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.
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-Schr\"odinger perturbation theory was used within a mean-field approach, which yields partially nonphysical results since the degeneracy between two adjacent Mott lobes is not taken into account. In order to correct such nonphysical 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 one 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 Mott insulator 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 a harmonic trap is added to the former calculations in a 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 finite hopping.
Bosonic atoms confined in optical lattices are described by the Bose-Hubbard model and can exist in two different phases, Mott insulator or superfluid, depending on the strength of the system parameters. In the vicinity of the phase boundary, there are degeneracies that occur between every two adjacent Mott lobes. Because of this, nondegenerate perturbation theory fails to give meaningful results for the condensate density: it predicts a phase transition in a point of the phase diagram where no transition occurs. Motivated by this, we develop two different degenerate perturbative methods to solve the degeneracy-related problems. Moreover, we study the one-dimensional repulsively interacting Bose gas under harmonic confinement, with special attention to the asymptotic behavior of the momentum distribution, which is a universal
k−4 decay characterized by the Tan's contact. The latter constitutes a direct signature of the short-range correlations in such an interacting system and provides valuable insights about the role of the interparticle interactions. We investigate the system constituted of
N interacting particles in the strongly interacting limit. In such a regime, the strong interparticle interaction makes the bosons behave similarly to the ideal Fermi gas. Because of the difficulty in analytically solving the system for
N particles at finite interaction, the Tonks-Girardeau regime provides a favorable scenario to probe the contact. Therefore, we are able to provide an analytical formula for the Tan's contact. Furthermore, we analyze the scaling properties of the Tan's contact in terms of
N in the high-temperature regime as well as in the strongly interacting regime. Finally, we compare our analytical calculations of the Tan's contact to quantum Monte Carlo simulations and discuss some fundamental differences between the canonical and the grand-canonical ensembles.
The Bose–Hubbard model describing interacting bosons in an optical lattice is reduced to a simple spin-1 XY model with single-ion anisotropy in the vicinity of the Mott phase. We propose a mean-field theory based on a constraint SU(3) pseudo-boson representation on the effective model to study the properties of the superfluid-Mott-insulator phase transition. By calculating the elementary excitation spectra and the average particle number fluctuation in the Brillouin zone center, we find that the energy gaps vanish continuously around ( J XY / J z ) c ≈0.175 and ( J XY / J z ) c ≈0.094 for 2D and 3D cubic lattices respectively, where the superfluid order parameters come up from zero and the Mott insulator state changes into a superfluid state.
We simulate quantum degenerate Bose gases at finite temperatures in a grand canonical ensemble and use the Bose-Hubbard model which can describe the dynamics of ultracold bosonic atoms in optical and magnetic lattices. We study ultracold atoms with the gauge P-representation in 1D with 1, 2 (double well), 3, 7 and 11 sites and study quantum correlations and other physical quantities at different temperatures across the superfluid to Mott insulator quantum phase transition. Moreover, using a truncated number-state basis and the gauge P-representation, we show that the Mott lobes are not present unless the temperature is below a certain value.
We use a diagrammatic hopping expansion to calculate finite-temperature Green functions of the Bose-Hubbard model which describes bosons in an optical lattice. This technique allows for a summation of subsets of diagrams, so the divergence of the Green function leads to non-perturbative results for the boundary between the superfluid and the Mott phase for finite temperatures. Whereas the first-order calculation reproduces the seminal mean-field result, the second order goes beyond and shifts the phase boundary in the immediate vicinity of the critical parameters determined by high-precision Monte-Carlo simulations of the Bose-Hubbard model. In addition, our Green’s function approach allows for calculating the excitation spectrum both for zero and finite temperature and for determining the effective masses of particles and holes.
We develop a projection operator formalism for studying both the zero-temperature equilibrium phase diagram and the nonequilibrium dynamics of the Bose-Hubbard model. Our work, which constitutes an extension of that of Trefzger and Sengupta [Phys. Rev. Lett. 106, 095702 (2011)], shows that the method provides an accurate description of the equilibrium zero-temperature phase diagram of the Bose-Hubbard model for several lattices in two and three dimensions. We show that the accuracy of this method increases with the coordination number
z0 of the lattice and reaches to within
0.5 of quantum Monte Carlo data for lattices with
z0=6. We compute the excitation spectra of the bosons using this method in the Mott and the superfluid phases and compare our results with mean-field theory. We also show that the same method may be used to analyze the nonequilibrium dynamics of the model both in the Mott phase and near the superfluid-insulator quantum critical point where the hopping amplitude
J and the on-site interaction
U satisfy ${z}_{0}J/U\ensuremath{\ll}1$. In particular, we study the nonequilibrium dynamics of the model both subsequent to a sudden quench of the hopping amplitude
J and during a ramp from
Ji to
Jf characterized by a ramp time $\ensuremath{\tau}$ and exponent $\ensuremath{\alpha}$: $J(t)={J}_{i}+({J}_{f}\ensuremath{-}{J}_{i}){(t/\ensuremath{\tau})}^{\ensuremath{\alpha}}$. We compute the wave function overlap
F, the residual energy
Q, the superfluid order parameter $\ensuremath{\Delta}(t)$, the equal-time order parameter correlation function
C(t), and the defect formation probability
P for the above-mentioned protocols and provide a comparison of our results to their mean-field counterparts. We find that
Q,
F, and
P do not exhibit the expected universal scaling. We explain this absence of universality and show that our results for linear ramps compare well with the recent experimental observations.