The bulk-edge correspondence for curved interfaces

The study of topological insulators is a central topic in condensed matter physics. These materials are insulating phases of matter, described by a Hamiltonian with a spectral gap, to which one can associate a quantized topological invariant: the Hall conductance. When two insulators with distinct topological invariants are glued together, protected gapless currents emerge along the interface: the material becomes a conductor. For straight interfaces, the interface conductance is also quantized and equals the difference of the bulk topological invariants. This fundamental result is called the bulk-edge correspondence.

The quantization of the bulk conductance (i.e. the fact that it takes values in a discrete set) was first observed in the quantum Hall effect [L81, TKNN, H82]. The characterization of Hall conductances as a Chern number marked the birth of topological phases of matter. Haldane [H88] demonstrated on a famous model of magnetic graphene that this phenomenon is not restricted to quantum Hall systems. Hatsugai [Hatsugai] then showed that topological materials support gapless states along their boundary; this is known today as the bulk-edge correspondence. Since then, the bulk-edge correspondence has been extended to various situations: see [KM05a, KM05b, FKM07] for $\mathbb{Z}_{2}$-topological insulators, [GT18, ST19] for Floquet topological systems, and [HR08, RH08, YGS15, DMV17, PDV19, F19] for physical fields beyond quantum science, such as photonics, accounstics and fluid mechanics. By now mathematical proofs of the bulk-edge correspondence have spanned a wide variety of situations: see for instance [Hatsugai, KS02, EG02, KS04, L15, Ku17, Br18, LT22, L23] for discrete models, [EGS05, LH11, GS18, ST19] for disordered systems, [GP13, ASV13, FSSWY20] for $\mathbb{Z}_{2}$-topological insulators, and [B19, D21b, D21, SW22a] for continuous Hamiltonians.

Various experimental results have suggested that the bulk-edge correspondence is an extremely stable principle: it does not depend on the fine details of the interface [WMTHXHW17, NHCR18, FSSWY20, TJCIBBP21]. However, with the exception of some $K$-theoretic approaches [Ku17, LT22, L23], most proofs of the bulk-edge correspondence have focused on straight interfaces. In this work, we provide a spectral proof of the bulk-edge correspondence for curved interfaces. We show equality between the edge conductance and an integer multiple of the Hall conductance; this goes beyond the aforementioned $K$-theoretic results where the bulk index takes the form of a geometric Hall conductance. We provide a physical interpretation of the emerging integer as an intersection number between the boundary and the measurement set. To the best of our knowledge, this is the first time that such a quantity emerges in the study of topological insulators.

We briefly review standard facts from condensed matter physics. Electronic propagation in a quantum material follows the Schrödinger equation with a selfadjoint operator $H$ on $\ell^{2}(\mathbb{Z}^{2},\mathbb{C}^{m})$. The operator $H$ describes the hopping of the electron between atomic sites. Its spectrum $\Sigma(H)$ characterizes the electronic nature of the material: $H$ is a conductor at energy $\lambda$ if and only if $\lambda\in\Sigma(H)$; and an insulator otherwise. In this paper, we will work with Hamiltonians that model limited hopping:

In (1.1), $d_{1}({\bm{x}},{\bm{y}})$ denotes the $\ell^{1}$-distance between ${\bm{x}},{\bm{y}}\in\mathbb{Z}^{2}$. When $H$ is an insulator at a certain energy, we can compute an invariant that characterizes the topological phase described by $H$: the Hall conductance. It is the intrinsic conductance of the material originating from the quantum Hall effect, see [TKNN].

The gap condition ${\mathcal{G}}\subset\Sigma(H)^{c}$ ensures that (1.2) is well-defined; see e.g. [EGS05] (or Proposition 1 below). This work aims to prove the bulk-edge correspondence: for edge systems interpolating between two insulators, the conductance of the edge is equal to the difference between the two Hall conductances. We turn to the definition of edge systems:

The condition (1.3) means that $H_{e}$ describes two insulators glued along the edge ${\partial}U$. To measure the conductance of $H_{e}$ along ${\partial}U$, we now discuss sets transverse to $U$.

In (1.4), $d_{1}({\bm{x}},{\partial}U)$ denotes the $\ell^{1}$-distance between ${\bm{x}}\in\mathbb{R}^{2}$ and the set ${\partial}U$. Transversality is a geometric condition on the relative position of the boundaries ${\partial}U$ and ${\partial}V$. It demands that these two sets get away from each other at a relatively mild rate, typically ${\Psi_{U,V}}({\bm{x}})\gtrsim|{\bm{x}}|^{\alpha}$ for some $\alpha>0$. Under the transversality condition (1.4), we can define the conductance of $H_{e}$ along ${\partial}U$ into the measurement set $V$:

In (1.5), $[H_{e},\mathds{1}_{V}]$ measures the number of particles moving into the set $V$ per unit time, while $\rho^{\prime}(H_{e})$ is an energy density in ${\mathcal{G}}$. As a result, $\sigma_{e}(H_{e})$ captures the expected charge moving along ${\partial}U$ into $V$ per unit time and per unit energy: it is the conductance of $H_{e}$ along ${\partial}U$. Transversality of $(U,V)$ guarantees that $\sigma_{e}(H_{e})$ is well defined; see Proposition 3 below. We also mention that $\sigma_{e}(H_{e})$ does not depend on $\rho$ satisfying (1.6) – this follows, for instance, from Theorem 1 below. Definitions 1 - 5 serve as the basis for our setup:

We are now ready to state our main result:

In rough terms, the integer $\mathcal{X}_{U,V}$ emerging in the formula (1.7), which we call the intersection number, counts algebraically how many times the boundary of $U$ (suitably oriented) enters $V$. See Figures 1 and 2 for a brief description on how to compute $\mathcal{X}_{U,V}$ and §5.1, §LABEL:sec-7.2 for detailed definitions.

Theorem 1 is a version of the bulk-edge correspondence that goes beyond half-planes: it applies to sets $(U,V)$ with arbitrary geometry. In particular, leveraging flexibility on $V$ improves our knowledge on edge currents. When ${\partial}U$ is disconnected, our result shows that each connected component of ${\partial}U$ supports a current with conductance $|\sigma_{b}(P_{+})-\sigma_{b}(P_{-})|$, propagating according to the orientation of ${\partial}U$; see Figures (2(a)) and (2(b)). Another application is when $U$ is a strip and $V$ is perpendicular to $U$: Theorem 1 implies that the edge conductance vanishes since the intersection number is $0$; see Figure (2(c)). Both these facts were heuristically known but rigorous proofs were missing.

Our proof of Theorem 1 consists of two independent components.

Part 1 (§2-§4). This part reduces the edge conductance $\sigma_{e}(H_{e})$ to a bulk quantity (i.e. that depends only on $H_{+}$, $H_{-}$). The key observation is that one can formally think of $\sigma_{e}(H_{e})$ as a singular trace:

We call it “singular” because the operator $[\rho(H_{e}),\mathds{1}_{V}]$ is not technically trace-class. However, if $W$ is a $R$-tubular neighborhood of ${\partial}U$ then $[\rho(H_{e})\mathds{1}_{W},\mathds{1}_{V}]$ is trace-class, and algebraic manipulations show that this trace vanishes. Therefore, as singular traces,

which is essentially a bulk term: for $R\gg 1$, $\rho(H_{e})\mathds{1}_{W^{c}}$ depends essentially on the bulk Hamiltonians $H_{+}$ and $H_{-}$.

At the heuristic level, our proof is based on the above informal observation but its mathematical execution is more sophisticated. We follow a strategy due to Elgart–Graf–Schenker [EGS05], tailored to the more complex geometry under focus here. To avoid trace-class issues, we inject the cutoff $\mathds{1}_{W}$ in the very first step: in the formula ${\operatorname{Tr}}(i\rho^{\prime}(H_{e})[H_{e},\mathds{1}_{V}])$ rather than in the ill-defined quantity ${\operatorname{Tr}}([\rho(H_{e}),\mathds{1}_{V}])$. Getting to the rigorous analogue of (1.9) produces commutators which we analyze using Helffer–Sjöstrand formulas and cyclicity. Edge terms vanish as predicted, and this eventually yields

see Theorem 2 below. We call the emerging qauntities $\sigma_{b}^{U,V}(P_{\pm})$ geometric bulk conductances.

Part 2 (§5-§LABEL:sec-7). In this part we reduce the geometric bulk conductances to integer multiples of Hall conductances (see Theorems 3 and LABEL:thm:2 below); we give an interpretation of the emerging integer as an intersection number between ${\partial}U$ and ${\partial}V$. The key observation is that $\sigma_{b}^{U,V}(P)$ is the trace of a commutator $[A,B]$ where $AB$ and $BA$ are not separately trace-class:

In particular, a compact perturbation $U^{\prime}$ of $U$ does not affect $\sigma_{b}^{U,V}(P)$, because

where the last identity comes from cyclicity, see Proposition 2 below.

When $U$ and $V$ are simple sets (i.e. their boundaries are connected), our strategy consists of using the robustness of the geometric bulk conductance to locally deform $U$ and $V$ to half-planes and recover the Hall conductance. In details, we construct sets $U_{n}$, $V_{n}$, that (up to permutation of $U$ and $V$) resemble $\{x_{2}>0\}$, $\{x_{1}>0\}$ in $\mathbb{D}_{2n}(0)$ (the disk of radius $n$, centered at the origin) and $U$,$V$ outside $\mathbb{D}_{4n}(0)$. Since $U_{n}$, $V_{n}$ resemble $U$, $V$ outside $\mathbb{D}_{4n}(0)$, the robustness of geometric bulk conductances (Proposition 2) guarantees that

On the other hand, $\mathds{1}_{U_{n}}({\bm{x}})$, $\mathds{1}_{V_{n}}({\bm{x}})$ approach $\mathds{1}_{\{x_{2}>0\}}({\bm{x}})$, $\mathds{1}_{\{x_{1}>0\}}({\bm{x}})$ when $n\to\infty$ for every ${\bm{x}}$. So formally,

The main challenge is to make (1.14) rigorous. We will rely on an application of the dominated convergence theorem on specifically designed deformations of $(U,V)$. Because (1.11) is a trace, the involved kernel is negligible outside large enough balls. Our construction of $U_{n},V_{n}$ preserve this negligibility uniformly, and this yield a rigorous proof of (1.14) for simple sets $U,V$; see Proposition 5.

In §LABEL:sec-7 we will use additivity properties of the bulk conductances to extend (1.14) to arbitrary transverse sets $U,V$, driving the emergence of the intersection number between $U$ and $V$:

See Theorem LABEL:thm:2. Theorem 1 will follow from combining (1.10) and (1.15).

The paper is organized as follows:

• •

  In §2, we provide basic estimates on short-range operators and transverse sets.

• •

  In §3, we introduce of bulk and edge conductance and detail their basic properties.

• •

  In §4, we reduce the edge conductance to the difference between geometric bulk conductances (Theorem 2).

• •

  In §5, we introduce the intersection number for simple transverse sets (transverse sets whose boundaries are connected).

• •

  In §6, we prove Theorem 3: for simple transverse sets, the geometric bulk conductance is equal to the intersection number times the Hall conductance.

• •

  In §LABEL:sec-7, we extend Theorem 3 to any pair of transverse sets. This leads to the emergence of the intersection number $\mathcal{X}_{U,V}$ and completes the proof of Theorem 1.

The bulk-edge correspondence has been extensively studied in the past thirty years. Various perspectives have been developed around it: $K$-theoretic approaches [KS02, KS04, T14, PS16, BKR, Ku17, Br18]; spectral flow methods [Hatsugai, Br18]; and tracial techniques [EG02, EGS05, GP13, GS18, GT18, ST19, B19, D21].

While many papers have focused on extending the bulk-edge correspondence to increasing degree of generality on the model – from Landau Hamiltonian to $\mathbb{Z}_{2}$-insulators, from discrete to continuous, from periodic to disordered – very few go beyond straight edges. The work [BBDKLW23] gave an explicit formula for edge states of an adiabatically modulated Dirac operator with a weakly curved interface. The result relies on semiclassical methods and gives precise results but is restricted to the adiabatic regime; see also [D22, B22, PPY22]. Our previous work [DZ23] shows the emergence of edge spectrum for topological insulators as long as both the domain and its complement contain arbitrarily large balls, but did not provide a formula for the conductance. Thiang [T20] derived similar results using $K$-theory and coarse geometry.

Later in [LT22, L23], Ludewig and Thiang proved a K-theoretic version of the bulk-edge correspondence on flasque spaces. These are spaces that satisfy coarse-geometric properties; in contrast with our condition (1.4), flasqueness [R96, Definition 9.3] can be challenging to check on concrete examples beyond perturbations of half-spaces. Their K-theoretic bulk-edge correspondence result [L23, Theorem V.4] and edge-traveling interpretation [L23, Theorem VII.7] together connect the edge conductance to the geometric bulk conductance (1.10). Our approach goes beyond [LT22, L23] by connecting the geometric conductances themselves to a topological marker independent of the geometry: the Hall conductance (1.2). This drove the emergence of the intersection number $\mathcal{X}_{U,V}$.

One natural question is whether our curved bulk-edge correspondence (1.7) holds for $\mathbb{Z}_{2}$-topological insulators. This would require a space-based trace formula for $\mathbb{Z}_{2}$-bulk indices, such as (1.2) for Hall insulators. At this point it is unclear whether such a formula is available or even possible; see for instance discussions in [LM19, FSSWY20].

Another interesting problem concerns the type of the edge spectrum; it is widely expected to include an absolutely continuous part. For instance, [BW22] proved that the edge spectrum is filled with absolutely continuous spectrum when the edge is straight, combining a form of the bulk-edge correspondence derived in [FSSWY20] with a result on the structure of unitary operators [MM21]. The present work provides the first step in the context of curved edges. In a follow-up project we will show that the emerging edge spectrum is absolutely continuous, extending the result of [BW22].

In [CB23], Chen and Bal showed that for a straight edge, the bulk index is equal to the difference between the number of modes reflected and transmitted. It would be very interesting to extend this result to curved edges. Yet this promises to be challenging, because the scattering techniques developed there rely on translational invariance.

Several authors [HM15, GMP17, BBDF18] have shown that the Hall conductance of interacting systems of particles is quantized (i.e. takes values in a discrete set). In our proof a geometric bulk conductance naturally emerges, see (1.10); it would be interesting to investigate whether the analogue of this quantity in interacting systems is quantized as well, and given by the integer $\mathcal{X}_{U,V}$ times the Hall conductance.

We gratefully acknowledge support from the National Science Foundation DMS 2054589 (AD) and the Pacific Institute for the Mathematical Sciences (XZ). The contents of this work are solely the responsibility of the authors and do not necessarily represent the official views of PIMS.

We use the following notations:

• •

  Regular font $x$, $x_{j}$, $z$ denotes scalars in $\mathbb{R}$, $\mathbb{Z}$, or $\mathbb{C}$.

• •

  Bold font ${\bm{x}}=(x_{1},x_{2}),{\bm{y}}=(y_{1},y_{2})$ denotes points in $\mathbb{Z}^{2}$.

• •

  $d(\cdot,\cdot)$ denote the Euclidean metric.

• •

  $d_{1}({\bm{x}},{\bm{y}}):=|x_{1}-y_{1}|+|x_{2}-y_{2}|$ denote the $\ell^{1}$-metric on $\mathbb{R}^{2}$.

• •

  $d_{l}({\bm{x}},{\bm{y}})=\ln(1+d_{1}({\bm{x}},{\bm{y}}))$ denote the logarithm metric on $\mathbb{R}^{2}$.

• •

  $\|\cdot\|$ and $\|\cdot\|_{1}$ denote the operator norm and the trace-class norm, respectively.

• •

  Throughout the paper, $C$ denotes a constant that is allowed to vary from line to line, that may depend on $\nu$ and $\rho$ (though this dependence is not emphasized). We will occasionally use $C_{N}$ to emphasize dependence on an integer $N$.

• •

  For $A\subset\mathbb{R}^{2}$, $A^{\mathcal{C}}$ denotes the open set $\operatorname{int}(A^{c})$.

• •

  $\mathbb{D}_{R}({\bm{z}}):=\{{\bm{x}}\in\mathbb{R}^{2}:d_{1}({\bm{x}},{\bm{z}})<R\}$ denotes the $\ell^{1}$-disk centered at $z$, of radius $R$.

• •

  ${\mathbb{H}}_{1}=\{x_{1}>0\}$ and ${\mathbb{H}}_{2}=\{x_{2}>0\}$.

• •

  If $K$ is an operator on $\ell^{2}(\mathbb{Z}^{2})$, $K({\bm{x}},{\bm{y}})$ denotes the Schwartz kernel of $K$.

Under short-range assumptions, we give here estimates on resolvents and product of operators. We start by introducing the notion of textitpolynomially-short-range (PSR) Hamiltonians.

Note that any ESR operator is also PSR. The next result shows that (1) resolvents of ESR for insulating energies are ESR; (2) functionals of ESR operators are PSR.

We now give estimates on products of ESR / PSR operators. We state them in a unifying way using the two metrics $d_{1},d_{\ell}$.

Note that $M_{d_{*},a}$ is independent of ${\bm{y}}\in\mathbb{Z}^{2}$ and

Recall that we say $U,V\subset\mathbb{R}^{2}$ if

Note that it is equivalent to

Meanwhile, because of the inequality

we have:

Therefore for any $N>\frac{2}{c}$, by (2.8)

The next result states that product of PSR operators that decay away from the boundaries of transverse sets $U,V$ are trace-class.

Because the right and upper half-planes are transverse sets, the Hall conductance $\sigma_{b}(H):=-i{\operatorname{Tr}}\big{(}P\big{[}[P,\mathds{1}_{\{x_{2}>0\}}],[P,\mathds{1}_{\{x_{1}>0\}}]\big{]}\big{)}$ is well-defined.

In particular, we have by (2.10) and (3.4)

Given two sets $A_{1},A_{2}$, we introduce the symmetric difference

One of the key properties of geometric bulk conductances is:

Now we can introduce the edge conductance.

For the rest of the paper, we consider an arbitrary but fixed pair of transverse sets $(U,V)$ in $\mathbb{R}^{2}$ and omit the superscripts $U,V$ from $\sigma_{e}^{U,V}(H_{e})$ for convenience.