Quantum rotor description of the Mott-insulator transition in the Bose-Hubbard model

We present the novel approach to the Bose-Hubbard model using the $\mathrm{U}(1)$ quantum rotor description. The effective action formalism allows us to formulate a problem in the phase only action and obtain an analytical formulas for the critical lines. We show that the nontrivial $\mathrm{U}(1)$ 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 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.