Improving mean-field theory for bosons in optical lattices via degenerate perturbation theory

Abstract

The objective of this paper is the theoretical description of the Mott-insulator to superfluid quantum phase transition of a Bose gas in an optical lattice. In former works the Rayleigh-Schrödinger perturbation theory was used within a mean-field approach, which yields partially non-physical results since the degeneracy between two adjacent Mott lobes is not taken into account. In order to correct such non-physical results we apply the Brillouin-Wigner perturbation theory to the mean-field approximation of the Bose-Hubbard model. Detailed explanations of how to use the Brillouin-Wigner theory are presented, including a graphical approach that allows to efficiently keep track of the respective analytic terms. To prove the validity of this computation, the results are compared with other works. Besides the analytic calculation of the phase boundary from Mottinsulator to superfluid phase, the condensate density is also determined by simultaneously solving two algebraic equations. The analytical and numerical results turn out to be physically meaningful and can cover a region of system parameters inaccessible until now. Our results are of particular interest provided an harmonic trap is added to the former calculations in an homogeneous system, in view of describing an experiment within the local density approximation. Thus, the paper represents an essential preparatory work for determining the experimentally observed wedding-cake structure of particle-density profile at both finite temperature and hopping.PACS numbers: 67.85.Hj,67.85.-d * martin kuebler@gmx.de †

Finite-temperature degenerate perturbation theory for bosons in optical lattices

“…It must be noted that for the zero-temperature regime, which is depicted in Fig. 4d, the results for the condensate densities are similar to those obtained in [19], which uses a Brillouin-Wigner treatment for the perturbation expansion followed by a proper diagonalization in order to calculate the system free energy. That approach differs from the one used in this work.…”

“…That approach differs from the one used in this work. While the Brillouin-Wigner approach in [19] is also able to correct the degeneracy problems from NDPT, it can only be applied to the zero-temperature case. On the other hand, the theory presented in this paper corrects degeneracy problems for both zero and finite temperatures, thus providing a relatively simple method for calculating the condensate density in a wide range of optical-lattice systems.…”

“…Likewise, [18] implemented a nearly degenerate perturbation theory for the zero-temperature case, which led to better results for the order parameter (OP) when compared to those from the RSPT calculations. More recently, Brillouin-Wigner perturbation theory was applied in order to correct such degeneracy-generated unphysical results at zero-temperature [19]. It turns out that nondegenerate finite-temperature perturbation theory, as applied in Refs.…”

A study on quantum gases: bosons in optical lattices and the one-dimensional interacting Bose gas

“…with the help of the projection operators. 185,187 To this end, we insert the unity operator 1 =P +Q on both sides of (4.1), yieldinĝ…”

“…Figure 11 -Condensate density from the one-state approach for n = 1 (negative ε/U , purple squares) and n = 2 (positive ε/U , red circles), with the hopping strength of Jz/U = 0.08. Source: KÜBLER et al 185 The condensate density Ψ * Ψ follows from iteratively solving both (4.40) and (4.41). The results are plotted in Fig.…”