  The effective low-energy models of the Hubbard model are usually derived from
perturbation theory. Here we derive the effective model of the 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) and
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 the Hubbard model and the projection
of the Hubbard model ground state into spin space and t-J space. The unitary
transformation we get from numerical optimization yields effective models that
are in line with perturbation theories. This numerical optimization method
starting from ground state provides another approach to analyze effective
low-energy models of strongly correlated electron systems.
