# Effective single-band models for strongly interacting fermions in an optical lattice

## TL;DR

This study numerically analyzes two interacting fermions in an optical lattice to validate effective Hamiltonians, revealing the conditions under which simplified models like the $t-J$ and Hubbard models accurately describe the system.

## Contribution

It demonstrates the applicability of effective single-band models for strongly interacting fermions in optical lattices and clarifies how model parameters relate to experimental conditions.

## Key findings

- The $t-J$ model accurately describes the system over a range of strong interactions.
- The superexchange term $J$ can be tuned through zero around unitarity.
- Near unitarity, a generalized Hubbard model with a small on-site energy is appropriate.

## Abstract

To test effective Hamiltonians for strongly interacting fermions in an optical lattice, we numerically find the energy spectrum for two fermions interacting across a Feshbach resonance in a double well potential. From the spectrum, we determine the range of detunings for which the system can be described by an effective lattice model, and how the model parameters are related to the experimental parameters. We find that for a range of strong interactions the system is well described by an effective $t-J$ model, and the effective superexchange term, $J$, can be smoothly tuned through zero on either side of unitarity. Right at and around unitarity, an effective one-band general Hubbard model is appropriate, with a finite and small on-site energy, due to a lattice-induced anharmonic coupling between atoms at the scattering threshold and a weakly bound Feshbach molecule in an excited center of mass state.