Nonsymmorphic Symmetry Protected Dirac, M\"obius, and Hourglass Fermions in Topological Materials

TL;DR

This paper reviews how nonsymmorphic symmetries in crystalline materials protect and give rise to exotic fermionic quasiparticles like Dirac, Möbius, and hourglass fermions, revealing new topological phases of matter.

Contribution

It classifies nonsymmorphic-symmetry-protected topological states and discusses their unique fermionic modes and surface states, expanding understanding of symmetry-driven topological phenomena.

Findings

Identification of four classes of nonsymmorphic-symmetry-protected topological states

Description of surface Dirac, Möbius, and hourglass fermions

Demonstration of symmetry and topology roles in quantum materials

Abstract

A lattice symmetry, if being nonsymmorphic, is defined by combining a point group symmetry with a fractional lattice translation that cannot be removed by changing the lattice origin. Nonsymmorphic symmetry has a substantial influence on both the connectivity and topological properties of electronic band structures in solid-state quantum materials. In this article, we review how nonsymmorphic crystalline symmetries can drive and further protect the emergence of exotic fermionic quasiparticles, including Dirac, M\"obius and hourglass fermions, that manifest as the defining energy band signatures for a plethora of gapless or gapped topological phases of matter. We first provide a classification of energy band crossings in crystalline solids, with an emphasis on symmetry-enforced band crossings that feature a nonsymmorphic-symmetry origin. In particular, we will discuss four distinct classes of nonsymmorphic-symmetry-protected topological states as well as their signature fermionic modes: (1) a $Z_2$ topological nonsymmorphic crystalline insulator with two-dimensional surface Dirac fermions; (2) a Dirac semimetal with three dimensional bulk-state Dirac nodes pinned at certain high symmetry momenta; (3) a topological M\"obius insulator with massless surface modes that resemble the topological structure of a M\"obius twist (dubbed as M\"obius fermions); (4) a time-reversal-invariant version of M\"obius insulators with hourglass-shaped massless surface states (dubbed as hourglass fermions). The emergence of the above exotic topological matter perfectly demonstrates the crucial roles of both symmetry and topology in understanding the colorful world of quantum materials.