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Microlocal Analysis of the Bulk-Edge Correspondence

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Abstract

The bulk-edge correspondence predicts that interfaces between topological insulators support robust currents. We prove this principle for PDEs that are periodic away from an interface. Our approach relies on semiclassical methods. It suggests novel perspectives for the analysis of topologically protected transport.

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Notes

  1. For instance, for magnetic Schrödinger operators (1.2), the Bethe–Sommerfeld conjecture holds [M97, K04, PS10]: both \(P_+\) and \(P_-\) have finitely many gaps. Thus, if \(\lambda _0\) is in the last gap of \(P_+\) or \(P_-\), then there are no higher-energy spectral gaps.

  2. These are however not concentrated in position. This seems to be a feature of the homogenization—rather than semiclassical—scaling.

  3. In [D19a, DW19], we defined the edge index as a spectral flow. Modulo a factor \(2\pi \), it equals (1.5)—see e.g. [ASV13, Proposition 3].

  4. Note that from (a), \(a_\alpha (x) = {\overline{a_\alpha (x)}}\) thus the ellipticity condition is equivalent to the more standard one \(\sum _{|\alpha |=2} a_\alpha (x) \xi ^\alpha \ge c |\xi |^2\).

  5. The power 6 is specific to the dimension \(n=2\); in general it is \(2n+2\).

  6. The number 5 is specific to \(n=2\); in general it is \(2n+1\).

  7. This space is denoted \(L_0\) in [GMS91] and [DS99, §13]. It is canonically identified with \(L^2({\mathbb {R}}^2)\), see Sect. 3.1.4.

  8. This reduces to \(\frac{h}{i}\{a,b\}\) when a or b is scalar-valued; however most operators considered below will be matrix and operator-valued.

  9. The numbers 2 and 5 are specific to dimension \(n=2\); in general they are n and \(2n+1\), respectively.

  10. Strictly speaking, [D93, Remark 1.3a] and [DS99, Lemma 13.29] are stated for symbols in \(S^{(22)}(1)\) that are compactly supported in x; the proof applies (with no change) to rapidly decaying symbols.

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Acknowledgements

I am grateful to F. Faure, J. Shapiro, M. I. Weinstein and M. Zworski for valuable discussions; and to the anonymous referee for their careful reading and constructive comments. I thankfully acknowledge support from NSF DMS-1440140 (MSRI, Fall 2019) and DMS-1800086, and from the Simons Foundation through M. I. Weinstein’s Math+X investigator award #376319.

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Appendix A

Appendix A

Lemma A.1

For any \(\nu \in {\mathbb {Z}}\), there exists \(a(\xi ) \in C^\infty ({\mathbb {R}}^2,{\mathbb {C}}^2)\) such that the line \({\mathbb {C}}a(\xi )\) is \((2\pi {\mathbb {Z}})^2\)-periodic in \(\xi \); and the vector bundle \({\mathbb {C}}a \rightarrow {{\mathbb {T}}}^2_*\) has Chern number \(-\nu \).

Proof

1. Fix \(\varepsilon > 0\), \(\alpha (\xi _1), \beta (\xi _1) \in C^\infty ({\mathbb {R}},{\mathbb {R}})\), both \(2\pi \)-periodic, with

$$\begin{aligned} \xi _1 \in [-1,1] \ \ \Rightarrow \ \ \alpha (\xi _1) = \xi _1 , \ \ \beta (\xi _1) = 0; \ \ \xi _1 \in [-\pi ,\pi ] \setminus [-1,1] \ \ \Rightarrow \beta (\xi _1) > 0. \end{aligned}$$

Let \(M_\varepsilon (\xi ) \in C^\infty \big ({\mathbb {R}}^2,M_2({\mathbb {C}})\big )\) be given by

$$\begin{aligned} M_\varepsilon (\xi ) \mathrel {{\mathop {=}\limits ^{{\mathrm{def}}}}}\left[ \begin{matrix} \alpha (\xi _1) &{} \beta (\xi _1) + \varepsilon e^{-i\nu \xi _2} \\ \beta (\xi _1) + \varepsilon e^{i\nu \xi _2} &{} -\alpha (\xi _1) \end{matrix} \right] . \end{aligned}$$

For any \(\xi \in {\mathbb {R}}^2\), \(M_\varepsilon (\xi )\) has a unique negative eigenvalue. Since \({\mathbb {R}}^2\) is contractible [M01, §1], \(M_\varepsilon (\xi )\) admits a normalized negative-energy eigenvector \(a_\varepsilon (\xi ) \in C^\infty ({\mathbb {R}}^2,{\mathbb {C}}^2)\). Since \(M_\varepsilon (\xi )\) is \((2\pi {\mathbb {Z}})^2\)-periodic, the eigenspace \({\mathbb {C}}a_\varepsilon (\xi )\) is \((2\pi {\mathbb {Z}})^2\)-periodic. Thus it induces a vector bundle \({\mathbb {C}}a_\varepsilon \rightarrow {{\mathbb {T}}}^2_*\).

2. The eigenprojector of M associated to the negative eigenvalue is

$$\begin{aligned} \pi _\varepsilon = {{\text {Id}}}- \dfrac{M_\varepsilon }{\sqrt{-\det M_\varepsilon }}. \end{aligned}$$

Thus the Berry curvature of \({\mathbb {C}}a_\varepsilon \rightarrow {{\mathbb {T}}}^2_*\) is

$$\begin{aligned} B_\varepsilon (\xi ) \mathrel {{\mathop {=}\limits ^{{\mathrm{def}}}}}-{{\text {Tr}}}_{{\mathbb {C}}^2} \left( \dfrac{M_\varepsilon }{\sqrt{-\det M_\varepsilon }} \left[ \dfrac{\partial }{\partial \xi _1} \dfrac{M_\varepsilon }{\sqrt{-\det M_\varepsilon }}, \dfrac{\partial }{\partial \xi _2} \dfrac{M_\varepsilon }{\sqrt{-\det M_\varepsilon }} \right] \right) . \end{aligned}$$

We observe that as \(\varepsilon \rightarrow 0\), the convergences

$$\begin{aligned} -\det M_\varepsilon = \alpha (\xi _1)^2 + |\beta (\xi _1)+\varepsilon e^{i\nu \xi _2}|^2 \rightarrow \alpha (\xi _1)^2+\beta (\xi _1)^2; \ \ \ \ \text {and} \ \ {\partial }_2 M_\varepsilon \rightarrow 0 \end{aligned}$$

are uniform. Moreover, for \(\xi _1 \in [-1,1]\), \(\alpha (\xi _1)^2+\beta (\xi _1)^2\) is bounded below by a positive constant. We deduce that \(B_\varepsilon (\xi ) \rightarrow 0\) uniformly away from \([-1,1] \times [-\pi ,\pi ]\).

3. When \(\xi _1 \in [-1,1]\), we have

$$\begin{aligned} M_\varepsilon (\xi ) \mathrel {{\mathop {=}\limits ^{{\mathrm{def}}}}}\left[ \begin{matrix} \xi _1 &{} \varepsilon e^{-i\nu \xi _2} \\ \varepsilon e^{i\nu \xi _2} &{} -\xi _1 \end{matrix} \right] . \end{aligned}$$

A direct calculation, see e.g. [D18, Lemma 6.3] shows that

$$\begin{aligned} B_\varepsilon (\xi _1) = \dfrac{i\varepsilon ^2\nu }{2(\xi _1^2 + \varepsilon ^2)^{3/2}}, \ \ \ \ \dfrac{i}{2\pi }\int _{[-1,1] \times [-\pi ,\pi ]} B_\varepsilon (\xi ) d\xi \rightarrow -\nu . \end{aligned}$$

It follows that if \(\varepsilon \) is sufficiently small, then the Chern number of \({\mathbb {C}}a_\varepsilon \rightarrow {{\mathbb {T}}}^2_*\) is \(-\nu \), as claimed. This completes the proof. \(\quad \square \)

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Drouot, A. Microlocal Analysis of the Bulk-Edge Correspondence. Commun. Math. Phys. 383, 2069–2112 (2021). https://doi.org/10.1007/s00220-020-03864-4

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