# Fermi sea topology and boundary geometry for free particles in one- and two-dimensional lattices

## TL;DR

This paper classifies the topological properties of Fermi seas for free spinless fermions on lattice-symmetric backgrounds in one and two dimensions, revealing a finite set of topological classes with physical implications.

## Contribution

It introduces a topological classification scheme for Fermi seas and their boundaries using orbifold geometries, expanding understanding of topological phases in lattice systems.

## Key findings

- Two topological classes in 1D: conductor and insulator.
- 17 classes in 2D, including disks, spheres, and more complex surfaces.
- Classification is robust under perturbative interactions.

## Abstract

Free gasses of spinless fermions moving on a lattice-symmetric geometric background are considered. Their topological properties at zero temperature can be used to classify their Fermi seas and associated boundaries. The flat orbifolds ${\Rb}^{d}/\Gamma$, where $\Gamma$ is the crystallographic group of symmetry in $d$-dimensional momentum space, are used to accomplish this task. Two topological classes exist for $d=1$: an interval, which is identified as a conductor, and a circumference, which corresponds to an insulator. The number of topological classes increases to 17 for $d=2$: 8 have the topology of a disk, that are generally recognized as conductors, and 4 correspond to a 2-sphere, matching insulators. Both sets eventually contain a finite number of conical singularities and reflection corners at the boundaries. The remaining cases in the listing relate to conductors (annulus, M\"obius strip) and insulators (2-torus, real projective plane, Klein bottle). Examples that fall under this list are given, along with physical interpretations of the singularities. It is anticipated that the findings of this classification will be robust under perturbative interactions due to its topological character.