Derivation of effective Hubbard models by unitary transformation based on numerical optimization using Hubbard ground state

The effective low-energy models of Hubbard model are usually derived from perturbation theory. Here we derive the effective model of Hubbard model in spin space and t-J space using a unitary transformation from numerical optimization. We represent the Hamiltonian as Matrix product state(MPO). We represent the unitary transformation using gates according to tensor network methods. We obtain this unitary transformation by optimizing the unitary transformation between the ground state of Hubbard model and the projection of Hubbard model ground state into spin space and t-J space. The ground state is computed using DMRG(Density matrix renormalization method). In half filling case, the optimization can reach a very high accuracy using a three-sites gate. We then analyze systems of different numbers of particles using t-J space. The unitary transformation we get from numerical optimization gives rise to effective models that are in line with perturbation theories. This numerical optimization method starting from ground state energy provides another approach to analyze effective low-energy models of strongly correlated electron systems.