Monte Carlo Diagonalization for Hubbard Model

The Hubbard model has often been studied with exact diagonalization (ED). This impurity solver is fundamentally limited by the exponential scaling of the Fock space. To address this problem, we introduce Monte Carlo diagonalization. Using a truncated Fock space constructed with a Monte Carlo approach, we reduce the size of the basis required to represent the Hamiltonian. We can then apply the Lanczos and band Lanczos algorithms in this truncated basis to find the ground state and the Green function. This results in a significant economy of resources and the capacity to break the $\sim$20-site limit of ED. Our results suggest that there is a threshold on the number of states needed to capture the physics. This allows us to reach clusters of size up to 32 sites and reproduce the expected physics.