Systematic nonperturbative approach for thermal averages in quantum many-body systems: The thermal-cluster-cumulant method.

Abstract

We present in this paper a systematic nonperturbative cluster-cumulant method for deriving averages of operators in quantum many-body systems. The method combines the advantages of the cumulant expansion scheme of thermodynamic perturbation theory, the approach of thermofield dynamics as a finite-temperature field theory, and the time-dependent coupled-cluster theory extended to imaginary time. We have generalized the concepts of in a nonperturbative manner and have posited on the statistical operator an exponential-like ansatz containing connected, size-extensive operators in the exponent. These latter cumulantlike operators have been termed cluster cumulants by us. For a compact treatment, we have derived an alternative field theory in which a time-ordered product is expanded in terms of thermal normal of operators and contractions-leading to a thermal Wick's theorem. The normal products are the finite-temperature analogs of the ordinary normal products and have zero averages. Operators in these products commute (anticommute) under permutations for bosons (fermions). This representation is shown to be unitarily related to the traditional thermofield dynamics formulation, but has the advantage of using only the physical variables. The imaginary-time evolution of the statistical operator is treated by our recently formulated time-dependent cluster-cumulant theory. The partition function is evaluated as an exponential of a connected quantity. As an illustrative example, we have computed the partition function of an anharmonic oscillator with equally weighted cubic and quartic perturbation for a wide range of coupling, extending to the strongly nonperturbative regime. We study the behavior of free energy in the low-temperature limit and verify numerically the validity of the Kohn-Luttinger theorem [Phys. Rev. 118, 41 (1960)] for this system. We also show that our formalism is a natural nonperturbative analog of the thermodynamic perturbative theory by showing that a perturbative solution of the thermal-cluster-cumulant equations generates a variation of the Bloch-Balian-de Dominicis theory.