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Continuous Topological Insulators Classification and Bulk Edge Correspondence

Guillaume Bal

TL;DR

The work addresses the mathematical classification of bulk and interface topological insulators realized as continuous PDE models in Euclidean spaces, focusing on robust, asymmetric transport at interfaces. It develops an operator-theoretic framework based on Fredholm indices, the Fedosov–Hörmander formula, domain-wall constructions, and the interface current observable $\sigma_I$ to define and compute bulk invariants and bulk-difference invariants (BDI). A central result is the bulk–edge correspondence, which equates $2\pi\sigma_I$ with the corresponding index data for elliptic operators in two dimensions and extends to higher dimensions, with stability under perturbations and clear caveats for non-elliptic cases such as shallow-water models. The theory is illustrated with concrete bulk models (Landau and Dirac), interface constructions, and applications to bilayer graphene, highlighting both the predictive power and limitations of the framework in describing edge transport in topological materials. The results provide rigorous connections between spectral flow, Chern numbers, and edge conductance, offering a versatile toolkit for analyzing topological transport beyond lattice systems, including continuum photonics and geophysical models.

Abstract

This paper reviews recent results on the classification of partial differential operators modeling bulk and interface topological insulators in Euclidean spaces. Our main objective is the mathematical analysis of the unusual, robust-to-perturbations, asymmetric transport that necessarily appears at interfaces separating topological insulators in different phases. The central element of the analysis is an interface-current-observable describing this asymmetry. We show that this observable may be computed explicitly by spectral flow when the interface Hamiltonian is explicitly diagonalizable. We review the classification of bulk phases for Landau and Dirac operators and provide a general classification of elliptic interface pseudo-differential operators by means of domain walls and a corresponding bulk-difference invariant (BDI). The BDI is simple to compute by the Fedosov-Hörmander formula implementing in a Euclidean setting an Atiyah-Singer index theory. A generalized bulk-edge correspondence then states that the interface current observable and the BDI agree on elliptic operators, whereas this is not necessarily the case for non-elliptic operators.

Continuous Topological Insulators Classification and Bulk Edge Correspondence

TL;DR

The work addresses the mathematical classification of bulk and interface topological insulators realized as continuous PDE models in Euclidean spaces, focusing on robust, asymmetric transport at interfaces. It develops an operator-theoretic framework based on Fredholm indices, the Fedosov–Hörmander formula, domain-wall constructions, and the interface current observable to define and compute bulk invariants and bulk-difference invariants (BDI). A central result is the bulk–edge correspondence, which equates with the corresponding index data for elliptic operators in two dimensions and extends to higher dimensions, with stability under perturbations and clear caveats for non-elliptic cases such as shallow-water models. The theory is illustrated with concrete bulk models (Landau and Dirac), interface constructions, and applications to bilayer graphene, highlighting both the predictive power and limitations of the framework in describing edge transport in topological materials. The results provide rigorous connections between spectral flow, Chern numbers, and edge conductance, offering a versatile toolkit for analyzing topological transport beyond lattice systems, including continuum photonics and geophysical models.

Abstract

This paper reviews recent results on the classification of partial differential operators modeling bulk and interface topological insulators in Euclidean spaces. Our main objective is the mathematical analysis of the unusual, robust-to-perturbations, asymmetric transport that necessarily appears at interfaces separating topological insulators in different phases. The central element of the analysis is an interface-current-observable describing this asymmetry. We show that this observable may be computed explicitly by spectral flow when the interface Hamiltonian is explicitly diagonalizable. We review the classification of bulk phases for Landau and Dirac operators and provide a general classification of elliptic interface pseudo-differential operators by means of domain walls and a corresponding bulk-difference invariant (BDI). The BDI is simple to compute by the Fedosov-Hörmander formula implementing in a Euclidean setting an Atiyah-Singer index theory. A generalized bulk-edge correspondence then states that the interface current observable and the BDI agree on elliptic operators, whereas this is not necessarily the case for non-elliptic operators.

Paper Structure

This paper contains 23 sections, 20 theorems, 107 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Bulk Hamiltonians
  3. Fredholm operators and modules and pairs of projections.
  4. Topology associated to Landau levels
  5. Geometric identity.
  6. Topology associated to Dirac bands.
  7. Dirac operators in arbitrary spatial dimensions
  8. Stability under perturbations and Helffer-Sjöstrand formula
  9. Interface Hamiltonian
  10. Interface current observable
  11. Interface observable and spectral flow.
  12. Application to continuous operators.
  13. Classification of elliptic pseudo-differential operators
  14. Weyl quantization and symbol classes.
  15. Classification by interface current observable.
  16. ...and 8 more sections

Key Result

Theorem 2.1

The operator $PUP$ is Fredholm on the range of $P$ and, moreover,

Figures (4)

  • Figure 1: Spectral of Dirac interface Hamiltonian with: Left: mass term $m(y)=y$ (only positive energies displayed for clearer presentation); Right: mass term $m(y)=\,{\rm sign}(y)$ with a unique branch of edge spectrum while filled (red) parts correspond to bulk spectrum.
  • Figure 2: Left: spectrum of $H$ when $f(y)=y$ with a spectral flow equal to $2$ as dictated by the BEC. Right: spectrum of $H$ when $f(y)=\,{\rm sign}(y)$.
  • Figure 3: Radial compactification of two Euclidean planes onto the unit sphere.
  • Figure 4: Left: bulk spectrum for $m(y)=1$. Right: Interface spectrum with Spectral Flow=-2.

Theorems & Definitions (20)

  • Theorem 2.1: avron1994
  • Lemma 2.2: Russo's criterion
  • Theorem 2.3: bal2019continuous
  • Theorem 2.4: bal2019continuous
  • Theorem 2.5
  • Proposition 2.6
  • Lemma 3.1
  • Corollary 3.2: Spectral flow
  • Corollary 3.3: Spectral flow and Index
  • Theorem 3.4: quinn2024asymmetric
  • ...and 10 more