[Paper Review] Generalization of Zak's phase for lattice models with non-centered inversion symmetry axis

A. M. Marques, R. G. Dias|arXiv (Cornell University)|Jul 19, 2017

Random Matrices and Applications3 citations

TL;DR

This paper generalizes Zak's phase for one-dimensional lattice models with inversion symmetry whose axis is not centered in the unit cell. It shows that the standard Zak’s phase loses quantization due to k-dependent matrix elements in the inversion operator, and proposes a corrected expression that restores π-quantized values by incorporating a k-dependent correction term, enabling topological invariants in systems with non-centered inversion symmetry.

ABSTRACT

We show how the presence of inversion symmetry in a one-dimensional (1D) lattice model is not a sufficient condition for a quantized Zak's phase. This is only the case when the inversion axis is at the center of the unit cell. When the inversion axis is not at the center, the modified inversion operator within the unit cell gains a k-dependence in some of its matrix elements which adds a correction term to the usual Zak's phase expression1, making it in general deviate from its quantized value. A general expression that recovers a quantized Zak's phase in a lattice model with a unit cell of arbitrary size and arbitrarily positioned inversion axis is provided in this paper, which relates the quantized value with the eigenvalues of a modified parity operator at the inversion invariant momenta.

Motivation & Objective

• To address the failure of standard Zak’s phase to remain quantized when inversion symmetry is present but not centered in the unit cell.
• To identify the origin of non-quantization as arising from k-dependent matrix elements in the unit-cell-level inversion operator.
• To derive a generalized, quantized Zak’s phase expression valid for arbitrary unit cell size and inversion axis position.
• To establish a modified parity operator at inversion-symmetric momenta that correctly predicts the quantized Zak’s phase.
• To provide a robust topological invariant applicable to 1D systems with non-centered inversion symmetry, including quasi-1D and ribbon systems.

Proposed method

• Derive the modified inversion operator within the unit cell, showing its k-dependence when the inversion axis is off-center.
• Identify the correction term in the Zak’s phase expression as arising from the non-Hermitian and k-dependent nature of the modified inversion operator.
• Use the Wilson loop formalism to express the Zak’s phase and derive the phase correction from the non-unitary evolution of the inversion operator across k-space.
• Introduce a modified parity operator at k = 0 and k = π, whose eigenvalues determine the corrected Zak’s phase.
• Apply the corrected Zak’s phase to model systems (SSH, t₁t₁t₂t₂, t₁t₁t₁t₂) and verify consistency with edge state physics and Wannier center calculations.
• Demonstrate that the phase shift between different inversion axis choices is exactly π, consistent with topological transitions.

Experimental results

Research questions

• RQ1 Under what conditions does the standard Zak’s phase fail to be quantized in one-dimensional lattice models with inversion symmetry?
• RQ2 How does the k-dependence of the unit-cell-level inversion operator affect the Zak’s phase and its quantization?
• RQ3 What correction term must be added to the Zak’s phase to restore π-quantization when the inversion axis is not centered in the unit cell?
• RQ4 How can the modified parity operator at inversion-symmetric momenta be used to predict the correct topological invariant in such systems?
• RQ5 Can the generalized Zak’s phase be applied to quasi-1D systems and ribbons with non-centered inversion symmetry?

Key findings

• The standard Zak’s phase is not quantized when the inversion axis is not centered in the unit cell due to k-dependent matrix elements in the inversion operator.
• A correction term proportional to the integral of |u_j,A(k)|² over k-space must be added to the Zak’s phase to restore π-quantization.
• The corrected Zak’s phase, denoted as ˜γ_j, is quantized and serves as a topological invariant regardless of the inversion axis position.
• For models like t₁t₁t₂t₂ with no centered inversion axis in any unit cell, the generalized Zak’s phase is essential for topological characterization.
• The phase difference between two inversion axis choices in the same unit cell is exactly π, consistent with topological transitions.
• Numerical verification confirms that edge states appear precisely when the corrected Zak’s phase shifts by π, matching predictions from the modified parity eigenvalues.