We study the boundary nature of trapped bosonic Mott insulators in optical square lattices, by performing quantum Monte Carlo simulation. We show that a finite superfluid density generally emerges in the incommensurate-filling (IC) boundary region around the bulk Mott state, irrespectively of the width of the IC region. Both off-diagonal and density correlation functions in the IC boundary region exhibit a nearly power-law decay. The power-law behavior and superfluidity are well developed below a characteristic temperature. These results indicate that a gapless boundary mode always emerges in any atomic Mott insulators on optical lattices. This further implies that if we consider a topological insulating state in Bose or Fermi atomic systems, its boundary possesses at least two gapless modes (or coupled modes) of an above IC edge state and the intrinsic topologically-protected edge state.
We consider a strongly repulsive fermionic gas in a two-dimensional optical lattice confined by a harmonic trapping potential. To address the strongly repulsive regime, we consider the $t\ensuremath{-}J$ Hamiltonian. The presence of the harmonic trapping potential enables the stabilization of coexisting and competing phases. In particular, at low temperatures, this allows the realization of a
d-wave superfluid region surrounded by purely (gapless) normal edges. Solving the Bogoliubov--de Gennes equations and comparing them with the local density approximation, we show that the proximity to the Mott insulator is revealed by a downturn of the Fermi liquid order parameter at the center of the trap where the
d-wave gap has a maximum. The density profile evolves linearly with distance.
We investigate the interplay of temperature and trap effects in particle systems at their quantum critical regime, such as cold bosonic atoms in optical lattices at the transitions between Mott-insulator and superfluid phases. The theoretical framework is provided by the one-dimensional Bose-Hubbard model in the presence of an external trapping potential, and the trap-size scaling theory describing the large trap-size behavior at a quantum critical point. We present numerical results for the low-temperature behavior of the particle density and the density-density correlation function at the Mott transitions, and within the gapless superfluid phase, in the hard-core limit.