Entanglement and the Mott insulator - superfluid phase transition in bosonic atom chains

We analyze the developing of bipartite and multipartite entanglement through the Mott-Insulator - Superfluid quantum phase transition. Starting from a Mott insulator state, where a filling factor $\nu = N/M = 1$ per lattice site is considered, we derive an exact expression for a completely connected graph configuration of bosons and show how bipartite and multipartite entanglement signals the phase transition predicted in previous works. Moreover, through the transition bipartite entanglement shows to be monogamous.