In this dissertation I present results for Bose-Hubbard model in 2D obtained with quantum Monte Carlo simulations by the Worm algorithm and Mean Field techniques. First I study a uniform system and I describe how the Mottinsulator-to-superfluid transition for a majority component is affected by the presence of a minority superfluid one. Then I focus on magnetic-like ground states known as checkerboard phase which can be obtained in the strongly interacting regime. I am able to establish constraints on parameters relevant for experimental setups which allows the considered phase to manifest
Motivated by the recent experiment on
p-orbital-band bosons in optical lattices, we study theoretically the quantum phases of Mott insulator and superfluidity in two dimensions. The system features a superfluid phase with transversely staggered orbital current at weak interaction and a Mott insulator phase with antiferro-orbital order at strong coupling and commensurate filling. We go beyond mean-field theory and derive from a microscopic model an effective action that is capable of describing both the
p-band Mott insulating and superfluid phases in strong coupling. We further calculate the excitation spectra near the quantum critical point and find two gapless modes away from the tip of the Mott lobe but four gapless modes at the tip. Our effective theory reveals how the phase coherence peak builds up in the Mott regime when approaching the critical point. We also discuss the finite-temperature phase transition of
p-band superfluidity.