Dönen ikili Bose Hubbard merdiveni

We analyze two leg Bose Hubbard model under uniform magnetic field within various methods. Before studying the model, we discuss the background on rotating Bose Einstein condensates, Bose Hubbard model and superfluid Mott insulator transition. We give a general overview of Density Matrix Renormalization Group (DMRG) theory and show some of the applications. Introducing two leg system Hamiltonian, we solve the single particle problem and find distinct structures above and belove a critical magnetic field $\alpha_c=0.21\pi$. Above this value of the field, it is found that system has travelling wave solutions. To see the effects of interactions, we use Gross Pitaevskii approximation. Spectrum of the system below the critical field and the change of $\alpha_c$ with the interaction strength are obtained for small interactions, i.e $Un/t<1$. To specify Mott insulator boundary, variational mean field theory and strong coupling perturbation (SCP) theories are used. The travelling wave solutions found in single particle spectrum above $\alpha_c$ is found to be persistent in mean field description. On the other hand, comparing with the strong coupling expansion results, it has been found that the mean field theory gives poor results, because of the one dimensional structure of the system. The change of the tip of the lobe where BKT transition takes place is found as a function of magnetic field by SCP. Finally we use DMRG to obtain the exact shape of the phase diagram. It is found that second order strong coupling perturbation theory gives very good results. System is found to display reenterant phase to Mott insulator. Looking at the infinite onsite interaction limit via DMRG, the critical value of the magnetic field is found to be exactly equal to the single particle solution. We have calculated the particle-hole gap for various fillings and different magnetic fields and found Fractional Quantum Hall like behaviors