Periodic Clifford symmetry algebras on flux lattices

TL;DR

This paper introduces a novel realization of real Clifford algebras in flux lattice models, revealing their connection to topological phases and enabling the design of complex topological states with potential experimental applications.

Contribution

It establishes a link between Clifford algebras and flux lattice symmetries, classifies degeneracies, and proposes new topological states in higher dimensions.

Findings

Real Clifford algebras are realized as projective symmetry algebras in flux lattices.

Degeneracy patterns determine topological state formation.

Novel topological phases like higher-order semimetals and M"{o}bius insulators are proposed.

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 $\pi$ 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 $3$D higher-order semimetal state with double-charged bulk nodal loops and hinge modes, a $4$D nodal surface semimetal with $3$D surface solid-ball zero modes, and $4$D M\"{o}bius 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 $\mathbb{Z}_2$ gauge fields and capability to simulate higher dimensional systems.