A Real-Space Formulation of the Zak Phase via Weyl m-Functions

TL;DR

This paper introduces a real-space formula for the Zak phase in one-dimensional periodic Jacobi operators using Weyl m-functions, bypassing Floquet-Bloch theory, and clarifies its boundary dependence and quantization under inversion symmetry.

Contribution

It presents a novel real-space formulation of the Zak phase based on Weyl m-functions, emphasizing boundary effects and classical quantization without Floquet-Bloch theory.

Findings

Derived a boundary-dependent real-space formula for the Zak phase

Revealed conditions for the quantization of the Zak phase in symmetric systems

Provided a new perspective on topological invariants in 1D periodic operators

Abstract

We establish a new, real-space formula for the Zak phase for one dimensional periodic Jacobi operators in terms of the Weyl $m_+$-function that does not rely on Floquet-Bloch theory. This novel representation highlights the dependence of the Zak phase on boundary terms. Moreover, we show how to recover the classical quantisation of the Zak phase for periodic Jacobi operators with inversion symmetric fundamental cells.