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.

• Received 17 November 2022
• Revised 20 February 2023
• Accepted 23 May 2023

DOI:https://doi.org/10.1103/PhysRevLett.130.256601