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The Zak phase in topologically insulating chains: invariants and quaternionic constraints

Federico Manzoni, Domenico Monaco, Gabriele Peluso

Abstract

In this work we investigate the topological content of the Zak phase in one-dimensional translation-invariant topological insulators endowed with time-reversal, particle-hole and/or chiral symmetries, extending results from \cite{Monaco_2023}. We analyze the extent to which the Zak phase captures the topology of all Altland--Zirnbauer--Cartan (AZC) symmetry classes in $1$D. Building on the framework of fibered Hamiltonians and spectral projections, we construct symmetric Bloch bases adapted to the discrete symmetries of the system and define a $\mathbb{Z}_2$-valued topological invariant $\mathrm{I}^{(\mathrm{AZC-class})}(H)$ obtained from the abelian Zak phase. Moreover, we demonstrate that in symmetry classes admitting a quaternionic structure, i.e. anti-unitary symmetries squaring to minus the identity, the Zak phase is further constrained, leading to the vanishing of the $\mathbb{Z}_2$ invariant mentioned above. This highlights the sensitivity of the Zak phase to additional geometric structures of the manifold of occupied energy states. As an application, we discuss the case of generalized Kitaev chains with arbitrary finite-range hopping and single or multiple chiral channels.

The Zak phase in topologically insulating chains: invariants and quaternionic constraints

Abstract

In this work we investigate the topological content of the Zak phase in one-dimensional translation-invariant topological insulators endowed with time-reversal, particle-hole and/or chiral symmetries, extending results from \cite{Monaco_2023}. We analyze the extent to which the Zak phase captures the topology of all Altland--Zirnbauer--Cartan (AZC) symmetry classes in D. Building on the framework of fibered Hamiltonians and spectral projections, we construct symmetric Bloch bases adapted to the discrete symmetries of the system and define a -valued topological invariant obtained from the abelian Zak phase. Moreover, we demonstrate that in symmetry classes admitting a quaternionic structure, i.e. anti-unitary symmetries squaring to minus the identity, the Zak phase is further constrained, leading to the vanishing of the invariant mentioned above. This highlights the sensitivity of the Zak phase to additional geometric structures of the manifold of occupied energy states. As an application, we discuss the case of generalized Kitaev chains with arbitrary finite-range hopping and single or multiple chiral channels.
Paper Structure (22 sections, 17 theorems, 135 equations, 3 tables)

This paper contains 22 sections, 17 theorems, 135 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Set up and preliminaries
  3. Lattice Hamiltonians, their Bloch--Floquet fibers and their spectrum
  4. Topological insulators and the ten-fold way
  5. The action of the symmetry operators on fibres
  6. Symmetric Bloch bases
  7. General definition
  8. Bloch bases in 1D through parallel transport
  9. The invariant
  10. Zak phase and its properties
  11. The $\mathbb{Z}_2$ invariant
  12. Symmetry classes with quaternionic structure
  13. Focus on the symmetry class BDI
  14. Arbitrary distant nearest neighbors Kitaev chain
  15. Multi-channel arbitrary distant nearest neighbors Kitaev chain
  16. ...and 7 more sections

Key Result

Proposition 2.1

The operator $\mathcal{F}_dH\mathcal{F}_d^{-1}$ is a fibered operator, whose fiber Hamiltonians are matrices $H_k \in \mathrm{M}_N(\mathbb{C})$ given by where $J:=\{j\in \mathbb{Z}^d \ : \ |j|_{\ell^1}\leq R\}$.

Theorems & Definitions (38)

  • Proposition 2.1: Fiber Hamiltonians
  • Proposition 2.2: Discrete symmetries and fiber operators
  • proof
  • Proposition 2.3: Discrete symmetries and spectral eigenprojections
  • Definition 3.1: Bloch basis
  • Definition 3.2: Symmetric Bloch basis
  • Theorem 3.1: Parallel transport
  • proof
  • Proposition 3.1: Discrete symmetries and parallel transport
  • proof
  • ...and 28 more