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Acta Physica Polonica A

Bose-Hubbard Model in the Rotating Frame of Reference

T. P. Polak, T. K. Kopeć

Published Aug 1, 2010• 0 citations
Publisher

Abstract

Following a novel experimental arrangement which can rotate a two-dimensional optical lattice at frequencies up to several kilohertz we discuss the ground state of the two-dimensional Bose-Hubbard Hamiltonian, relevant for rotating gaseous Bose-Einstein condensates, by employing U(1) quantum rotor approach and the topologically constrained path integral.Ultracold atoms in such a rotating lattice can be used for the direct quantum simulation of strongly correlated systems under large effective magnetic fields.We derive an effective quantum action for the Bose-Hubbard model, which enables a non-perturbative treatment of the zero-temperature phase transition in the rotating frame.We calculate the ground-state phase diagram, analytically deriving maximum repulsive energy for several rational values of the frustration rotation parameter f = 0, 1/2, 1/3, 1/4, and 1/6 for the square and triangular lattice.Performed calculations revealed strong non-monotonical dependence of the critical ratio of the kinetic energy to the repulsive on-site energy, that separates the global coherent from the insulating state, on topology of the lattice.

DOI
10.12693/aphyspola.118.279

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Following a novel experimental arrangement which can rotate a two-dimensional optical lattice at frequencies up to several kilohertz we discuss the ground state of the two-dimensional Bose-Hubbard Hamiltonian, relevant for rotating gaseous Bose-Einstein condensates, by employing U(1) quantum rotor approach and the topologically constrained path integral.Ultracold atoms in such a rotating lattice can be used for the direct quantum simulation of strongly correlated systems under large effective magnetic fields.We derive an effective quantum action for the Bose-Hubbard model, which enables a non-perturbative treatment of the zero-temperature phase transition in the rotating frame.We calculate the ground-state phase diagram, analytically deriving maximum repulsive energy for several rational values of the frustration rotation parameter f = 0, 1/2, 1/3, 1/4, and 1/6 for the square and triangular lattice.Performed calculations revealed strong non-monotonical dependence of the critical ratio of the kinetic energy to the repulsive on-site energy, that separates the global coherent from the insulating state, on topology of the lattice.
We discuss the ground state of the two-dimensional Bose-Hubbard (BH) Hamiltonian, relevant for rotating gaseous Bose-Einstein condensates, by employing U(1) quantum rotor approach and the topologically constrained path integral that includes a summation over U(1) topological charge. We derive an effective quantum action for the BH model, which enables a nonperturbative treatment of the zero-temperature phase transition. We calculate the ground-state phase diagram, analytically deriving maximum repulsive energy for several rational values of the frustration rotation parameter f=0f=0f=0, 1/2, 1/3, 1/4, and 1/6 for the square and triangular lattices, which improves upon previous theoretical treatments. The ground state of the rotating Bose-Einstein condensates on a triangular lattice appears to be most stable against the effects of rotation. Performed calculations revealed strong dependence of the critical ratio of the kinetic energy to the repulsive on-site energy, which separates the global coherent from the insulating state, on topology of the lattice.
We discuss the finite-temperature phase diagram in the three-dimensional Bose–Hubbard (BH) model in the strong correlation regime, relevant for Bose–Einstein condensates in optical lattices, by employing a quantum rotor approach. In systems with strong on-site repulsive interactions, the rotor U (1) phase variable dual to the local boson density emerges as an important collective field. After establishing the connection between the rotor construction and the on-site interaction in the BH model a robust effective action formalism is developed which allows us to study the superfluid phase transition in various temperature-interaction regimes.
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.
Abstract We study the behavior of interacting ultracold bosons in a two-dimensional optical lattice described by the Bose–Hubbard (BH) model. We apply the quantum rotor approach to reduce the BH model to a (semiclassical) effective Hamiltonian in which the phases of local order parameters play the role of dynamical variables. This leads to the effective quantum XY model, which is used to calculate the Berezinskii–Kosterlitz–Thouless transition temperature T BKT as a function of model parameters. We find that T BKT is monotonically dependent on the kinetic energy of atoms and the value of the chemical potential.
The study of geometrically frustrated many-body quantum systems is of central importance to uncover novel quantum mechanical effects. We design a scheme where ultracold bosons trapped in a one-dimensional state-dependent optical lattice are modeled by a frustrated Bose-Hubbard Hamiltonian. A derivation of the Hamiltonian parameters based on Cesium atoms, further show large tunability of contact and nearest-neighbor interactions. For pure contact repulsion, we discover the presence of two phases peculiar to frustrated quantum magnets: the bond-order-wave insulator with broken inversion symmetry and a chiral superfluid. When the nearest-neighbor repulsion becomes sizable, a further density-wave insulator with broken translational symmetry can appear. We show that the phase transition between the two spontaneously symmetry-broken phases is continuous, thus representing a one-dimensional deconfined quantum critical point not captured by the Landau-Ginzburg-Wilson symmetry-breaking paradigm. Our results provide a solid ground to unveil the novel quantum physics induced by the interplay of nonlocal interactions, geometrical frustration, and quantum fluctuations.