Analytic bulk-edge connection in circular-symmetric models

TL;DR

This paper introduces an analytic method to connect bulk and edge eigenfunctions in two-dimensional circular-symmetric models, revealing phase relations and spectral transitions, and comparing with topological bulk-edge correspondence.

Contribution

It develops a systematic analytic continuation approach to map bulk modes to edge modes, providing detailed phase relation analysis and spectral transition insights.

Findings

Bulk and edge modes are spectrally separated.

Transitions occur from delocalized bulk to localized edge modes.

Phase relations change during spectral transitions.

Abstract

We propose a systematic analysis of the eigenfunctions of two-band systems in two dimensions with a circular edge. Our approach is based on an analytic continuation of the wavenumber, which yields a mapping from the bulk modes to the edge modes. Phase relations of the eigenfunctions are described by their mapping onto a three-dimensional field of unit vectors. This mapping is studied in detail for a two-band Laplacian model and a Dirac model. The direction of the unit vector identifies the phase relation of the eigenfunctions and enables us to distinguish between the upper band, the lower band and the edge spectrum. Bulk and edge modes are spectrally separated, which results in two transitions from delocalized bulk modes to localized edge modes. These transitions are accompanied by transitions of the phase relations. Our analytic approach is compared with the topological bulk-edge correspondence, which is based on the Chern number of the bulk.