Expansion of the Gibbs potential for quantum many-body systems: General formalism with applications to the spin glass and the weakly non-ideal Bose gas

TL;DR

This paper develops a systematic expansion of the Gibbs potential for quantum many-body systems, providing a unified formalism with explicit calculations for spin glasses and Bose gases, capturing beyond mean-field effects.

Contribution

It introduces a general formalism for expanding the Gibbs potential in quantum systems, including a projector technique and applications to spin glasses and Bose gases.

Findings

Calculated Onsager terms for spin glass and Bose gas.

Derived new TAP-like equations for spin glass.

Achieved agreement with existing results at Bose-Einstein condensation temperature.

Abstract

For general quantum systems the power expansion of the Gibbs potential and consequently the power expansion of the self energy is derived in terms of the interaction strength. Employing a generalization of the projector technique a compact representation of the general terms of the expansion results. The general aspects of the approach are discussed with special emphasis on the effects characteristic for quantum systems. The expansion is systematic and leads directly to contributions beyond mean-field of all thermodynamic quantities. These features are explicitly demonstrated and illustrated for two non-trivial systems, the infinite range quantum spin glass and the weakly interacting Bose gas. The Onsager terms of both systems are calculated, which represent the first beyond mean-field contributions. For the spin glass new TAP-like equations are presented and discussed in the paramagnetic region. The investigation of the Bose gas leads to a beyond mean-field thermodynamic description. At the Bose-Einstein condensation temperature complete agreement is found with the results presented recently by alternative techniques.