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

I vaguely understand that first term is about transition from vertex j to i ( destroyed at j and created at i ) and second term will result in $U$ when acted on state where two particle occupy 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 energy, and eigenfunction of the $H$?

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Are you talking about fermions or bosons ? That looks like bosons (since there are only one species), but double occupancy is used in discussions about fermions (you can put as many bosons as you want on one site). – Adam Jul 3 '14 at 19:29

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.

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Hi Jiang-min, we usually expect answers to be self-contained and not require content from the linked source. Can you update your answer to at least sketch the general idea out so that your answer stands on its own? – Brandon Enright Jul 3 '14 at 19:17

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