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

Ken Shiozaki

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 $η$-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 $\mathbb{Z}_2$ 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.

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

Abstract

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

Paper Structure

This paper contains 24 sections, 6 theorems, 98 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Real bundle and real Berry connection
  4. $K$-groups and known topological invariants
  5. Construction of $\mathbb{Z}_2$ invariant
  6. Spin-CS action
  7. $\mathbb{Z}_2$ invariant
  8. Spin structure dependence of $\nu[q,{\sigma}]$
  9. Alternative formulations of the $\mathbb{Z}_2$ invariant
  10. Direct sum case
  11. Tensor product case
  12. Applications
  13. $1 \times 1$ model
  14. Example of a model detected only by the $\nu$-invariant
  15. Other models
  16. ...and 9 more sections

Key Result

Lemma 3.1

If $w_1(P)=0$, one can choose a four-dimensional spin manifold $X$ with boundary $\partial X=M$ together with an extension $\tilde{A}$ of the $so(n)$ connection $A$ from $M$ to $X$. In this case, the spin-CS action is written as where $\tilde{F}=d\tilde{A}+\tilde{A}^2$ is the curvature. This expression is independent of the particular choice of $X$ and $\tilde{A}$. Moreover, reducing modulo $2\pi

Theorems & Definitions (7)

  • Lemma 3.1
  • Definition 3.2
  • Proposition 3.3
  • Lemma 3.4
  • Corollary 3.5
  • Proposition 3.6: Proposition 1.7 in JenquinClassical2005
  • Proposition 4.1