Dynamical role of eigenvalues for a large circular domain wall

Determine how the eigenvalues of the two-dimensional Dirac operator with circular domain wall mass profile \(\kappa=1-2\mathbf{1}_{B_R(0)}\) in \((-1,1)\) contribute to wave dynamics as the radius \(R\) becomes large.

Background

For the circular domain wall κ=1−21BR(0)\kappa=1-2\mathbf{1}_{B_R(0)}, the spectrum in the bulk gap (−1,1)(-1,1) is discrete for each finite RR, consisting solely of eigenvalues. Nevertheless, numerical simulations in the paper display effective transport along the circular interface.

The unresolved issue is to characterize the mechanism by which a discrete family of eigenstates produces transport, and to relate the eigenvalue dynamics for large but finite RR to the edge-state transport established for asymptotically straight interfaces.

References

Therefore, two questions have emerged: \item What is the distribution of eigenvalues of D in (-1,1)? \item How do eigenvalues of D in (-1,1) contribute to dynamics? We will respond to these questions in a future work.

— Reflectionless edge states in Dirac models of topological insulators  (2609.16949 - Drouot, 15 Sep 2026) in Section 1, subsection “Further investigations”