Similarities between the $t-J$ and Hubbard models in weakly correlated regimes

TL;DR

This study demonstrates that the $t-J$ model effectively approximates the Hubbard model across all correlation strengths in weakly correlated, far-from-half-filling regimes, offering computational advantages.

Contribution

It establishes a variational mapping between the Hubbard and $t-J$ models over the entire correlation range, providing a new perspective on their relationship.

Findings

The $t-J$ Hamiltonian can be seen as an effective model of the Hubbard Hamiltonian across all correlation strengths.

Numerical comparisons show good agreement in energies, charge gaps, and bond orders.

Using the $t-J$ model is computationally advantageous in low-filled systems.

Abstract

We present a comparative study of the Hubbard and $t-J$ models far away from half-filling. We show that, at such fillings the $t-J$ Hamiltonian can be seen as an effective model of the repulsive Hubbard Hamiltonian over the whole range of correlation strength. Indeed, the $|t/U| \in [0,+\infty [$ range of the Hubbard model can be mapped onto the finite range $|J/t| \in [1, 0 ]$ of the $t-J$ model, provided that the effective exchange parameter $J$ is defined variationally as the local singlet-triplet excitation energy. In this picture the uncorrelated limit U=0 is associated with the super-symmetric point $J=-2|t|$ and the infinitely correlated $U=+\infty$ limit with the usual J=0 limit. A numerical comparison between the two models is presented using different macroscopic and microscopic properties such as energies, charge gaps and bond orders on a quarter-filled infinite chain. The usage of the $t-J$ Hamiltonian in low-filled systems can therefore be a good alternative to the Hubbard model in large time-consuming calculations.