Topological zero modes and Dirac points protected by spatial symmetry and chiral symmetry

TL;DR

This paper introduces a new class of topologically protected zero energy modes and Dirac points arising from the coexistence of chiral and spatial symmetries, expanding understanding of symmetry-protected topological phases.

Contribution

It demonstrates that spatial symmetries can protect Dirac nodes and zero modes independently of the winding number, with applications to various lattice systems and topological charges.

Findings

Dirac nodes in honeycomb and square lattices are protected by rotation symmetries.

Symmetry-protected band touching points can exist even with zero winding number.

Examples of three-dimensional Dirac semimetals with protected band touchings are provided.

Abstract

We explore a new class of topologically stable zero energy modes which are protected by coexisting chiral and spatial symmetries. If a chiral symmetric Hamiltonian has an additional spatial symmetry such as reflection, inversion and rotation, the Hamiltonian can be separated into independent chiral-symmetric subsystems by the eigenvalue of the space symmetry operator. Each subsystem supports chiral zero energy modes when a topological index assigned to the block is nonzero. By applying the argument to Bloch electron systems, we detect band touching at symmetric points in the Brillouin zone. In particular, we show that Dirac nodes appearing in honeycomb lattice (e.g. graphene) and in half-flux square lattice are protected by three-fold and two-fold rotation symmetry, respectively. We also present several examples of Dirac semimetal with isolated band-touching points in three-dimensional $k$-space, which are protected by combined symmetry of rotation and reflection. The zero mode protection by spatial symmetry is distinct from that by the conventional winding number. We demonstrate that symmetry-protected band touching points emerge even though the winding number is zero. Finally, we identify relevant topological charges assigned to the gapless points.