A bulk--edge correspondence for two-dimensional self-adjoint boundary and interface problems
Abstract
This paper analyzes the quantized asymmetric transport along the boundary of a two-dimensional topological insulator occupying a half plane, or along the interface separating two insulators. In contrast with problems posed on the Euclidean plane, terminating a sample requires a choice of self-adjointrealization, HL, and we show that a quantized transport depends on that choice. The physical observable modeling the asymmetric transport is the edge, or interface, current σI[HL]=Tr i[HL,P]φ'(HL). It is shown to be well defined along two routes. For Shapiro--Lopatinski elliptic realizations, we use estimates from Boutet de Monvel's calculus to show that the current observable is a well-defined trace, that ℐ=2πσI[HL] is a Fredholm index, and that it is stable under deformations and compactly supported changes of the operator and of its termination. For translation invariant operators along the edge, a spectral analysis defines ℐ when the edge spectrum escapes a bulk gap at large momenta. ℐ is then also a spectral flow and a Maslov index. Our main theorem is a bulk--edge correspondence ℐ=cR−cRL+η(L∞,LmΦ(+))−η(L∞,LmΦ(−)), We determine the range of ℐ. We extend the formula to interfaces with arbitrary transmission conditions. At the transparent condition, the invariant equals the bulk-difference invariant of the Euclidean plane. Eight examples illustrate the theoretical findings.
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BibTeX
@misc{bal2026bulkedge,
title = {{A bulk--edge correspondence for two-dimensional self-adjoint boundary and interface problems}},
author = {Guillaume Bal},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05616},
eprint = {2610.05616},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05616},
}
RIS
TY - UNPB
TI - A bulk--edge correspondence for two-dimensional self-adjoint boundary and interface problems
AU - Guillaume Bal
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05616
UR - https://arxiv.org/abs/2610.05616
AB - This paper analyzes the quantized asymmetric transport along the boundary of a two-dimensional topological insulator occupying a half plane, or along the interface separating two insulators. In contrast with problems posed on the Euclidean plane, terminating a sample requires a choice of self-adjointrealization, H_L, and we show that a quantized transport depends on that choice. The physical observable modeling the asymmetric transport is the edge, or interface, current σ_I[H_L]=Tr i[H_L,P]φ'(H_L). It is shown to be well defined along two routes. For Shapiro--Lopatinski elliptic realizations, we use estimates from Boutet de Monvel's calculus to show that the current observable is a well-defined trace, that ℐ=2πσ_I[H_L] is a Fredholm index, and that it is stable under deformations and compactly supported changes of the operator and of its termination. For translation invariant operators along the edge, a spectral analysis defines ℐ when the edge spectrum escapes a bulk gap at large momenta. ℐ is then also a spectral flow and a Maslov index. Our main theorem is a bulk--edge correspondence ℐ=cR−cR_L+η(L_∞,Lm^Φ(+))−η(L_∞,Lm^Φ(−)), We determine the range of ℐ. We extend the formula to interfaces with arbitrary transmission conditions. At the transparent condition, the invariant equals the bulk-difference invariant of the Euclidean plane. Eight examples illustrate the theoretical findings.
ER -