Thermodynamics of interacting many-level systems from random-phase-approximation Green's functions

Standard-basis-operator (SBO) Green's function equations of motion are developed in the random-phase approximation (RPA) for a general Hamiltonian characterizing interacting systems having discrete energy levels. It is shown that the RPA equations have the same algebraic structure as the chain-diagram equations in the perturbation theory of SBO Green's functions. The earlier problems of redundancy and failure to satisfy the monotopic restrictions in the calculation of autocorrelation functions are resolved. Ratios of RPA autocorrelation functions can always be determined from commutator equations from which thermal average energy-level occupation probabilities $P_{\beta }$ associated with state $|\beta 〉$ are obtained. Using the properties of the SBO, a very simple derivation of an explicit formula for $P_{\beta }$ is given for an important class of Hamiltonians, which includes the Hubbard $s$-band model and the Heisenberg ferromagnet. This derivation completely avoids the use of complicated methods such as the moment generator used to solve equations coupling average moments of the spin operator ${\hat{S}}_Z$ in ferromagnetic and antiferromagnetic systems.