# Finite-temperature degenerate perturbation theory for bosons in optical lattices

## TL;DR

This paper develops a finite-temperature degenerate perturbation theory for bosons in optical lattices to accurately describe the Mott-insulator to superfluid phase transition, overcoming limitations of previous non-degenerate methods.

## Contribution

It introduces a degenerate perturbation theory approach based on projection operators to resolve degeneracy issues near the phase transition in the Bose-Hubbard model.

## Key findings

- Provides physically consistent order parameters near the phase transition.
- Addresses degeneracy problems in mean-field approximations.
- Improves theoretical understanding of phase behavior in optical lattice bosons.

## Abstract

Bosonic atoms confined in optical lattices can exist in two different phases, Mott-insulator and superfluid, depending on the strength of the system parameters, such as the on-site interaction between particles and the hopping parameter. This work is motivated by the fact that non-degenerate perturbation theory applied to the mean-field approximation of the Bose-Hubbard Hamiltonian at zero and finite temperature fails to give consistent results in the vicinity of the Mott-insulator-superfluid phase transition, e.g., the order parameter calculated via non-degenerate perturbation theory reveals an unphysical behavior between neighboring Mott lobes, which is an explicit consequence of degeneracy problems that artificially arise from such a treatment. Therefore, in order to fix this problem, we propose a finite-temperature degenerate perturbation theory approach based on a projection operator formalism which ends up solving such degeneracy problems in order to obtain physically consistent results for the order parameter near the phase transition.