I have numerically constructed a Hamiltonian matrix. I am currently finding the ground state by full diagonalisation of the matrix (with the GSL library) and finding the most negative eigenvalue and its associated eigenvector.
This is very slow and can surely be done better. I thought about just finding the eigenvalues and the solving the linear equation associated with the most negative eigenvalue for the eigenvector but this is probably not much quicker?
I am wondering if any one knows of any faster numerical routines for finding the ground state associated with a Hamiltonian matrix. If it makes any difference this is for the unit filled 1D Bose Hubbard Model in second quantised form.
i.e. $\hat{H} = -J\sum_{<ij>}\hat{b}_{i}^{\dagger}\hat{b}_{j} + \frac{U}{2}\sum_{i}\hat{n}_{i}(\hat{n}_{i} – 1)$
Also I have, already, fairly good estimates for the ground state energy (within 5%) and wavefunction if that may help any possible routines.
Many thanks.
Best Answer
Much of (perhaps most of) the entire field of theoretical condensed matter physics is dedicated to solving this problem. I don't think you'll be able to find a comprehensive answer on Stack Exchange.