The Zak phase in topologically insulating chains: invariants and limitations

TL;DR

This paper examines the Zak phase in 1D topological insulators with various symmetries, revealing its limitations and how it partially captures topological invariants across different symmetry classes.

Contribution

It extends the analysis of the Zak phase to all AZC symmetry classes in 1D, constructing symmetric bases and defining a $ ext{Z}_2$ invariant, highlighting its constraints and partial topological information.

Findings

Zak phase defines a $ ext{Z}_2$ invariant in certain symmetry classes.

In quaternionic symmetry classes, the Zak phase's $ ext{Z}_2$ invariant vanishes.

Zak phase only partially captures the topological phases of generalized Kitaev chains.

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 well as its limitations in being an effective marker for topological phases of insulating chains. As an example, we discuss the case of generalized Kitaev chains with arbitrary finite-range hopping and single or multiple chiral channels, and show how the Zak phase only retains partial information about their different topological phases.