Nonsymmorphic symmetry-required band crossings in topological semimetals

TL;DR

This paper demonstrates that nonsymmorphic symmetries can enforce bulk band crossings in topological semimetals, leading to reduced-dimensional Fermi surfaces and novel topological charges, with implications for understanding band degeneracies.

Contribution

It introduces a framework showing how nonsymmorphic symmetries enforce band crossings and characterizes their topological charges, expanding the understanding of topological semimetals.

Findings

Nonsymmorphic symmetries enforce unavoidable bulk band crossings.

Band crossings are located at high-symmetry points with symmetry-determined positions.

A global $ ext{Z}_2$ topological charge characterizes these degeneracies.

Abstract

We show that for two-band systems nonsymmorphic symmetries may enforce the existence of band crossings in the bulk, which realize Fermi surfaces of reduced dimensionality. We find that these unavoidable crossings originate from the momentum dependence of the nonsymmorphic symmetry, which puts strong restrictions on the global structure of the band configurations. Three different types of nonsymmorphic symmetries are considered: (i) a unitary nonsymmorphic symmetry, (ii) a nonsymmorphic magnetic symmetry, and (iii) a nonsymmorphic symmetry combined with inversion. For nonsymmorphic symmetries of the latter two types, the band crossings are located at high-symmetry points of the Brillouin zone, with their exact positions being determined by the algebra of the symmetry operators. To characterize these band degeneracies we introduce a \emph{global} topological charge and show that it is of $\mathbb{Z}_2$ type, which is in contrast to the \emph{local} topological charge of Fermi points in, say, Weyl semimetals. To illustrate these concepts, we discuss the $\pi$-flux state as well as the SSH model at its critical point and show that these two models fit nicely into our general framework of nonsymmorphic two-band systems.