Unitary fermions and Luscher's formula on a crystal

TL;DR

This paper develops a model for two-particle scattering in a periodic lattice, connecting lattice and continuum descriptions, and provides a method to extract scattering information from finite lattice simulations.

Contribution

It introduces a continuum model respecting lattice symmetry for s-wave contact interactions and relates lattice energy shifts to continuum scattering parameters.

Findings

Energy shifts match between lattice and continuum schemes.

Results are valid across weak to unitary interaction regimes.

Applicable for extracting scattering data from finite lattice simulations.

Abstract

We consider the low-energy particle-particle scattering properties in a periodic simple cubic crystal. In particular, we investigate the relation between the two-body scattering length and the energy shift experienced by the lowest-lying unbound state when this is placed in a periodic finite box. We introduce a continuum model for s-wave contact interactions that respects the symmetry of the Brillouin zone in its regularisation and renormalisation procedures, and corresponds to the na{\"i}ve continuum limit of the Hubbard model. The energy shifts are found to be identical to those obtained in the usual spherically symmetric renormalisation scheme upon resolving an important subtlety regarding the cutoff procedure. We then particularize to the Hubbard model, and find that for large finite lattices the results are identical to those obtained in the continuum limit. The results reported here are valid in the weak, intermediate and unitary limits, and can be used for the extraction of scattering information ,via exact diagonalisation or Monte Carlo methods, of two-body systems in realistic periodic lattices.