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arXiv:0804.3040 · Physical Review A

Effect of anisotropic hopping on the Bose-Hubbard model phase diagram: Strong-coupling perturbation theory on a square lattice

James K. Freericks

Published Jul 18, 2008• 4 citations
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Abstract

There has been a recent resurgence of experimental efforts to quantitatively determine the phase diagram of the Bose-Hubbard model by carefully analyzing experiments with ultracold bosonic atoms on an optical lattice. In many realizations of these experiments, the hopping amplitudes are not homogeneous throughout the lattice, but instead, the lattice has an anisotropy where hopping along one direction is not exactly equal to hopping along a perpendicular direction. In this contribution, we examine how an anisotropy in the hopping matrix elements affects the Mott lobes of the Bose-Hubbard model. For weak anisotropy, we find the phase diagram is only slightly modified when expressed in terms of the average hopping, while for strong anisotropy, one expects to ultimately see dimensional crossover effects.

DOI
10.1103/physreva.78.013624

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There has been a recent resurgence of experimental efforts to quantitatively determine the phase diagram of the Bose-Hubbard model by carefully analyzing experiments with ultracold bosonic atoms on an optical lattice. In many realizations of these experiments, the hopping amplitudes are not homogeneous throughout the lattice, but instead, the lattice has an anisotropy where hopping along one direction is not exactly equal to hopping along a perpendicular direction. In this contribution, we examine how an anisotropy in the hopping matrix elements affects the Mott lobes of the Bose-Hubbard model. For weak anisotropy, we find the phase diagram is only slightly modified when expressed in terms of the average hopping, while for strong anisotropy, one expects to ultimately see dimensional crossover effects.
We compute the ground-state phase diagram of the two-dimensional (2D) Bose-Hubbard model with anisotropic hopping using quantum Monte Carlo simulations, connecting the one-dimensional (1D) to the 2D system. We find that the tip of the lobe lies on a curve controlled by the 1D limit over the full anisotropy range, while the universality class is always the same as in the isotropic 2D system. This behavior can be derived analytically from the lowest renormalization-group equations and has a shape typical for the underlying Kosterlitz-Thouless transition in one dimension. We also compute the phase boundary of the Mott lobe at unit density for strong anisotropy and compare it to the 1D system. Our calculations shed light on recent cold gas experiments monitoring the dynamics of an expanding cloud.
In a quantum many-body system, dimensionality plays a critical role in determining the type of quantum phase transition. To study the quantum system during dimensional crossover, we studied the Bose-Hubbard model on a cubic lattice with anisotropic hopping by using the high-order symbolic strong-coupling expansion method. The analytic series expanded boundaries between the Mott insulator and the superfluid phase up to eighth order are calculated. The critical exponents are extracted by the Pad\'e resummation method, which clearly shows dimensional crossover behavior. Meanwhile, the critical points at commensurate filling can also be obtained, and they match well with the prediction of renormalization group theory. Finally, the scaling of the gap energy and the whole phase diagram are given, and they can be taken as the benchmark for experiment and numerical simulations in future study.
We present a strong-coupling expansion of the Bose-Hubbard model which describes both the superfluid and the Mott phases of ultracold bosonic atoms in an optical lattice. By performing two successive Hubbard-Stratonovich transformations of the intersite hopping term, we derive an effective action which provides a suitable starting point to study the strong-coupling limit of the Bose-Hubbard model. This action can be analyzed by taking into account Gaussian fluctuations about the mean-field approximation as in the Bogoliubov theory of the weakly interacting Bose gas. In the Mott phase, we reproduce results of previous mean-field theories and also calculate the momentum distribution function. In the superfluid phase, we find a gapless spectrum and compare our results with the Bogoliubov theory.
Building on the work of Fisher et al. [Phys. Rev. B 40, 546 (1989)], we develop the perturbation theory for the Bose-Hubbard model and apply it to calculate the effects of a degenerate gas of spin-polarized fermions interacting by contact interactions with the constituent bosons. For the single-band Bose-Hubbard model, we find that the net effect of the screening of the boson on-site interaction by the fermions is to suppress the Mott-insulating lobes in the Bose-Hubbard phase diagram. For the more general multiband model, we find that, in addition to the fermion screening effects, the virtual excitations of the bosons to the higher Bloch bands, coupled with the contact interactions with the fermions, result in an effective increase (decrease) of the boson on-site repulsion (hopping parameter). If the higher-band renormalization of the boson parameters is dominant over the fermion screening of the interaction, the Mott-insulating lobes in the Bose-Hubbard phase diagram are enhanced for either sign of the Bose-Fermi interactions, consistent with the recent experiments.
We study the occupation numbers and number fluctuations of ultracold atoms in deep optical lattices for finite-temperatures within the Bose-Hubbard model. Simple analytical expressions for the mean occupation number and number fluctuations are obtained in the weak-hopping regime using an interpolation between results from different perturbation approaches in the Mott-insulator and superfluid phases. With this approach the magnitude of number fluctuations under a wide range of experimental conditions can be estimated and the properties of the finite-temperature phase diagram can be studied. These analytical results are compared to exact one-dimensional numerical calculations using a finite temperature variant of the density-matrix renormalization group (DMRG) method and found to have a high degree of accuracy. We find very good agreement, also in the crossover ``thermal'' region. We also analyze the influence of finite temperature on the behavior of the system in the vicinity of the zero-temperature phase transition, in one, two, and three dimensions.
To analyze the ground-state phase diagram of Bose-Bose mixtures loaded into ddd-dimensional hypercubic optical lattices, we perform a strong-coupling power-series expansion in the kinetic energy term (plus a scaling analysis) for the two-species Bose-Hubbard model with onsite boson-boson interactions. We consider both repulsive and attractive interspecies interaction, and obtain an analytical expression for the phase boundary between the incompressible Mott insulator and the compressible superfluid phase up to third order in the hoppings. In particular, we find a re-entrant quantum phase transition from paired superfluid (superfluidity of composite bosons, i.e., Bose-Bose pairs) to Mott insulator and again to a paired superfluid in all one, two, and three dimensions, when the interspecies interaction is sufficiently large and attractive. We hope that some of our results could be tested with ultracold atomic systems.