Ground-State of the Bose-Hubbard Model

The Bose-Hubbard Model represents a s simple theoretical model to describe the physics of interacting Boson systems. In particular it has proved to be an effective description of a number of physical systems such as arrays of Josephson arrays as well as dilute alkali gases in optical lattices. Here we wish to study the ground-state of this system using two disparate but related moments calculational schemes: the Lanczos (tridiagonal) method as well as a Generalized moments approach. The Hamiltonian to be studied is given by (in second-quantized notation): \[H = -t\sum _{ }b_{i}^{\dag }b_{j} +\frac{U}{2}\sum _{i}n_{i}\left (n_{i} -1\right ) -\mu \sum _{i}n_{i} . \] Here $i$ is summed over all lattice sites, and~$ $ denotes summation over all neighbhoring sites $i$ and $j$, while $b_{i}^{\dag }$ and $b_{i}$ are bosonic creation and annihilation operators. $n_{i} =$$b_{i}^{\dag }b_{i}$ gives the number of particles on site $i$. Parameter $t$ is the hopping amplitude, describing mobility of bosons in the lattice. Parameter $U$ describes the on-site interaction, repulsive, if $U >0$, and attractive for $U <0$. $\mu $ is the chemical potential. Both the ground-state energy and energy gap are evaluated as a function of $t$, $U$ and $\mu$.