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

Habib Ammari, Clemens Thalhammer

TL;DR

This paper derives a real-space representation of the Zak phase for 1D periodic Jacobi operators by expressing the Berry connection in terms of the half-line Weyl $m$-function $m_+$. The main result shows that $\mathcal{A}_n(k) = \frac{\lambda_n'(k)}{2}\frac{\Re(m_+'(\lambda+i0))}{\Im(m_+(\lambda+i0))} - \frac{1}{2}$ and provides a Zak-phase formula $\gamma_n = \int_{\lambda_{n,\min}}^{\lambda_{n,\max}} \frac{\Re(m_+'(\lambda+i0))}{\Im(m_+(\lambda+i0))}\,d\lambda - \pi$, which depends only on boundary data and not on Bloch theory. The authors verify consistency with the classical Zak phase in SSH and Rice-Mele models and show how inversion-symmetric cells yield quantised Zak phases via a real-space, symmetry-based argument. The approach clarifies the bulk-boundary interplay and offers a computational tool linking spectral data to geometric phases, with potential implications for surface impedance and topological classification in 1D systems.

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.

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

TL;DR

This paper derives a real-space representation of the Zak phase for 1D periodic Jacobi operators by expressing the Berry connection in terms of the half-line Weyl -function . The main result shows that and provides a Zak-phase formula , which depends only on boundary data and not on Bloch theory. The authors verify consistency with the classical Zak phase in SSH and Rice-Mele models and show how inversion-symmetric cells yield quantised Zak phases via a real-space, symmetry-based argument. The approach clarifies the bulk-boundary interplay and offers a computational tool linking spectral data to geometric phases, with potential implications for surface impedance and topological classification in 1D systems.

Abstract

We establish a new, real-space formula for the Zak phase for one dimensional periodic Jacobi operators in terms of the Weyl -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.
Paper Structure (11 sections, 3 theorems, 80 equations, 2 figures)

This paper contains 11 sections, 3 theorems, 80 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Result
  4. Three Examples
  5. Su–Schrieffer–Heeger Model
  6. Rice-Mele Model
  7. Mirror Symmetric Fundamental Cells
  8. Proof of Theorem \ref{['thm: main']}
  9. Concluding Remarks
  10. Acknowledgements
  11. Code Availability

Key Result

Lemma 2.4

Let $\mathcal{J}$ be a $p$-periodic Jacobi operator. Then there exists some continuous function $f$ from $\mathbb{C}$ into $\mathbb{C}^{p\times p}$, called the symbol of the operators $T(f)$ or $L(f)$ that are respectively the associated Toeplitz and Laurent operators to $\mathcal{J}$, such that $\m

Figures (2)

  • Figure 1: Comparison between formula \ref{['eq: zak_new']} and a standard implementation for various parameters. The grey lines mark where the Berry phase becomes degenerate due to equal or vanishing hopping terms.
  • Figure 2: Computation for symmetric trimer setup with ${\bm{b}^{(1)}} = {\bm{b}^{(3)}}=0, {\bm{b}^{(2)}}=1, {\bm{a}^{(1)}}={\bm{a}^{(2)}}=1.2$ and ${\bm{a}^{(3)}}=1.5$. Using the new formula and $N=500$ discretisation points, the Zak phase of the first band is computed to be $3.1284$, while the standard implementation yields a value of $\pi$.

Theorems & Definitions (9)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4
  • Theorem 3.1
  • Definition 4.1
  • Lemma 4.2
  • Definition 4.3
  • proof