltonian is formulated in terms of these operators. The Hamiltonian contains terms which express the interaction of the ion with the crystal field and the external fields as well as terms which arise from the mutual interaction of ions. Using the doubletime temperature-dependent Green's-function technique, the equations of motion of the Green's functions of standard-basis operators are developed in the random-phase decoupling approximation. It is demonstrated that the temperature-dependent correlation functions of standard-basis operators, which are obtained from associated spectral Green's functions, lead to a set of equations that can be solved for the occupation probabilities of the single-ion energy levels. Hence, one can calculate the thermal-average expectation value of any quantum-mechanical operator representing a microscopic observable of a single ion, or a pair of ions (correlations). An important feature of the standard-basis matrix-operator formalism is that the single-ion terms, such as crystal field, molecular field, or external fields, are always treated exactly in any Green's-function decoupling scheme. In contrast, Green's-function methods which use angular momentum operators often necessarily treat single-ion terms in a decoupling approximation. As an example, the general standard-basis operator theory is applied to the Heisenberg ferromagnet in the presence of uniaxial single-ion crystal-field anisotropy, which has received extensive theoretical attention previously, with widely varying results.