A bulk--edge correspondence for two-dimensional self-adjoint boundary and interface problems

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 ${\sigma }_I[H_L]=\text{Tr}i[H_L,P]{\phi }^{\mathrm{\prime }}(H_L)$. 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 $\mathcal{I}=2\pi {\sigma }_I[H_L]$ 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 $\mathcal{I}$ when the edge spectrum escapes a bulk gap at large momenta. $\mathcal{I}$ is then also a spectral flow and a Maslov index. Our main theorem is a bulk–edge correspondence $$\mathcal{I}=\text{cR}-{\text{cR}}_L+\eta (L_{\mathrm{\infty }},{\text{Lm}}^{\mathrm{\Phi }}(+))-\eta (L_{\mathrm{\infty }},{\text{Lm}}^{\mathrm{\Phi }}(-)),$$ We determine the range of $\mathcal{I}$. 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.