Hubbard Model Hamitonian

Question

$$H = -\sum\limits_{i,j} A_{ij} c_i^{\dagger} c_j + \frac{U}{2} \sum\limits_i(c_i^\dagger c_i)(c_i^\dagger c_i -1)$$ is defined to be a Hamiltonian for modelling the quantum random walk of identical particles on a graph (i.e., the Hubbard Model). A particle can make a transition from one vertex to another if there is an edge between them and a double-occupancy charge $U$ is imposed. $A$ is the adjacency matrix of the finite graph.

I vaguely understand that the first term is about transitioning from vertex j to i (i.e. destroyed at j and created at i) and the second term will result in $U$ when acted on a state where two particles occupy the same state.

How do I make this understanding rigorous and obtain the matrix elements of $H$ when there are two bosonic particles in some vertices initially? What corresponds to the energy, and eigenfunctions of $H$?

Answer 1

http://arxiv.org/abs/1102.4006

if you want to do exact diagonalization, the paper above might be useful for you.

To obtain the matrix representation of the Hamiltonian, the basic idea is straightforward. First, enumerate the basis vectors; Second, act your Hamiltonian on each basis vector, see what basis vectors will be generated. In the second step, you can establish the Hamiltonian matrix column by column.

Once you have erected your Hamiltonian matrix, just diagonalize it numerically, say using Matlab.