Quantum phases of a spin-1 bose gas in an optical lattice : A focus on mean field and perturbative approaches

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The thesis investigates the ground state properties of a spin-1 ultracold Bose gas in an optical lattice. We have employed a spin-1 Bose Hubbard model (SBHM) which includes an additional spin dependent interaction potential due to the presence of the hyperfine degrees of freedom. The primary motivation is to explore various quantum phases, such as, Mott insulator (MI), superfluid (SF) and the density ordered phases in the SBHM for either sign (positive or negative) of the spin dependent interaction potential. The phase diagrams are obtained using a site decoupling mean field approximation (MFA) by considering the SBHM in presence of different types of the interaction potential. The formation of the singlet and nematic MI phases and their transition to the SF phase have been carefully scrutinized by analyzing the behaviour of the order parameter and the spin eigenvalues. All the numerical mean field phase diagrams are then compared with the analytical phase diagrams obtained by using a strong coupling perturbative expansion (PMFA) technique. The effects of an on-site random disorder in the SBHM results in a glassy phase, known as the Bose glass (BG) phase in addition to the MI and SF phases. The site inhomogeneities in the MFA is taken care of by defining an indicator which is the fraction of lattice sites having finite SF order parameter and non-integer occupation densities. The transition between these three phases shows a percolation phenomena similar to the statistical mechanics and the phase diagrams are obtained based on the appearance of the SF percolating cluster computed using Hoshen–Kopelman (HK) algorithm. Further, the inclusion of a synthetic and an external magnetic fields is found to be competing against each other on the formation of the spin singlet MI phase. The synthetic magnetic field tries to stabilize the insulating phase while the external magnetic field destabilizes the even MI lobes by suppressing the spin singlet pair formation.
Supervisor: Saurabh Basu