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Edge spectrum for truncated $\mathbb{Z}_2$-insulators

Alexis Drouot, Jacob Shapiro, Xiaowen Zhu

TL;DR

The paper addresses bulk-edge gap filling for fermionic time-reversal-invariant 2D insulators with a Z2 index. It introduces a local, origin-independent trace formula for the Z2-index and proves center-independence and local-determinacy of the index, enabling the index to be read from large but finite regions. Using a contradiction strategy akin to previous quantum Hall work, it shows that when two bulks with different Z2 indices meet an interface occupying arbitrarily large balls, the edge spectrum necessarily fills the bulk gap. The geometric condition on the interface region is shown to be essential: if the interface geometry is restricted (e.g., within a finite-width strip), one can construct insulating edge configurations, highlighting the delicate interplay between geometry and topological protection in Z2-insulators.

Abstract

Fermionic time-reversal-invariant insulators in two dimensions -- class AII in the Kitaev table -- come in two different topological phases. These are characterized by a $\mathbb{Z}_2$-index: the Fu-Kane-Mele index. We prove that if two such insulators with different indices occupy regions containing arbitrarily large balls, then the spectrum of the resulting operator fills the bulk spectral gap. Our argument follows a proof by contradiction developed in an earlier work by two of the authors for quantum Hall systems. It boils down to showing that the $\mathbb{Z}_2$-index can be computed only from bulk information in sufficiently large balls. This is achieved via a result of independent interest: a local trace formula for the $\mathbb{Z}_2$-index.

Edge spectrum for truncated $\mathbb{Z}_2$-insulators

TL;DR

The paper addresses bulk-edge gap filling for fermionic time-reversal-invariant 2D insulators with a Z2 index. It introduces a local, origin-independent trace formula for the Z2-index and proves center-independence and local-determinacy of the index, enabling the index to be read from large but finite regions. Using a contradiction strategy akin to previous quantum Hall work, it shows that when two bulks with different Z2 indices meet an interface occupying arbitrarily large balls, the edge spectrum necessarily fills the bulk gap. The geometric condition on the interface region is shown to be essential: if the interface geometry is restricted (e.g., within a finite-width strip), one can construct insulating edge configurations, highlighting the delicate interplay between geometry and topological protection in Z2-insulators.

Abstract

Fermionic time-reversal-invariant insulators in two dimensions -- class AII in the Kitaev table -- come in two different topological phases. These are characterized by a -index: the Fu-Kane-Mele index. We prove that if two such insulators with different indices occupy regions containing arbitrarily large balls, then the spectrum of the resulting operator fills the bulk spectral gap. Our argument follows a proof by contradiction developed in an earlier work by two of the authors for quantum Hall systems. It boils down to showing that the -index can be computed only from bulk information in sufficiently large balls. This is achieved via a result of independent interest: a local trace formula for the -index.
Paper Structure (14 sections, 8 theorems, 66 equations, 1 figure)

This paper contains 14 sections, 8 theorems, 66 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Time-reversal-invariant insulators
  3. Interface systems
  4. Strategy
  5. Trace formulas for the $\mathbb{Z}_2$-index and the proof of \ref{['prop:5']}.
  6. Notation
  7. Center-independence
  8. Local-determinancy
  9. Proof of \ref{['thm:4']}
  10. About the geometric condition on $\Omega$
  11. Wallace and Haldane models
  12. Construction of $H_\pm$
  13. Construction of $H_e$
  14. Spectrum at $0$

Key Result

Theorem 1

Assume that $H$ models an interface system at energy $E_F$ in the sense of def:edge (in particular, $\Omega$, $\Omega^c$ contain arbitrarily large balls and $\mathcal{I}(H_+) \neq \mathcal{I}(H_-)$). Then $E_F \in \operatorname{Spec}(H)$.

Figures (1)

  • Figure 1: The eigenvalues of $B_1,B_2$ (blue, red) and the contour ${\partial} S$ (green). It is a rectangle passing through the point $\lambda_0$. Its distance to the spectra of $B_1, B_2$ is half the length of the gap, hence at least $2^{-4} (\|B_1\|+1)^{-1}$. It length is at most $4$.

Theorems & Definitions (22)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 1
  • Proposition 1
  • Remark 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 12 more