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arXiv:1303.1642 · Journal of Statistical Mechanics Theory and Experiment

Phase transitions in three-dimensional bosonic systems in optical lattices

H Kleinert, Z Narzikulov, Abdulla Rakhimov

Published Jan 16, 2014• 6 citations
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Abstract

We formulate the collective quantum field theory for three-dimensional bosonic optical lattices and evaluate its consequences in a mean-field approximation to two collective fields, proposed by Cooper et al (2011 Phys. Rev. A 83 053622), and in a lowest-order variational perturbation theory. It is shown that the present mean-field approximation predicts some essential features of the experimentally observed dependence of the critical temperature on the coupling strength and a second-order quantum phase transition. In contrast to a recent prediction for atomic gases given by Cooper et al , our finding indicates no superfluid state with zero condensate fraction.

DOI
10.1088/1742-5468/2014/01/p01003

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We formulate the collective quantum field theory for three-dimensional bosonic optical lattices and evaluate its consequences in a mean-field approximation to two collective fields, proposed by Cooper et al (2011 Phys. Rev. A 83 053622), and in a lowest-order variational perturbation theory. It is shown that the present mean-field approximation predicts some essential features of the experimentally observed dependence of the critical temperature on the coupling strength and a second-order quantum phase transition. In contrast to a recent prediction for atomic gases given by Cooper et al , our finding indicates no superfluid state with zero condensate fraction.
New results are presented for bosons defined on the three-dimensional (3D) cubic lattice at very low temperatures. Calculations are made within the mean-filed (or Hartree) theory applied to the so-called O(2) model. The analysis is used to calculate the zero-temperature phase diagram and determine the critical coupling strength λ for a quantum phase transition. Properties of the zero temperature quantum state, such as: the energy per site, the order parameter, the first possible excitation, and the test single-site phase wave function, are displayed and discussed. Their behavior predicts a transition from an ordered (superfluid) to a disordered (normal/insulating) phase at a critical value of the coupling constant λc = 6.
We study the quantum phase transition from a superfluid to a Mott insulator in optical lattices using a Bose-Hubbard Hamiltonian. For this purpose we develop a field theoretical approach in terms of path integral formalism to calculate the second-order quantum corrections to the energy density as well as to the superfluid fraction in cubic optical lattices. Using the present approach, the condensate fraction and ground-state energy are calculated as functions of the sss-wave scattering length. In contrast to the Bogoliubov model, which is, technically speaking, a one-loop approximation, we carry the calculation up to two loops and improve the result further by variational perturbation theory. The result suggests that the quantum phase transition exists.
We work out two different analytical methods for calculating the boundary of the Mott-insulator--superfluid (MI-SF) quantum phase transition for scalar bosons in cubic optical lattices of arbitrary dimension at zero temperature which improve upon the seminal mean-field result. The first one is a variational method, which is inspired by variational perturbation theory, whereas the second one is based on the field-theoretic concept of effective potential. Within both analytical approaches we achieve a considerable improvement of the location of the MI-SF quantum phase transition for the first Mott lobe in excellent agreement with recent numerical results from quantum Monte Carlo simulations in two and three dimensions. Thus, our analytical results for the whole quantum phase diagram can be regarded as being essentially exact for all practical purposes.
We investigate the superfluid--Mott-insulator quantum phase transition of spin-1 bosons in an optical lattice created by pairs of counterpropagating linearly polarized laser beams, driving an Fg=1{F}_{g}=1Fg​=1 to Fe=1{F}_{e}=1Fe​=1 internal atomic transition. The whole parameter space of the resulting two-component Bose-Hubbard model is studied. We find that the phase transition is not always second order as in the case of spinless bosons, but can be first order in certain regions of the parameter space. The calculations are done in the mean-field approximation by means of exact numerical diagonalization as well as within the framework of perturbation theory.
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.
In this paper, the quantum phase transition between the superfluid state and the Mott-insulator state is studied based on an extended Bose-Hubbard model with two- and three-body on-site interactions. By employing the mean-field approximation we find the extension of the insulating ``lobes'' and the existence of a fixed point in three-dimensional phase space. We investigate the link between experimental parameters and theoretical variables. The possibility to observe our results through some experimental effects in optically trapped Bose-Einstein condensates is also discussed.