# Mobile Spin Impurity in an Optical Lattice

## TL;DR

This paper explores the behavior of a spin impurity in a one-dimensional optical lattice, revealing a transition from a tight-binding model to a multi-well potential and eventually to a topological Su-Schrieffer-Heeger model as the lattice filling increases.

## Contribution

It demonstrates how impurity Hamiltonians in a 1D optical lattice evolve with filling, connecting simple tight-binding models to topological models beyond the Hubbard approximation.

## Key findings

- At low filling, impurity behaves as a single-band tight-binding model.
- At higher filling, impurity experiences a multi-well potential.
- Full filling of bands realizes the Su-Schrieffer-Heeger model.

## Abstract

We investigate the Fermi polaron problem in a spin-1/2 Fermi gas in an optical lattice for the limit of both strong repulsive contact interactions and one dimension. In this limit, a polaronic-like behaviour is not expected, and the physics is that of a magnon or impurity. While the charge degrees of freedom of the system are frozen, the resulting tight-binding Hamiltonian for the impurity's spin exhibits an intriguing structure that strongly depends on the filling factor of the lattice potential. This filling dependency also transfers to the nature of the interactions for the case of two magnons and the important spin balanced case. At low filling, and up until near unit filling, the single impurity Hamiltonian faithfully reproduces a single-band, quasi-homogeneous tight-binding problem. As the filling is increased and the second band of the single particle spectrum of the periodic potential is progressively filled, the impurity Hamiltonian, at low energies, describes a single particle trapped in a multi-well potential. Interestingly, once the first two bands are fully filled, the impurity Hamiltonian is a near-perfect realisation of the Su-Schrieffer-Heeger model. Our studies, which go well beyond the single-band approximation, that is, the Hubbard model, pave the way for the realisation of interacting one-dimensional models of condensed matter physics.