${\mathbb{Z}}_2$ topological invariant in three-dimensional PT- and PC-symmetric class CI band structures

Abstract

We construct a previously missing ${\mathbb{Z}}_2$ topological invariant for three-dimensional band structures in symmetry class CI defined by parity-time (PT) and parity-particle-hole (PC) symmetries. PT symmetry allows one to define a real Berry connection and, based on the $\eta$-invariant, a spin-Chern--Simons (spin-CS) action. We show that PC symmetry quantizes the spin-CS action to $\{0,2\pi \}$ with $4\pi$ periodicity, thereby yielding a well-defined ${\mathbb{Z}}_2$ invariant. This invariant is additive under direct sums of isolated band structures, reduces to a known ${\mathbb{Z}}_2$ index when a global Takagi factorization exists, and in general depends on the choice of spin structure. Finally, we demonstrate lattice models in which this newly introduced ${\mathbb{Z}}_2$ invariant distinguishes topological phases that cannot be detected by the previously known topological indices.

arXiv comment18 pages. typos corrected