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arXiv:0802.3774 · Physical Review A

Quasiperiodic Bose-Hubbard model and localization in one-dimensional cold atomic gases

Guillaume Roux, Thomas Barthel, Ian P. McCulloch, Corinna Kollath, Ulrich Schollwöck, Thierry Giamarchi

Published Aug 22, 2008• 203 citations
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Abstract

We compute the phase diagram of the one-dimensional Bose-Hubbard model with a quasiperiodic potential by means of the density-matrix renormalization group technique. This model describes the physics of cold atoms loaded in an optical lattice in the presence of a superlattice potential whose wavelength is incommensurate with the main lattice wavelength. After discussing the conditions under which the model can be realized experimentally, the study of the density vs the chemical potential curves for a nontrapped system unveils the existence of gapped phases at incommensurate densities interpreted as incommensurate charge-density-wave phases. Furthermore, a localization transition is known to occur above a critical value of the potential depth ${V}_{2}$ in the case of free and hard-core bosons. We extend these results to soft-core bosons for which the phase diagrams at fixed densities display new features compared with the phase diagrams known for random box distribution disorder. In particular, a direct transition from the superfluid phase to the Mott-insulating phase is found at finite ${V}_{2}$. Evidence for reentrances of the superfluid phase upon increasing interactions is presented. We finally comment on different ways to probe the emergent quantum phases and most importantly, the existence of a critical value for the localization transition. The latter feature can be investigated by looking at the expansion of the cloud after releasing the trap.

DOI
10.1103/physreva.78.023628

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We compute the phase diagram of the one-dimensional Bose-Hubbard model with a quasiperiodic potential by means of the density-matrix renormalization group technique. This model describes the physics of cold atoms loaded in an optical lattice in the presence of a superlattice potential whose wavelength is incommensurate with the main lattice wavelength. After discussing the conditions under which the model can be realized experimentally, the study of the density vs the chemical potential curves for a nontrapped system unveils the existence of gapped phases at incommensurate densities interpreted as incommensurate charge-density-wave phases. Furthermore, a localization transition is known to occur above a critical value of the potential depth V2{V}_{2}V2​ in the case of free and hard-core bosons. We extend these results to soft-core bosons for which the phase diagrams at fixed densities display new features compared with the phase diagrams known for random box distribution disorder. In particular, a direct transition from the superfluid phase to the Mott-insulating phase is found at finite V2{V}_{2}V2​. Evidence for reentrances of the superfluid phase upon increasing interactions is presented. We finally comment on different ways to probe the emergent quantum phases and most importantly, the existence of a critical value for the localization transition. The latter feature can be investigated by looking at the expansion of the cloud after releasing the trap.
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We analyze topological properties of the one-dimensional Bose-Hubbard model with a quasiperiodic superlattice potential. This system can be realized in interacting ultracold bosons in optical lattice in the presence of an incommensurate superlattice potential. We first analyze the quasiperiodic superlattice made by the cosine function, which we call Harper-like Bose-Hubbard model. We compute the Chern number and observe a gap-closing behavior as the interaction strength UUU is changed. Also, we discuss the bulk-edge correspondence in our system. Furthermore, we explore the phase diagram as a function of UUU and a continuous deformation parameter βββ between the Harper-like model and another important quasiperiodic lattice, the Fibonacci model. We numerically confirm that the incommensurate charge density wave (ICDW) phase is topologically non-trivial and it is topologically equivalent in the whole ICDW region.
Ultra cold polar bosons in a disordered lattice potential, described by the extended Bose–Hubbard model, display a rich phase diagram. In the case of uniform random disorder one finds two insulating quantum phases—the Mott-insulator and the Haldane insulator—in addition to a superfluid and a Bose glass phase. In the case of a quasiperiodic potential, further phases are found, e.g. the incommensurate density wave, adiabatically connected to the Haldane insulator. For the case of weak random disorder we determine the phase boundaries using a perturbative bosonization approach. We then calculate the entanglement spectrum for both types of disorder, showing that it provides a good indication of the various phases.
We determine the phase diagram and the momentum distribution for a one-dimensional Bose gas with repulsive short-range interactions in the presence of a two-color lattice potential, with an incommensurate ratio among the respective wavelengths, by using a combined numerical (density matrix renormalization group) and analytical (bosonization) analysis. The system displays a delocalized (superfluid) phase at small values of the intensity of the secondary lattice V2{\mathit{V}}_{2}V2​ and a localized (Bose-glasslike) phase at larger intensity V2{\mathit{V}}_{2}V2​. We analyze the localization transition as a function of the height V2{\mathit{V}}_{2}V2​ beyond the known limits of free and hard-core bosons. We find that weak repulsive interactions disfavor the localized phase, i.e., they increase the critical value of V2{\mathit{V}}_{2}V2​ at which localization occurs. In the case of integer filling of the primary lattice, the phase diagram at fixed density displays, in addition to a transition from a superfluid to a Bose glass phase, a transition to a Mott-insulating state for not too large V2{\mathit{V}}_{2}V2​ and large repulsion. We also analyze the emergence of a Bose-glass phase by looking at the evolution of the Mott-insulator lobes when increasing V2{\mathit{V}}_{2}V2​. The Mott lobes shrink and disappear above a critical value of V2{\mathit{V}}_{2}V2​. Finally, we characterize the superfluid phase by the momentum distribution, and show that it displays a power-law decay at small momenta typical of Luttinger liquids, with an exponent depending on the combined effect of the interactions and of the secondary lattice. In addition, we observe two side peaks that are due to the diffraction of the Bose gas by the second lattice. This latter feature could be observed in current experiments as characteristics of pseudo-random Bose systems.
Ultracold atomic gases in optical lattices provide an unique framework to study quantum phenomena in strongly correlated systems. In addition to the precise control over all relevant parameters in the experiment, these experiments can be mapped to a fundamental model from solid-state physics. For the bosonic version of this model, the so-called Bose-Hubbard model, a phase transition from a superfluid to a Mott insulator was theoretically predicted and later experimentally observed in an ultracold gas of Rb-87 atoms in three-dimensional as well as in one-dimensional optical lattices. Apart from homogeneous optical lattices one can introduce more complex lattice topologies such as two-color superlattices which give rise to a rich phase diagram including more exotic phases like the Bose-glass. We employ various powerful many-body techniques like exact diagonalization in complete and truncated Hilbert spaces and the Density-Matrix Renormalization Group (DMRG) algorithm to study the phase diagrams of the one-dimensional Bose-Hubbard and the one-dimensional Bose-Fermi-Hubbard Hamiltonian. Most theoretical studies of these systems discuss the phase diagrams with respect to the generic parameters of the Hubbard model. These Hubbard parameters, however, depend non-trivially on the control parameters used in experiments. The focus of this work is on the ab-initio calculation of the phase diagram of ultracold Rb-87 in one-dimensional optical superlattices starting directly from the experimental setup. To this end, we first employ band-structure calculations to extract the Hubbard parameters from the experimental parameters. Then, we use state-of-the-art DMRG techniques to solve the many-body problem for realistic particle numbers and lattice sizes that occur in experiments. Our results show that by using the intensities of the two laser fields forming the two-color superlattice as control parameters while keeping all other experimental parameters fixed, it is possible to access all relevant quantum phases of the system. Furthermore, we found out that the critical values of the laser intensities for the different phase transitions depend strongly on a third parameter that has to be included for a realistic description of the experiment. This third parameter is the strength of a harmonic trapping potential which accounts for the Gaussian shape of the laser fields and an additional magnetic potential used to confine the atoms in the center of the trap.
We use quantum Monte Carlo simulations and exact diagonalization to explore the phase diagram of the Bose-Hubbard model with an additional superlattice potential. We first analyze the properties of superfluid and insulating phases present in the hard-core limit where an exact analytic treatment is possible via the Jordan-Wigner transformation. The extension to finite on-site interactions is achieved by means of quantum Monte Carlo simulations. We determine insulator and superfluid phase diagrams as functions of the on-site repulsive interaction, superlattice potential strength, and filling, finding that insulators with fractional occupation numbers, which are present in the hard-core case, extend deep into the soft-core region. Furthermore, at integer fillings, we find that the competition between the on-site repulsion and the superlattice potential can produce a phase transition between a Mott insulator and a charge-density-wave insulator, with an intermediate superfluid phase. Our results are relevant to the behavior of ultracold atoms in optical superlattices which are beginning to be studied experimentally.