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

## TL;DR

This paper introduces a numerical optimization approach using tensor network methods to derive effective low-energy models of the Hubbard model, mapping it to the t-J model via ground state unitary transformations.

## Contribution

It presents a novel numerical method to derive effective models from the ground state, complementing traditional perturbation theory approaches.

## Key findings

- Unitary transformations align with perturbation theory results.
- Effective models accurately represent low-energy physics.
- Method offers a new perspective for strongly correlated systems.

## Abstract

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.