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arXiv:1308.2229 · Physical Review A

Ground-state phase diagram of the two-dimensional Bose-Hubbard model with anisotropic hopping

Janik Schönmeier-Kromer, Lode Pollet

Published Feb 7, 2014• 16 citations
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Abstract

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.

DOI
10.1103/physreva.89.023605

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We study the two-dimensional Bose-Hubbard model with anisotropic hopping. Focusing on the effects of anisotropy on superfluid properties such as the helicity modulus and the normal-to-superfluid [Berezinskii-Kosterlitz-Thouless (BKT)] transition temperature, two different approaches are compared: large-scale quantum Monte Carlo simulations and the self-consistent harmonic approximation (SCHA). For the latter, two different formulations are considered, one applying near the isotropic limit and the other applying in the extremely anisotropic limit. Thus we find that the SCHA provides a reasonable description of superfluid properties of this system provided the appropriate type of formulation is employed. The accuracy of the SCHA in the extremely anisotropic limit, where the BKT transition temperature is tuned to zero (i.e., at a quantum critical point) and therefore quantum fluctuations play a dominant role, is particularly striking.
The first reliable analytic calculation of the phase diagram of the Bose gas on a d -dimensional lattice with on-site repulsion is presented. In one dimension, the analytic calculation is in excellent agreement with the numerical Monte Carlo results. In higher dimensions, the deviations from the Monte Carlo calculations are larger, but the correct shape of the Mott-insulator lobes is still obtained. Explicit expressions for the energy of the Mott and the "defect" phase are given in a strong-coupling expansion.
We review the physics of the Bose–Hubbard model with disorder in the chemical potential focusing on recently published analytical arguments in combination with quantum Monte Carlo simulations. Apart from the superfluid and Mott insulator phases that can occur in this system without disorder, disorder allows for an additional phase, called the Bose glass phase. The topology of the phase diagram is subject to strong theorems proving that the Bose Glass phase must intervene between the superfluid and the Mott insulator and implying a Griffiths transition between the Mott insulator and the Bose glass. The full phase diagrams in 3d and 2d are discussed, and we zoom in on the insensitivity of the transition line between the superfluid and the Bose glass in the close vicinity of the tip of the Mott insulator lobe. We briefly comment on the established and remaining questions in the 1d case, and give a short overview of numerical work on related models.