# Variational Approach to Many-Body Problems Incorporating Many-Body Effects at Finite Temperature

## TL;DR

This paper introduces a finite-temperature variational method that self-consistently includes many-body correlations, revealing enhanced dynamical processes and potential qualitative changes in quasiparticle properties in weakly interacting Bose-Einstein condensates.

## Contribution

A novel finite-temperature variational formalism that incorporates many-body effects through Green's functions and self-consistent parameters.

## Key findings

- Many-body effects lower free energy compared to mean-field approximations.
- Enhanced pair creation/annihilation processes at long wavelengths.
- 3/2-body correlations significantly affect quasiparticle self-energy.

## Abstract

We develop a variational approach at finite temperature that incorporates many-body correlation self-consistently. The grand potential is constructed in terms of Green's function expressed by the variational parameters. We apply this formalism to weakly interacting Bose-Einstein condensates to incorporate the dynamical 3/2-body processes, which are considered important in the dynamical properties. The processes lower the free energy below the mean-field Hartree--Fock--Bogoliubov's value in the same way as a previous zero-temperature formalism. From our numerical results, the pair creation or annihilation processes neglected in the Popov--Shohno approximation are enhanced, particularly in the long wavelength region, owing to the many-body effects. Because the 3/2-body correlations give a finite contribution to the self-energy of quasiparticles, they may change the microscopic properties qualitatively, even in the weak-coupling region.