$\mathbb{Z}_2$ topological invariants from the Green's function diagonal zeros

Abstract: We investigate the relationship between the analytical properties of the Green's function and $\mathbb{Z}_2$ topological insulators, focusing on three-dimensional inversion-symmetric systems. We show that the diagonal zeros of the Green's function in the orbital basis provide a direct and visual way to calculate the strong and weak $\mathbb{Z}_2$ topological invariants. We introduce the surface of crossings of diagonal zeros in the Brillouin zone, and show that it separates TRIMs of opposite parity in two-band models, enabling the visual computation of the $\mathbb{Z}_2$ invariants by counting the relevant TRIMs on either side. In three-band systems, a similar property holds in every case except when a trivial band is added in the band gap of a non-trivial two-band system, reminiscent of the band topology of fragile topological insulators. Our work could open avenues to experimental measurements of $\mathbb{Z}_2$ topological invariants using ARPES measurements.

Analysis of $\mathbb{Z}_2$ Topological Invariants via Green Function's Diagonal Zeros

In their exploration of topological insulators, Florian Simon and Corentin Morice delve into the intricacies of $\mathbb{Z}_2$ topological invariants, focusing on the analytical properties of Green's functions in inversion-symmetric systems. Their work introduces a promising methodology that leverages the diagonal zeros of the Green function to compute these invariants, potentially facilitating experimental measurement, notably through ARPES techniques.

Overview and Context

Topological insulators represent states of matter characterized by nontrivial topological properties that remain invariant under continuous deformations. Historically, calculating topological invariants has relied heavily on simplified theories often inadequate for realistic, interacting systems. This paper aims to bridge this gap by focusing on a direct association between Green function properties and topological invariants, providing an approach that circumvents the typical limitations of non-interacting models.

Simon and Morice propose using diagonal zeros of the Green function matrix as markers for determining $\mathbb{Z}_2$ invariants. Their methodology is especially relevant in three-dimensional, time-reversal, and inversion-symmetric systems. In such systems, the Green function matrix is analyzed in the orbital basis, where specific diagonal zeros distinguish between trivial and nontrivial topological states.

Methodological Insights

The authors elaborate on three core theoretical concepts:

1. Zero-Pole Representation: The Green function, expressed in terms of its eigenvalues and eigenvectors, reveals that the diagonal components signal the presence of topological states through their zeros.

2. Eigenvector-Eigenvalue Relationship: This relationship confirms that zero-pole touchings, which occur when diagonal zeros coincide with poles, are indicative of topological changes in the system.

3. Cauchy Interlacing Inequality: This mathematical principle restricts the alignment of zeros and poles, thereby helping to identify significant zero-pole touchings and further elucidating topological states.

Through these theories, the structure of the Green function can be analyzed visually, allowing researchers to compute strong and weak $\mathbb{Z}_2$ invariants by observing the separation of TRIMs (Time-Reversal Invariant Momenta) with opposite parities in the Brillouin zone.

Numerical Results and Examples

Applying their theoretical framework, the authors specifically focus on two-band systems to exemplify the concepts. They demonstrate that in such models, bands of opposite parity are separated by zero-zero crossings, forming surfaces within the Brillouin Zone. These surfaces inherently define regions of differing topological phases, enabling one to visually assess the $\mathbb{Z}_2$ invariants.

For practical visualization, the authors discuss the Wilson-Dirac model, presenting three distinct topological phases. The visual identification of zero surfaces correlates with the phase character (trivial, weak, or strong insulator), verifying the efficacy of their approach.

Implications and Future Directions

Simon and Morice's approach offers a potentially transformative tool for experimentalists by suggesting the possibility of direct measurement of topological invariants from Green function zeros through ARPES. This aligns well with emerging interests in connecting theoretical predictions with measurable physical phenomena.

For future research, applying this method to multiband systems could illuminate the complexities introduced by band interdependencies and fragile topological states. Additionally, exploring its applicability in correlated and non-crystalline systems could further expand its utility, deeply enriching our understanding of topological physics.

Conclusion

This paper introduces a novel perspective on calculating $\mathbb{Z}_2$ topological invariants via Green function diagonal zeros, providing a methodological gateway between theoretical predictions and experimental realizations. The results move towards diminishing longstanding limitations in topological insulator research, fostering potential advancements in both practical measurements and theoretical modeling.