Periodic Clifford symmetry algebras on flux lattices

Abstract

Real Clifford algebras play a fundamental role in the eight real Altland-Zirnbauer symmetry classes and the classification tables of topological phases. Here, we present another elegant realization of real Clifford algebras in the d-dimensional spinless rectangular lattices with π flux per plaquette. Due to the T -invariant flux configuration, real Clifford algebras are realized as projective symmetry algebras of lattice symmetries. Remarkably, d mod 8 exactly corresponds to the eight Morita equivalence classes of real Clifford algebras with eightfold Bott periodicity, resembling the eight real Altland-Zirnbauer classes. The representation theory of Clifford algebras determines the degree of degeneracy of band structures, both at generic k points and at high-symmetry points of the Brillouin zone. Particularly, we demonstrate that the large degeneracy at high-symmetry points offers a rich resource for forming novel topological states by various dimerization patterns, including a 3D higherorder semimetal state with double-charged bulk nodal loops and hinge modes, a 4D nodal surface semimetal with 3D surface solid-ball zero modes, and 4D Möbius topological insulators with a eightfold surface nodal point or a fourfold surface nodal ring. Our theory can be experimentally realized in artificial crystals by their engineerable Z2 gauge fields and capability to simulate higher dimensional systems.