We present an approach to the Bose-Hubbard model using the U͑1͒ quantum rotor description. The effective action formalism allows us to formulate a problem in the phase only action and obtain analytical formulas for the critical lines. We show that the nontrivial U͑1͒ phase field configurations have an impact on the phase diagrams. The topological character of the quantum field is governed by terms of the integer charges-winding numbers. The comparison of presented results to recently obtained quantum Monte Carlo numerical calculations suggests that the competition between quantum effects in strongly interacting boson systems is correctly captured by our model. PACS number͑s͒: 05.30.Jp, 03.75.Lm, 03.75.Nt † ͔ = ␦ ij , n i = a i † a i is the boson number operator on the site PHYSICAL REVIEW B 76, 094503 ͑2007͒

Topological phase engineering of neutral bosons loaded in an optical lattice opens a new window for manipulating of transport phenomena in such systems. Exploiting the Bose Hubbard model and using the magnetic Kubo formula proposed in this paper we show that the optical conductivity abruptly changes for different flux densities in the Mott phase. Especially, when the frequency of the applied field corresponds to the on-site boson interaction energy, we observe insulator or metallic behavior for a given Hofstadter spectrum. We also prove, that for different synthetic magnetic field configurations, the critical conductivity at the tip of the lobe is non-universal and depends on the energy minima of the spectrum. In the case of 1/2 and 1/3 flux per plaquette, our results are in good agreement with those of the previous Monte Carlo (MC) study. Moreover, we show that for half magnetic-flux through the cell the critical conductivity suddenly changes in the presence of a superlattice potential with uniaxial periodicity.

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We discuss the finite-temperature phase diagram in the three-dimensional Bose-Hubbard (BH) model in the strong correlation regime, relevant for Bose-Einstein condensates in optical lattices, by employing a quantum rotor approach. In systems with strong on site repulsive interactions, the rotor U(1) phase variable dual to the local boson density emerges as an important collective field. After establishing the connection between the rotor construction and the the on-site interaction in the BH model the robust effective action formalism is developed which allows us to study the superfluid phase transition in various temperature-interaction regimes.

We study the ground-state phase diagram of a mixture of bosonic and fermionic cold atoms confined on twoand three-dimensional optical lattices. The coupling between bosonic fluctuations and fermionic atoms can be attractive or repulsive and has similarities with electron-phonon coupling in crystals. We investigate behavior of the mixtures in the limit, where the Bogoliubov sound velocity that dictates bosonic dynamics is comparable to the Fermi velocity, hence the retardation effects are an important part of the physics. The dynamic Lindhard response function of the fermionic density to changes in the bosonic number of particles above some critical frequency can alter the sign, and consequently the interspecies interaction between particles becomes repulsive in contrast to the static limit (instantaneous and always attractive). Considering the above, we show that the structure of the phase diagrams crucially depends on the difference in masses of the bosons and fermions. We discuss the situations where integrating out the fermionic field provides an additional interaction that can decrease or increase bosonic coherence.