General Theory of Momentum-Space Nonsymmorphic Symmetry

TL;DR

This paper introduces a new theoretical framework for momentum-space nonsymmorphic symmetries (k-NSGs) in crystals, expanding the understanding of crystal symmetries and their role in topological phases.

Contribution

It develops a general theory linking real-space symmorphic groups to momentum-space nonsymmorphic groups via projective representations, applicable in any dimension.

Findings

All k-NSGs can be realized by gauge fluxes over real-space lattices.

The theory extends the classification of crystalline topological phases.

Provides a method to identify real-space groups corresponding to momentum-space nonsymmorphic groups.

Abstract

As a fundamental concept of all crystals, space groups are partitioned into symmorphic groups and nonsymmorphic groups. Each nonsymmorphic group contains glide reflections or screw rotations with fractional lattice translations, which are absent in symmorphic groups. Although nonsymmorphic groups ubiquitously exist on real-space lattices, on the reciprocal lattices in momentum space, the ordinary theory only allows symmorphic groups. In this work, we develop a novel theory for momentum-space nonsymmorphic space groups (k-NSGs), utilizing the projective representations of space groups. The theory is quite general: Given any k-NSGs in any dimensions, it can identify the real-space symmorphic space groups (r-SSGs) and construct the corresponding projective representation of the r-SSG that leads to the k-NSG. To demonstrate the broad applicability of our theory, we show these projective representations and therefore all k-NSGs can be realized by gauge fluxes over real-space lattices. Our work fundamentally extends the framework of crystal symmetry, and therefore can accordingly extend any theory based on crystal symmetry, for instance the classification crystalline topological phases.