📝 SummarXiv

Abstract

Free gases of spinless fermions moving on a geometric background with lattice symmetries are considered. Their Fermi seas and corresponding boundaries may be classified according to their topological properties at zero temperature. This is accomplished by considering the flat orbifolds $R^{d}/\Gamma$, with $\Gamma$ being the crystallographic group of symmetry in $d$-dimensional momentum space. For $d=1$, there are 2 topological classes: a circumference, corresponding to an insulator and an interval, identified as a conductor. For $d=2$, the number of topological classes extends to 17: there are 8 with the topology of a disk identified as conductors and 4 corresponding to a 2-sphere matching insulators, both sets eventually including finite numbers of conical singularities and reflection corners at the boundaries. The rest of the listing includes single cases corresponding to insulators (2-torus, real projective plane, Klein bottle) and conductors (annulus, M\"obius strip). Physical interpretations of the singularities are provided, as well as examples that fit within this listing. Given the topological nature of this classification, its results are expected to be robust against small perturbative interactions.