Numerical study of the trapped and extended Bose-Hubbard models
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The Bose-Hubbard model describes the physics of a system of bosonic ultracold atoms in an optical lattice, in which a phase transition is present between a superfluid phase and a Mott insulator one. The exact solution of this Hamiltonian is only feasible to find the ground-state of small systems, while other techniques (as mean-field schemes or quantum Monte Carlo) are necessary to study systems of larger size. As a first application, we study the trapped model - relevant for the comparison with current experiments - through an inhomogeneous mean-field scheme. We describe some signatures of the phase crossover between superfluid and Mott insulator. In particular, the visibility of the quasimomentum distribution shows some kinks as a function of the lattice depth; we describe these features and we link them with the ones observed in other works in the literature. As a second application, we use quantum Monte Carlo techniques to study the one-dimensional Bose-Hubbard model with long-range interactions and we focus on the appearance of the Haldane insulating phase, distinguishable from the Mott one through the presence of non-local hidden order. Non-local correlation functions are also used to describe the difference between the superfluid phase and the Mott insulator one.