Abstract

Topological band theory was recently extended to crystals hosting synthetic Z2 fluxes within unit cells, with the possible symmetry-compatible flux arrangements classified by second group cohomology for all two-dimensional space groups [1]. Notably, the fluxes enforce the space group to be projectively represented, enabling new topological band features, such as Möbius insulators or Klein-bottle Brillouin zones [2]. However, such analyses have not been extended to crystals hosting non-Abelian gauge fields.

Separately, noncommutativity of translations has been introduced to crystalline structures through a construction dubbed Cayley-Schreier lattice [3]. The noncommutativity is achieved by decorating each site of a Bravais lattice with a collection of orbitals: one for each element of a discrete group G, which acts as a gauge group. However, this research has similarly focused on the Abelian choices G = ZN, leaving non-Abelian generalizations unexplored.

In this talk, we first review the notions of synthetic gauge structures and Cayley–Schreier lattices, and then present our generalization of the Cayley–Schreier construction to non-Abelian gauge groups [4]. Specifically, choosing G as the quaternion group Q8 = {±1, ±i, ±j, ±k} (which serves as a discretized proxy for SU(2) and where i, j, k are pairwise anticommuting elements) allows us to simulate known spinful topological insulators in one and two spatial dimensions. The presented construction can be conveniently realized in synthetic platforms using only real hopping amplitudes between the orbitals. Our work sets the stage to investigate topological band theory in the presence of non-Abelian gauge fields.

References:

[1] Z. Y. Chen, Z. Zhang, S. A. Yang, and Y. X. Zhao, Classification of time-reversal-invariant crystals with gauge structures, Nat. Commun. 14, 743 (2023).

[2] T. Li, J. Du, Q. Zhang, Y. Li, X. Fan, F. Zhang, and C. Qiu, Acoustic Möbius Insulators from Projective Symmetry, Phys. Rev. Lett. 128, 116803 (2022); Z. Y. Chen, S. A. Yang, and Y. X. Zhao, Brillouin Klein bottle from artificial gauge fields, Nat. Commun. 13, 2215 (2022).

[3] M. Marciani, Translation Groups for arbitrary Gauge Fields in Synthetic Crystals with real hopping amplitudes, arXiv:2508.08461 (2025).

[4] Z. Guba, R.-J. Slager, L. K. Upreti, and T. Bzdušek, Topological non-Abelian Gauge Structures in Cayley-Schreier Lattices, arXiv: 2509.25316 (2025).