Classification of time-reversal-invariant crystals with gauge structures

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.

62 pages, 11 pages for main text + 51 pages for supplementary information

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• Topological Acoustics
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• Artificial Gauge Field Engineered Excited-State Topology: Control of Dynamical Evolution of Localized Spinons
• Topological constraints on the electronic band structure of hexagonal lattice in a magnetic field
• Negative exchange interaction in Si quantum dot arrays via valley-phase induced ${\mathbb{Z}}_2$ gauge field
• Single $\pi$-flux hosting topological defect modes in bilayer acoustic metamaterials