Finite-temperature degenerate perturbation theory for bosons in optical lattices

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 nondegenerate perturbation theory applied to the mean-field approximation of the Bose-Hubbard Hamiltonian at both 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 nondegenerate 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. This problem also appears in other bosonic systems that present similar Mott-lobe structures. 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.

Figure 1

Phase diagram for the inverse temperatures $\beta =5/U$ (dotted-dashed black), $\beta =10/U$ (dashed red), $\beta =30/U$ (dotted green), and $\beta \to \infty$ (continuous blue).

Figure 2

Condensate density (a) from (20) and particle density (b) from (21) via NDPT as functions of $\mu /U$ for $tz/U=0.2$ as well as $\beta =5/U$ (dotted-dashed black), $\beta =10/U$ (dashed red), $\beta =30/U$ (dotted green), and $\beta \to \infty$ (continuous blue).

Figure 3

Condensate densities as functions of $\mu /U$ evaluated from FTDPT via (36) for four different temperatures: (a) $\beta =5/U$, (b) $\beta =10/U$, (c) $\beta =30/U$, and (d) $T=0$. Different point styles correspond to different hopping: $tz/U=0.2$ (blue circles), $tz/U=0.15$ (orange squares), $tz/U=0.1$ (green rhombuses), $tz/U=0.05$ (red triangles), and $tz/U=0.01$ (purple inverted triangles).

Figure 4

Comparison between the condensate densities calculated via FTDPT (points) and NDPT (lines) for the temperatures $\beta =30/U$ (left panel) and $T=0$ (right panel) for $tz/U=0.2$ (blue circles and continuous blue lines), $tz/U=0.15$ (orange squares and dashed orange lines), and $tz/U=0.1$ (green rhombuses and dotted green lines). Panels (a) and (b) correspond to the region between $n=0$ and 1, while (c) and (d) correspond to the region between the first and second lobes.

Figure 5

Equation of state for the hopping strengths (a) $tz/U=0.05$ and (b) $tz/U=0.1$ and the temperatures $T=0$ (continuous blue), $\beta =30/U$ (dotted green), $\beta =10/U$ (dashed red), and $\beta =5/U$ (dotted-dashed black).