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

TL;DR

The paper develops a symmetry-based, topological classification of Fermi seas for free spinless fermions on lattice backgrounds by mapping momentum-space geometry to flat orbifolds ${\mathbb{R}}^{d}/Γ$, where Γ is a crystallographic group. In 1D, the Fermi sea topology reduces to either an interval (conductor) or a circumference (insulator), corresponding to ${\mathbb{R}}/{\mathbb{Z}}$ and ${\mathbb{R}}/{\mathbb{D}_\infty}$ orbifolds; in 2D, 17 orbifolds arise from the 17 wallpaper groups, yielding seven fundamental topologies (S^2, D^2, ${\mathbb{R}}P^2$, T^2, K, A, M) with insulators and conductors distinguished, and constrained by conical/reflection singularities via the Euler characteristic $\chi(O)$. The work highlights how nesting and Van Hove singularities relate to boundary features and how the approach extends to 3D (outlined but not completed), with a clear path for applying the results to real materials and perturbative interactions due to the topological nature of the classifications.

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 ${\Rb}^{d}/Γ$, where $Γ$ 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, Möbius 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.