Classification of time-reversal-invariant crystals with gauge structures

TL;DR

This paper develops a comprehensive classification of projective symmetry algebras in time-reversal-invariant 2D crystals, revealing new physical signatures and providing a foundation for exploring novel topological states in artificial crystals.

Contribution

It establishes a unified classification of all projective symmetry algebras in 2D crystals, a largely unexplored area, and identifies key physical signatures arising from these structures.

Findings

Classified all 458 projective symmetry algebras in 2D crystals

Identified three physical signatures: momentum shifts, nontrivial Zak phase, eight-fold nodal point

Provided a theoretical foundation for artificial crystal topological phenomena

Abstract

A peculiar feature of quantum states is that they may embody so-called projective representations of symmetries rather than ordinary representations. Projective representations of space groups-the defining symmetry of crystals-remain largely unexplored. Despite recent advances in artificial crystals, whose intrinsic gauge structures necessarily require a projective description, a unified theory is yet to be established. Here, we establish such a unified theory by exhaustively classifying and representing all 458 projective symmetry algebras of time-reversal-invariant crystals from 17 wallpaper groups in two dimensions-189 of which are algebraically non-equivalent. We discover three physical signatures resulting from projective symmetry algebras, including the shift of high-symmetry momenta, an enforced nontrivial Zak phase, and a spinless eight-fold nodal point. Our work offers a theoretical foundation for the field of artificial crystals and opens the door to a wealth of topological states and phenomena beyond the existing paradigms.