ℤ₂ topological invariant in three-dimensional PT- and PC-symmetric class CI band structures

2025/09/24 by Shiozaki, Ken
#FOS: Physical sciences #Mesoscale and Nanoscale Physics (cond-mat.mes-hall)

paper · doi:10.48550/arxiv.2509.19825

Abstract

We construct a previously missing ℤ₂ 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 η-invariant, a spin-Chern--Simons (spin-CS) action. We show that PC symmetry quantizes the spin-CS action to \0,2π\ with 4π periodicity, thereby yielding a well-defined ℤ₂ invariant. This invariant is additive under direct sums of isolated band structures, reduces to a known ℤ₂ 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 ℤ₂ invariant distinguishes topological phases that cannot be detected by the previously known topological indices.

Citations

• A Discrete Formulation of Second Stiefel-Whitney Class for Band Theory
• Classical Chern-Simons on manifolds with spin structure

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