In the presentation, I will mainly exhibit two of our recent works.
In the first work, we investigate the extended hard-core Bose-Hubbard model (positive hopping-coefficient) on the triangular lattice as a function of spatial anisotropy with respect to both hopping and nearest-neighbor interaction strength [1]. At half-filling the system can be tuned from decoupled one-dimensional chains to a two-dimensional solid phase with alternating density order by adjusting the anisotropic coupling. At intermediate anisotropy, however, frustration effects dominate and an incommensurate supersolid phase emerges, which is characterized by incommensurate density order as well as an anisotropic superfluid density. We demonstrate that this intermediate phase results from the proliferation of topological defects in the form of quantum bosonic domain walls. Whereafter I will also discuss properties of the negative-hopping side where we develop a “parallel tempering” algorithm of DMRG to solve the problem of “metastable states” in the calculations.
In the other work, we firstly show what the form of the time-average Hamiltonian is in general. After a short introduction to two our relevant works, I will turn to the artificially gauged Bose-Hubbard model on the triangular lattice. We systematically study its phase diagram by using large-scale two-dimensional density-matrix renormalization group (DMRG) method, infinite-DMRG method and also the process-chain algorithm of the strong-coupling expansion. The numerical results self-consistently point out that the artificial gauge-field strongly affect the type of the transitions from the Mott-insulating phase to the superfluid phase. Especially, the transition becomes discontinuous at high-symmetry points because of breaking a discrete symmetry. Other properties of criticality are discussed too.