# Exact diagonalization to resolve ground state degeneracies

I am studying a perturbed Toric Code model that is not analytically solvable. On a torus the ground state degeneracy of the unperturbed model is 4. Once we turn on the perturbation there is a change in the ground state degeneracy. I would like to detect this change in ground state degeneracy numerically using exact diagonalization techniques.

On my computer I have stored the action of the Hamiltonian on a set of basis states. So if you give me some state $\left|\psi\right\rangle$ I can give you $\hat{H}\left|\psi\right\rangle$ in terms of the basis states. Now I used this information to compute the spectrum using the Lanczos algorithm and the Jacobi-Davidson algorithm. While the spectrum itself is correctly reproduced neither of these algorithms reproduces the degeneracy of the ground state correctly - not even in the unperturbed case.

Hence the following question: What are common exact diagonalization algorithms for this type of many-body system that correctly resolve the degeneracy of the ground state?

I am looking forward to your responses!

• The Lanczos algorithm may have trouble resolving states that are very close or degenerate in energy. Are you re-orthogonalizing the states you get in the Lanczos recursion process? – delete000 May 1 '14 at 7:48
• Our sister site, Computational Science, may be a good place to ask this question if you need more detail than what we can provide. – Emilio Pisanty Jun 9 '14 at 4:53
• Just curious, why is it important to do this numerically? Wouldn't it be more straightforward to do some group theory and get how the perturbation affects the degeneracy that way? – Kevin Driscoll Mar 15 '16 at 12:31