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Bulk-edge correspondence in finite photonic structure

Jiayu Qiu, Hai Zhang

TL;DR

The paper proves a rigorous bulk-edge correspondence for finite two-dimensional photonic structures by equating the bulk gap Chern number with an edge index defined on a bounded domain with Dirichlet boundary conditions. The core method expresses both invariants through Green-function representations, then shows that boundary-induced corrections vanish in the large-domain limit via exponential decay of Green functions in spectral gaps. A detailed set of estimates—on Green functions, trace-class kernels, and interband transition functions—underpins the equivalence, with the edge observable interpreted physically as the circulation of electromagnetic energy along the boundary. The framework not only provides a rigorous foundation for finite-size topological photonic devices but also offers pathways to extensions to other elliptic operators and disordered systems, while highlighting the role of energy conservation in the BE mechanism.

Abstract

In this work, we establish the bulk-edge correspondence principle for finite two-dimensional photonic structures. Specifically, we focus on the divergence-form operator with periodic coefficients and prove the equality between the well-known gap Chern number (the bulk invariant) and an edge index defined via a trace formula for the operator restricted to a finite domain with Dirichlet boundary conditions. We demonstrate that the edge index characterizes the circulation of electromagnetic energy along the system's boundary, and the BEC principle is a consequence of energy conservation. The proof leverages Green function techniques and can be extended to other systems. These results provide a rigorous theoretical foundation for designing robust topological photonic devices with finite geometries, complementing recent advances in discrete models.

Bulk-edge correspondence in finite photonic structure

TL;DR

The paper proves a rigorous bulk-edge correspondence for finite two-dimensional photonic structures by equating the bulk gap Chern number with an edge index defined on a bounded domain with Dirichlet boundary conditions. The core method expresses both invariants through Green-function representations, then shows that boundary-induced corrections vanish in the large-domain limit via exponential decay of Green functions in spectral gaps. A detailed set of estimates—on Green functions, trace-class kernels, and interband transition functions—underpins the equivalence, with the edge observable interpreted physically as the circulation of electromagnetic energy along the boundary. The framework not only provides a rigorous foundation for finite-size topological photonic devices but also offers pathways to extensions to other elliptic operators and disordered systems, while highlighting the role of energy conservation in the BE mechanism.

Abstract

In this work, we establish the bulk-edge correspondence principle for finite two-dimensional photonic structures. Specifically, we focus on the divergence-form operator with periodic coefficients and prove the equality between the well-known gap Chern number (the bulk invariant) and an edge index defined via a trace formula for the operator restricted to a finite domain with Dirichlet boundary conditions. We demonstrate that the edge index characterizes the circulation of electromagnetic energy along the system's boundary, and the BEC principle is a consequence of energy conservation. The proof leverages Green function techniques and can be extended to other systems. These results provide a rigorous theoretical foundation for designing robust topological photonic devices with finite geometries, complementing recent advances in discrete models.
Paper Structure (27 sections, 27 theorems, 216 equations, 1 figure)

This paper contains 27 sections, 27 theorems, 216 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Main results
  4. Outline of the Proof of Theorem \ref{['thm_bec']}
  5. Relation to previous works and perspectives
  6. Preliminary
  7. Notations and conventions
  8. Almost-analytic extension and Hellfer-Sjöstrand formula
  9. Trace-class properties of singular integral operators
  10. $L^p$-estimate of second-order divergence-form operators with complex coefficients
  11. Estimate of Green functions
  12. Estimate of the Green function $G_{\sharp}(x,x^{\prime};z)$ and $G_{L}(x,x^{\prime};z)$
  13. Estimate of the quasi-periodic Green function $G_{\sharp}(x,x^{\prime};z,\kappa)$
  14. Representation of bulk index using Green function
  15. Band transition functions
  16. ...and 12 more sections

Key Result

Theorem 1.2

Assuming eq_GL_singularities_1-eq_G_sharp_singularities_2 on the singularities of Green functions, we have

Figures (1)

  • Figure 1: A finite sample consists of dielectric rods with PEC boundary.

Theorems & Definitions (44)

  • Definition 1.1: edge index
  • Theorem 1.2
  • Corollary 1.3
  • proof
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • proof : Proof of Theorem \ref{['prop_Chern_number_expression']} and \ref{['prop_edge_index_expression']}
  • Corollary 2.1
  • Proposition 2.2: reed1972methods
  • ...and 34 more