• fully frustrated Bose-Hubbard diamond chain

Bose-Hubbard models provide an effective description of a bosonic gas loaded in an optical lattice. In this thesis we study a Bose-Hubbard model describing spinless particles at zero temperature in a fully frustrated quasi-one-dimensional diamond chain. Frustration means that all the nearest neighbor terms of the Hamiltonian cannot be minimized simultaneously; furthermore, in our model, in absence of interactions frustration leads to strongly localized states, named Aharonov-Bohm cages. However we consider a model including interactions as well and using two numerical approaches, namely the Exact Diagonalization and the Density-Matrix Renormalization Group, we study the phases displayed by the system. Actually we study an effective model, derived by the original one assuming that interactions are sufficiently week to let all the particles be in the lowest band state relative to the noninteracting case, i.e. we use the lowest band approximation.