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The effective low-energy models of the Hubbard model are usually derived fromperturbation theory. Here we derive the effective model of the Hubbard model inspin space and t-J space using a unitary transformation from numericaloptimization. We represent the Hamiltonian as Matrix product state(MPO) andrepresent the unitary transformation using gates according to tensor networkmethods. We obtain this unitary transformation by optimizing the unitarytransformation between the ground state of the Hubbard model and the projectionof the Hubbard model ground state into spin space and t-J space. The unitarytransformation we get from numerical optimization yields effective models thatare in line with perturbation theories. This numerical optimization methodstarting from ground state provides another approach to analyze effectivelow-energy models of strongly correlated electron systems.

Mapping the Hubbard model to the t-J model using ground state unitary transformations