This paper proposes a classification of elliptic (pseudo-)differential Hamiltonians describing topological insulators and superconductors in Euclidean space by means of domain walls. Augmenting a given Hamiltonian by one or several domain walls results in confinement that naturally yields a Fredholm operator, whose index is taken as the topological charge of the system. The index is computed explicitly in terms of the symbol of the Hamiltonian by a Fedosov–Hörmander formula, which implements in Euclidean spaces an Atiyah–Singer index theorem. For Hamiltonians admitting an appropriate decomposition in a Clifford algebra, the index is given by the easily computable topological degree of a naturally associated map. A practically important property of topological insulators is the asymmetric transport observed along one-dimensional lines generated by the domain walls. This asymmetry is captured by the edge conductivity, a physical observable of the system. We prove that the edge conductivity is quantized and given by the index of a second Fredholm operator of the Toeplitz type. We also prove topological charge conservation by stating that the two aforementioned indices agree. This result generalizes to higher dimensions and higher-order topological insulators, the bulk-edge correspondence of two-dimensional materials. We apply this procedure to evaluate the topological charge of several classical examples of (standard and higher-order) topological insulators and superconductors in one, two, and three spatial dimensions.

## I. INTRODUCTION

This paper considers topological insulators modeled by single particle Hamiltonians in the form of (pseudo-)differential operators (PDOs). See, e.g., Refs. 11, 43, 47, 48, 51, 56, and 57 for background and details on these materials and topological phases of matter. The partial differential systems we consider here naturally appear in a variety of contexts, which include heuristic descriptions of topological insulators and superconductors,^{11,56,57} macroscopic approximations of more accurate microscopic models such as, for instance, tight-binding^{11,43} or modulated periodic Schrödinger equations,^{33,34} as well as models of atmospheric fluid flows.^{24,54}

Two-dimensional topological insulators enjoy the following striking property. When two insulators in different topological phases are brought together, the interface separating them becomes *conducting*, i.e., no longer insulating. Moreover, transport along the interface is *asymmetric* with an asymmetry quantized, and stable against perturbations. This guaranteed asymmetric transport in the presence of impurities is one of the main practical interests of such materials. It is also surprising since the non-trivial topology may be interpreted as an obstruction^{4,47} to Anderson localization, which states that transport is exponentially suppressed in the presence of random fluctuations.

Two main mechanisms of confinement lead to insulators. The first one is the magnetic confinement found in the integer quantum Hall effect, the first example of a topological insulator.^{1,2,10,11} Asymmetric transport along interfaces separating different magnetic confinements is analyzed for Schrödinger and Dirac equations in, e.g., Refs. 26 and 50. In such models, the magnetic confinement leads to *flat bands* of the essential spectrum (the Landau levels). The classification and many of the mathematical results we present below *do not* apply (potentially with false predictions) in the presence of such flat bands.

We are interested here in the second mechanism of confinement based on what we will refer to as *mass* terms. Mass terms typically take the form of one or more scalar functions. In two space dimensions, a scalar function *m*(*x*) may also act as a *transition* between the two insulating materials: the domains *m* > 0 and *m* < 0 are insulating, while the (vicinity of the) interface described as the 0-level set *m*^{−1}(0) and separating the two bulk phases is conducting. We refer to such transition functions *m*(*x*) as *domain walls*.

It remains to define and compute topological invariants that characterize such an asymmetry. While such computations may be carried out explicitly for some models, typically by means of spectral flows,^{3,6,26,41,50} they remain notoriously difficult for more complex systems. A general principle called a *bulk-edge correspondence* relates the transport asymmetry along the interface to invariants associated with the bulk properties of the insulator in the regions ±*m*(*x*) > 0. The computation of such *bulk invariants*, typically by means of Chern numbers and winding numbers, is often much easier. Bulk-edge correspondences have been derived mathematically in a number of settings for discrete^{29,35,38,47,53} as well as continuous Hamiltonians;^{6,19,20,27,44,49} see also Refs. 30 and 56 for a bulk-boundary correspondence and a notion of topological charge conservation similar to the one we will describe in this paper. Directly related to this work is the correspondence established in Refs. 6 and 49, where the asymmetric transport is related to a *bulk-difference invariant* evaluated as a simple integral [the Fedosov–Hörmander formula in (10) below]. We remark that bulk phases for domains with *m* > 0 or with *m* < 0 constant may not be defined for natural differential operators such as the Dirac operator.^{3,6} In such settings, one may define a bulk-difference invariant^{6} that combines the properties of both domains *m* > 0 and *m* < 0. Heuristically, this indicates that it is easier to define phase transitions rather than absolute (bulk) phases.

This paper aims to generalize the above correspondence to arbitrary dimensions and for a large class of *elliptic* pseudo-differential Hamiltonians. The topological classification of each Hamiltonian is performed in two different ways. We start from a given Hamiltonian in *d* dimensions denoted by *H*_{k}, where 0 ≤ *k* ≤ *d* − 1 refers to the number of *confined* dimensions. Therefore, *H*_{0} with *k* = 0 may be a bulk *metallic* system (a gapless Hamiltonian such as a Dirac model for transport in graphene). In dimension *d* = 2, *k* = 1 may correspond to a Hamiltonian describing a topological transition between two insulators modeled by *m*(*x*_{1}) > 0 and *m*(*x*_{1}) < 0, respectively. We wish to *test* the topological properties of *H*_{k} by assessing its response to the addition of *domain walls*. When *k* ≤ *d* − 2, we first construct a Hamiltonian *H*_{d−1} by appropriately adjoining *d* − *k* − 1 domain walls to it. The resulting Hamiltonian *H*_{d−1} models a system that is confined in all but one dimension, say *x*_{d}. We, therefore, expect asymmetric transport to occur along the corresponding *d*th dimension. We will introduce an edge conductivity *σ*_{I}(*H*_{d−1}) in (11) in Sec. III; this is a physical observable of the system associated with the asymmetric transport. We will show that 2*πσ*_{I} is equal to the index of a Fredholm operator *T*, which is a Toeplitz operator. Such Fredholm operators and the structures of Fredholm modules (or spectral triples) naturally appear in topological classifications.^{6,19,20,27,44,47,49} As in the two-dimensional setting, the computation of such an index is difficult in practice.

Following Refs. 6 and 49, we, therefore, test the topology of *H*_{k} by implementing a final domain wall *m*(*x*_{d}) in the *d*th dimension and introduce an operator *F* = *H*_{d−1} − *im*(*x*_{d}). We will show that *F* is a (non-Hermitian) Fredholm operator. The *topological charge* of *H*_{k} is then *defined* as the Index *F* = dim Ker *F* − dim Ker *F**. The main advantage of this classification is its computational simplicity: the index of *F* is given by an explicit integral of its symbol, the Fedosov–Hörmander formula (10) below. Moreover, the mapping from *H*_{k} to *F* is *local* in the sense that *F* = *γ*_{1} ⊗ *H*_{k} + *γ*_{2} for *γ*_{1} a constant matrix and *γ*_{2} an explicit multiplication operator in the physical variables. The evaluation of (10) is thus directly related to the symbol of a given Hamiltonian *H*_{k} of interest. For technical reasons that will allow us to apply the pseudo-differential calculus and index theory presented in Ref. 40, Chap. 19, we *assume* here that all (smooth) domain walls are unbounded. Since transport is exponentially suppressed away from the 0-level sets of domain walls, this is a reasonable assumption in many practical settings. See Refs. 6 and 49 for a theory in two dimensions involving bounded domain walls.

The above constructions describe two topological charges associated with *H*_{k} and given by the indices of the Fredholm operators *F* and *T*. As a generalization of the bulk-edge correspondence, the main result of this paper is that the topological charge is *conserved* through a series of deformations leading to an equality of the two indices. This shows that the physically relevant asymmetric transport characterized by the edge conductivity and the Fredholm operator *T* may be estimated by the much simpler integral (10) associated with *F*.

As a final theoretical result, we show that for specific Hamiltonians, the integral (10) admits an explicit expression given as the topological degree of a natural map associated with *H*_{k}. Such degrees, which naturally take integral values, are often quite easy to compute in problems of interest. We will see that the construction of *H*_{d−1} and *F* from *H*_{k} in arbitrary dimensions requires a structure of Clifford algebras in order to define appropriately *orthogonal* domain walls. When *H*_{k} itself admits a natural decomposition in a Clifford algebra, then we will associate with *H*_{k} a map h^{k} whose topological degree is straightforward to compute for several practical Hamiltonians.

The classification based on domain walls may be compared with other classification mechanisms. Topological phases of matter are heuristically characterized by non-trivial topologies of Hamiltonians in dual, Fourier, variables.^{11,47,56} This non-trivial topology may be tested in several ways. Linear response theory in two dimensions tests a domain by applying, e.g., a linear electric field in one direction and assessing the resulting transport in the transverse direction leading to the notion of Hall conductivity.^{2,11,36} While physically different, adding to a metallic operator *H*_{0} in two dimensions a domain wall in a given direction and testing asymmetric transport in the transverse direction play a similar classifying role. An advantage of the classification based on domain walls is its natural generalization to arbitrary dimensions and the explicit Fedosov–Hörmander formula satisfied by the index. Note that the latter is also used in a different context (and with a different algebra) of operators in Euclidean space by Refs. 18 and 21 to test the topology of a physical potential with appropriate behavior at infinity in the physical variables using a Dirac operator.

The rest of the paper is structured as follows: the definition and construction of the operators *H*_{k}, *H*_{d−1}, *F*, and *T* require an unavoidable amount of pseudo-differential and functional calculus. The conservation of topological charge, as in the index theory developed in Ref. 40, Chap. 19, requires us to consider semiclassical transformations and an associated semiclassical calculus. The notation and required results, which may be found in more detail in Refs. 15, 25, and 40, are collected in the Appendix.

The functional setting and construction of the operator *H*_{d−1} and *F* using confinement by domain walls are presented in Sec. II. The main result of this section is the index formula in Ref. 40, Chap. 19 recalled in Theorem 2.3. The definition of the edge conductivity *σ*_{I} associated with *H*_{d−1} is given in (11) in Sec. III. We define a corresponding Fredholm operator in Toeplitz form and show in Theorem 3.2 that 2*πσ*_{I} = Index *T*. Several lemmas in that section also show the stability of 2*πσ*_{I} = Index *T* against a number of continuous deformations of the Hamiltonian *H*_{k}.

The conservation of topological charge Index *F* = Index *T*, the main theoretical result of this paper, is proved in Theorem 4.1 of Sec. IV. Finally, the computation of the integral (10) by means of the topological degree of the map h^{k} associated with Hamiltonians *H*_{k} in appropriate Clifford algebra form is carried out in Theorem 5.1 in Sec. V.

Section VI is devoted to several applications of the theoretical results. It details the computation of the topological charge of generalizations of Dirac systems of equations in Clifford algebra form, possibly including magnetic contributions, as well as some Hamiltonians not in Clifford algebra form. An application to higher-order insulators in three spatial dimensions shows, in particular, how two domain walls may be defined to construct a coaxial cable with an arbitrary number of topologically protected propagating modes. These models find applications in, e.g., graphene- and bilayer graphene-based topological insulators, topological superconductors, and topological atmospheric flows.

## II. FREDHOLM OPERATOR AND TOPOLOGICAL CHARGE

This section introduces classes of elliptic pseudo-differential Hamiltonians, classifies them by means of domain wall confinements, and defines their topological charge as the index of a Fredholm operator naturally associated with each Hamiltonian. The advantage of this classification is the explicit integral (10) that computes the topological charge. This Fedosov–Hörmander formula, presented in Theorem 2.3 below, implements in Euclidean geometry an Atiyah–Singer theory.^{12}

### A. Classification of Dirac operators in low dimensions

To illustrate how the topology of Hamiltonians is tested by domain walls, we present the constructions for Dirac operators, which are generic models for band crossings^{11,27,32–34,51} and arguably the simplest models for topological phases of matter.

Consider first a one-dimensional material and asymmetric transport modeled by the Hamiltonian *H*_{0} = *D*_{x} with *D*_{x} = −*i∂*_{x}, which may be seen as an unbounded self-adjoint operator on $L2(R)$ with domain $D(H0)=H1(R)$. This operator admits purely absolutely continuous spectrum $\sigma (H0)=\sigma ac(H0)=R$ and displays asymmetric transport along the *x* axis in the sense that solutions of (*D*_{t} + *D*_{x})*u* = 0 with initial condition *u*(0, *x*) = *u*_{0}(*t*) are given by *u*(*t*, *x*) = *u*_{0}(*x* − *t*).

To generate *confinement* in the vicinity of *x* = 0, we introduce the domain wall *m*(*x*) = *x* and the operator $F=Dx\u2212ix=\u2212ia$ with $a=\u2202x+x$ an annihilation operator. The operator *F* is now a Fredholm operator from its domain of definition $D(F)={f\u2208L2(R);f\u2032\u2208L2(R)\u2009and\u2009xf\u2208L2(R)}$ to $L2(R)$. Moreover, we verify that Index *F* = dim Ker *F* − dim Ker *F** = 1 with kernel of *F* spanned by the eigenfunction $e\u221212x2$. We *define* Index *F* as the *topological charge* of *H*_{0}.

Consider next the two-dimensional version of the above example, where *H*_{0} = *D*_{1}*σ*_{1} + *D*_{2}*σ*_{2} with *D*_{j} = −*i∂*_{j} for *j* = 1, 2 and *σ*_{1,2,3} are the standard Pauli matrices. The operator *H*_{0} appears as a generic low-energy description of energy band crossings and is ubiquitous in works on topological insulators. We classify *H*_{0} by augmenting it with a domain wall along one direction and assessing the resulting asymmetric transport in the transverse direction. We implement a domain wall *m*_{1}(*x*_{1}) = *x*_{1} along the first variable by introducing *H*_{1} = *H*_{0} + *x*_{1}*σ*_{3}. This models includes insulating regions *x*_{1} > 0 and *x*_{1} < 0 while transport remains possible in the vicinity of the interface *x*_{1} = 0.

To obtain confinement in the second variable, we introduce the second domain wall *m*_{2}(*x*_{2}) = *x*_{2}. Associated to *H*_{0} and *H*_{1} is then the operator *F* = *H*_{1} − *ix*_{2}. This is again a Fredholm operator from its domain of definition to $L2(R2)\u2297C2$ and we verify that Index *F* = 1, which defines the topological charge associated with *H*_{0} (and *H*_{1}). The kernel of *F* is spanned by the spinor $e\u221212|x|2(1,i)t$, while the kernel of *F** is trivial.

This construction generalizes to higher dimensions in a straightforward way, except for the fact that the construction of the domain walls requires additional degrees of freedom as dimension increases. Note that in dimension *d* = 1, we have $F*F=Dx2+x2+c1$ with *c*_{1} = −1 here. In dimension *d* = 2, we observe that $F*F=(D12+D22+x12+x22+c2)I2$ with *I*_{2} the 2 × 2 identity matrix. The latter is obtained because (*σ*_{1}, *σ*_{2}, *σ*_{3}, *i*) satisfy appropriate *orthogonality* properties. In particular, *σ*_{1,2,3} satisfy *σ*_{i}*σ*_{j} + *σ*_{j}*σ*_{i} = 2*δ*_{ij}*I*_{2}. Ensuring this property in higher spatial dimensions requires enlarging the size of the *spinors* so they generate an appropriate representation of the Clifford algebra $Cld(C)$ (Ref. 12, Chap. 17).

Consider in $R3$ the Weyl Hamiltonian *H*_{0} = *D*_{1}*σ*_{1} + *D*_{2}*σ*_{2} + *D*_{3}*σ*_{3}. As an operator acting on spinors in $C2$, the latter operator is stable against gap opening by domain walls.^{11} We, therefore, introduce the operator *H*_{1} = *σ*_{1} ⊗ *H*_{0} + *σ*_{2} ⊗ *I*_{2}*x*_{1} with a domain wall *m*_{1}(*x*_{1}) = *x*_{1} in the first direction but now acting on spinors in $C4$. The operator *H*_{1} thus admits surface states concentrated in the vicinity of *x*_{1} = 0, as does the operator *H*_{0} in the two-dimensional setting. Its topology is then characterized by asymmetric transport in the third dimension after a second domain wall in the *x*_{2} direction is introduced: *H*_{2} = *H*_{1} + *σ*_{3} ⊗ *I*_{2}*x*_{2}.

Confinement in the last variable is imposed by the domain wall *m*_{3}(*x*_{3}) = *x*_{3}. We introduce the operator

This is a Fredholm operator [from its domain of definition to $L2(R3)\u2297C4$]. We verify (and will show in greater generality in Sec. VI) that the topological charge of *H*_{0} is Index *F* = −1, with a sign change here reflecting the fact that indices depend on the orientation of the Clifford matrices used to construct the operators *H*_{j} as well as the orientation of the domain walls. The kernel of *F** has for eigenfunction the spinor $e\u221212|x|2(1,\u22121,\u22121,\u22121)t$. The *topological charge* of *H*_{0}, of *H*_{1}, and of *H*_{2} is *defined* as Index *F*.

Note that in the above three-dimensional model, *H*_{2} models *higher-order* topological insulators with transport occurring along a *hinge*, or co-dimension two manifold.^{52} The classification of *H*_{2} by domain walls is arguably simpler than one based on bulk phases that may be hard to identify.

### B. Pseudo-differential elliptic operators

We now define the classes of pseudo-differential operators we wish to classify. Such classes naturally generalize the examples of Dirac operators seen in the preceding paragraphs.

Consider a spatial dimension *d* ≥ 1 and operators defined on functions of the Euclidean space $Rd$. We denote by $\xi \u2208Rd$ the dual (Fourier) variable and $X=(x,\xi )\u2208R2d$ the phase space variable. The algebras of pseudo-differential operators (PDOs) we consider are written in Weyl quantization as

for *a*(*X*), a matrix-valued *symbol* in $M(n)$, the space of complex *n* × *n* matrices, and *ψ*(*x*) a spinor with values in $Cn$. The value of *n* is problem-dependent. For the Fredholm operator (1), we verify that *F* = Op *a* for *a*(*X*) = *σ*_{1} ⊗ (*ξ*_{1}*σ*_{1} + *ξ*_{2}*σ*_{2} + *ξ*_{2}*σ*_{3}) + *x*_{1}*σ*_{2} ⊗ *I*_{2} + *x*_{2}*σ*_{3} ⊗ *I*_{2} − *ix*_{3}*I*_{4} with thus *n* = 4. Relevant notation and results on pseudo-differential operators and functional calculus are collected in the Appendix.

The starting point is an operator *H*_{k} = Op *a*_{k} with *a*_{k} a given matrix-valued symbol in $M(nk)$ interpreted as confining the first *k* variables. Our first aim is to construct the operators *H* ≔ *H*_{d−1} = Op *a*_{d−1} and *F* = Op *a* with *F* a Fredholm operator. This requires introducing the following notation and functional setting.

We decompose the spatial variables $x=(xk\u2032,xk\u2032\u2032)$ with $xk\u2032\u2208Rk$ and $xk\u2032\u2032\u2208Rd\u2212k$. We use the notation $\u27e8y\u27e9=1+|y|2$ and $\u27e8y1,y2\u27e9=1+|y1|2+|y2|2$ and define the weights

For a given spinor dimension *n* = *n*_{k} with 0 ≤ *k* ≤ *d*, and an order *m* > 0, we denote by $Skm=Skm[nk]$ the class of symbols *a*_{k} such that for each *d*-dimensional multi-indices *α* and *β*, there is a constant *C*_{α,β} such that for each component *b* of $ak\u2208M(nk)$, we have

Here, $wkm=(wk)m$. We define the space of symbols $S\u0303m$ as $Sdm$ but acting on spinors of (lower) dimension *n*_{d−1} instead of *n*_{d}. Here *m* is the order of the operator.

For the two-dimensional Dirac operator, we find *m* = 1, *H*_{0} = Op *a*_{0} for *a*_{0} = *ξ*_{1}*σ*_{1} + *ξ*_{2}*σ*_{2}, while *H*_{1} = Op *a*_{1} for *a*_{1} = *a*_{0} + *x*_{1}*σ*_{3} and *F* = Op *a* for *a* = *a*_{1} − *ix*_{2}. For *n*_{0} = *n*_{1} = 2, we observe that $aj\u2208Sj1$ for *j* = 0, 1, while $a\u2208S\u03031$.

We impose a number of assumptions on *H*_{k} and *a*_{k}. The first one is *Hermitian symmetry* $ak=ak*$. A second symmetry is the *chiral symmetry* (6) below when *d* + *k* is even. This symmetry ensures the (potential) topological non-triviality of the Hamiltonian. Finally, we assume all our Hamiltonians to be *elliptic*, a necessary assumption in any index theory.^{12} Ellipticity is defined as the constraint

In other words, all eigenvalues of the Hermitian matrix *a*_{k}(*X*) are bounded away from 0 by at least $Cwkm(X)$ for *X* outside of a compact set in $R2d$. Since $ak\u2208Skm$, all (positive and negative) eigenvalues of *a*_{k} are of order $wkm(X)$ away from a compact set.

We denote by $ESkm$ the *elliptic symbols* in $Skm$ and $ES\u0303m$ the elliptic symbols in $S\u0303m$.

Associated to the spaces of symbols $Skm$ and $S\u0303m$ are Hilbert spaces $Hkm$ and $H\u0303m$ defined in (A5) in the Appendix. These spaces are constructed so that for $ak\u2208ESkm$, we obtain that *H*_{k} = Op *a*_{k} is an unbounded self-adjoint operator on $Hk0=L2(Rd)\u2297M(nk)$ with domain of definition $D(Opak)=Hkm$, while for $a\u2208ES\u0303m$, we obtain that *F* = Op *a* is an unbounded operator with domain of definition $D(F)=H\u0303m$. For any $ak\u2208ESkm$, the space $Hkm$ is defined explicitly as $Hkm=(Opak+i)\u22121Hk0$.

For the Dirac operators in dimension *d* = 3, we have, for instance, the Hilbert space $Hk1={\psi \u2208L2(R3;Cnk);(Dj\psi )1\u2264j\u22643\u2208L2(R3;Cnk)\u2009and\u2009(xj\psi )1\u2264j\u2264k\u2208L2(R3;Cnk)}$ with *n*_{0} = 2 and *n*_{1} = *n*_{2} = *n*_{3} = 4.

### C. Classification by domain walls

We start from a given elliptic (self-adjoint) operator *H*_{k} = Op *a*_{k} for $ak\u2208ESkm$ with *m* > 0. By ellipticity assumption, *H*_{k} is an unbounded self-adjoint operator with domain $D(Hk)=Hkm$ and acts of spinors in $Cnk$. The ellipticity of *H*_{k} and the construction of the weight $wk(X)=\u27e8xk\u2032,\xi \u27e9$ imply that the first *k* variables parameterized by $xk\u2032$ are confined in the sense that each singular value of *a*_{k} is large for $|xk\u2032|$ large.

To obtain a non-trivial topological classification, the operator *H*_{k} needs to satisfy a chiral symmetry when *d* + *k* is even (complex class AIII^{47}). When *d* + *k* is odd, no symmetry is imposed beyond the Hermitian structure.

Assume first that *d* + *k* is even with *k* ≤ *d* − 2. Recall that *σ*_{1,2,3} are the Pauli matrices, a set of Hermitian 2 × 2 matrices such that *σ*_{i}*σ*_{j} + *σ*_{j}*σ*_{i} = 2*δ*_{ij} and *σ*_{1}*σ*_{2} = *iσ*_{3}. Using the notation $\sigma \xb1=12(\sigma 1\xb1i\sigma 2)$, the *chiral symmetry* takes in a suitable basis the following form:

We next introduce the *domain walls*

They are constructed to have the same asymptotic homogeneity of order *m* as the Hamiltonian *H*_{k}. We then define the new spinor dimension *n*_{k+1} = *n*_{k} and the augmented Hamiltonian

This implements a domain wall in the variable *x*_{k+1}.

Assume now *d* + *k* is odd. We define the new spinor dimension *n*_{k+1} = 2*n*_{k} and the augmented Hamiltonian,

The operator *H*_{k+1} satisfies the chiral symmetry of the form (6), as requested since *d* + *k* + 1 is now even.

We denote by *a*_{k+1} the symbol of *H*_{k+1} = Op *a*_{k+1} and observe that *a*_{k+1} = *a*_{k} + *m*_{k+1}*σ*_{3} ⊗ *I* when *d* + *k* is even and *a*_{k+1} = *σ*_{1} ⊗ *a*_{k} + *m*_{k+1}*σ*_{2} ⊗ *I* when *d* + *k* is odd.

The procedure is iterated until *H*_{d} has been constructed. Note that $ak+2(X)\u2208M(2nk)$ with dimension of the spinor space on which the matrices act that doubles every time *k* is raised to *k* + 2. Since 2*d* is even, *H*_{d} = *σ*_{−} ⊗ *F** + *σ*_{+} ⊗ *F* for an operator *F* = *F*_{d} = *H*_{d−1} − *im*_{d} =: Op *a*, or equivalently, *a* = *a*_{d−1} − *im*_{d}.

For 0 < *l* ≤ *d* − *k*, the intermediate Hamiltonians all have the form

where for some integer *p* = *p*(*l*, *k*) and for some matrices *γ*_{j} such that {*γ*_{i}, *γ*_{j}} ≔ *γ*_{i}*γ*_{j} + *γ*_{j}*γ*_{i} = 0 for all *i* ≠ *j* in {0, …, *l*}, we have

We now show that all operators *H*_{l} are elliptic and that *H*_{d} and *F* are Fredholm operators.

*Let* $ak\u2208ESkm$ *and* *H*_{k} = Op *a*_{k} *satisfying the chiral symmetry* *(6)* *when* *d* + *k* *is even. Then* $aj\u2208ESjm$ *for all* *k* ≤ *j* ≤ *d* *and* $a\u2208ES\u0303m$*.*

Let *a*_{k+l} be the symbol of *H*_{k+l} = Op *a*_{k+l} for 0 < *l* ≤ *d* − *k*. By construction and commutativity {*γ*_{i}, *γ*_{j}} = 0, we obtain that $ak+l2=I\u2297ak2+\u2211j=1lmk+l2\u2297I$ with *I* identity matrices with appropriate dimensions. This shows that *a*_{k+l} satisfies the ellipticity condition (5) for the weight *w*_{k+l}(*X*). The decay properties for derivatives of *a*_{k+l} in (4) with *k* replaced by *k* + *l* follow from the corresponding properties for *a*_{k}. That $a\u2208ES\u0303m$ comes from the corresponding result for *a*_{d} and the construction of *F*.

Let $\Lambda =\u2212\Delta +|x|2+1=Op\u27e8x,\xi \u27e92$ be an elliptic self-adjoint operator, which by construction, maps $H(wd\u22121m)$ to $H(1)=L2(Rd)$.^{45}

*The operators* *H*_{d} *and* *F* *are Fredholm operators from* $Hdm$ *to* $Hd0$ *and* $H\u0303m$ *to* $H\u03030$*, respectively. Equivalently,* Λ^{−m}*H*_{d} *and* Λ^{−m}*F* *are Fredholm operators on* $Hd0$ *and* $H\u03030$*, respectively.*

The construction of *H*_{l} implements *l* − *k* domain walls to test the topology of the operator *H*_{k}. When *l* = *d*, the operator *H*_{d} has *d* confined variables and is, as we saw, a Fredholm operator, i.e., an operator that admits left and right inverse modulo compact operators (Ref. 40, Chap. 19). The operator *H*_{d} is self-adjoint, and so its index vanishes. However, it satisfies the chiral symmetry (6), and the corresponding operator *F* = *H*_{d−1} − *im*_{d} is also Fredholm. Its index may not vanish, and we *define* the *topological charge*^{56} of *H*_{k} as Index *F*.

The intermediate operator *H*_{d−1} is physically relevant with *d* − 1 confined spatial variables (in the vicinity of $xd\u22121\u2032=0$) and transport allowed along the direction $xd\u22121\u2032\u2032=xd$. As we show in Secs. III and IV, this transport is asymmetric and quantized by the topological charge of *H*_{k}.

### D. Topological charge and integral formulation

We next apply Ref. 40, Theorem 19.3.1′ (see also Ref. 31) to the operator Λ^{−m}*F* to obtain that the index of *F*, which equals that of Λ^{−m}*F* since Λ^{−m} has trivial index, is given by the following Fedosov–Hörmander formula:

*For*

*F*= Op

*a*

*, we have*

*Here,*

*R*

*is a sufficiently large constant so that*

*a*

*is invertible outside of the ball of radius*

*R*,

*and the orientation of*$R2d$

*and that induced on*$SR2d\u22121$

*is chosen so that*

*dξ*

_{1}∧

*dx*

_{1}∧ ⋯ ∧

*dξ*

_{d}∧

*dx*

_{d}> 0

*.*

*(Ref. 40, Theorem 19.3.1′).* Comes from the approximation of symbols in *S*(*M*, *g*^{s}) by symbols in *S*(*M*, *g*^{i}) and the index theorem (Ref. 40, Theorem 19.3.1) proving (10) for symbols in *S*(*M*, *g*^{i}). The approximation is described in Ref. 40, Lemma 19.3.3. See the Appendix for notation on the metrics *g*^{i} and *g*^{s}. We use a similar approximation in Lemma 3.3 below to prove topological charge conservation in Theorem 4.1.

*(Ref. 40, Theorem 19.3.1′).* Applies to Λ^{−m}*F* and not *F* directly. However, the index of Λ^{−tm}*F* is independent of *t* ∈ [0, 1] since the index of Λ^{−tm} is trivial and the index of a product is the sum of the indices (Ref. 40, Chap. 19). The corresponding symbols *a*_{t} of Λ^{−tm}*F* are uniformly invertible for |*X*| ≥ *R* for *R* sufficiently large and smooth in *t*. The integral (10) is, therefore, the same for *a* = *a*_{0} and *a* = *a*_{1}. This may be proved as follows:

Let M and N be smooth closed manifolds. Here, $M=SR2d\u22121$ and $N=GL(nd\u22121;C)$. For *t* ∈ [0, 1], let *i*_{t}: M → M × [0, 1] defined by *i*_{t}(*x*) = (*x*, *t*). Let *a*_{0} and *a*_{1} be homotopic smooth maps from M to N, and let *a*: M × [0, 1] → N be the smooth homotopy map such that *a*_{0} = *a*◦*i*_{0} and *a*_{1} = *a*◦*i*_{1}.

*p*≥ 1, there exists by Ref. 42, Lemma 17.9, a homotopy operator

*h*: Ω

^{p}(M × [0, 1]) to Ω

^{p−1}(M) such that for each

*p*-form $\omega \u0303\u2208\Omega p(M\xd7[0,1])$, then

*ω*∈ Ω

^{p}(N) be a closed

*p*-form on N. In our application, this is $\omega =tr(A\u22121dA)\u2227(2d\u22121)$. Then we have that $\omega \u0303=a*\omega $ is a closed

*p*-form on M × [0, 1] since

*da**

*ω*=

*a**

*dω*= 0. Using the above homotopy operator, we thus have

*p*-form on M. By the Stokes theorem, this means that for a top-degree form, $\u222bMa1*\omega =\u222bMa0*\omega $. As a result, the right-hand side in (10) is the same for

*a*the symbol of

*F*or that of Λ

^{−m}

*F*.

## III. PHYSICAL OBSERVABLE AND TOEPLITZ OPERATOR

While systematic and explicit, the classification of *H*_{k} in Sec. II based on the index of *F* is *a priori* unrelated to any physical observable. We now present a second topological classification based on a physical observable that characterizes the asymmetric transport of the operator *H*_{d−1} introduced in Sec. II C.

The operator *H*_{d−1} confines in *d* − 1 directions while allowing transport in the remaining dimension parameterized by *x*_{d}. The following edge conductivity quantifies asymmetric transport in that direction. Let *H* ≔ *H*_{d−1} and $\phi \u2208S[0,1]$ a smooth non-decreasing switch function and $P=P(xd)\u2208S[0,1]$ a smooth spatial switch function.

Here and below, a function $f:R\u2192R$ is called a *switch function* $f\u2208S[0,1]$ if *f* is bounded measurable and there are *x*_{L} and *x*_{R} in $R$ such that *f*(*x*) = 0 for *x* < *x*_{L} and *f*(*x*) = 1 for *x* > *x*_{R}. We denote by $C\u221eS[0,1]$ the subset of smooth switch functions.

We define the *edge conductivity*

Here, *φ*′(*H*) defined by functional calculus with *φ*′(*h*) ≥ 0 being a spectral density (integrating to 1). We assume that *i*[*H*, *P*]*φ*′(*H*) is a trace-class operator (i.e., a compact operator with summable singular values). The commutator is defined as [*A*, *B*] = *AB* − *BA*. The edge conductivity *σ*_{I} has been used to model edge transport and derive a bulk-edge correspondence for two dimensional materials in a number of contexts (see, e.g., Refs. 6, 27, 29, 36, 38, 47, and 49).

It may be given the following interpretation. Let *ψ*(*t*) = *e*^{−itH}*ψ* be a solution of the Schrödinger equation *i∂*_{t}*ψ*(*t*) = *Hψ*(*t*) with initial condition *ψ*, and let *P* be a Heaviside function defined as *P*(*x*_{d}) = 1 for $xd>x\u0303$ while *P*(*x*_{d}) = 0 for $xd<x\u0303$ for some $x\u0303\u2208R$. Then $\u27e8P\u27e9(t)\u2254(\psi (t),P\psi (t))$ is interpreted as the mass of *ψ*(*t*) in the half space on the right of the hyperplane $xd=x\u0303$. Its derivative $ddt\u27e8P\u27e9=(\psi (t),i[H,P]\psi (t))=Tri[H,P]\psi (t)\psi (t)*$ thus describes current crossing the hyperplane $xd=x\u0303$. We formally replace the density *ψ*(*t*)*ψ*(*t*)* by the spectral density *φ*′(*H*). This heuristically gives the interpretation of *σ*_{I} as the rate of signal propagating from the left to the right of the hyperplane $xd=x\u0303$ per unit time for a density of states in the system given by the spectral density *φ*′(*H*).

The main objective of this section is to prove that for $ak\u2208ESkm$, then *i*[*H*_{d−1}, *P*]*φ*′(*H*_{d−1}) is indeed a trace-class operator so that *σ*_{I}(*H*_{d−1}) is well-defined. We next relate the edge conductivity to the index of the Toeplitz operator $T\u2254P\u0303U(H)P\u0303RanP\u0303$ for $P\u0303$ an orthogonal projector in $S[0,1]$ and *U*(*H*) = *e*^{2πiφ(H)}. In particular, we show that $2\pi \sigma I=IndexT\u2208Z$ so that *σ*_{I} is indeed quantized. Finally, we prove that *σ*_{I} and the index of *T* are stable against a number of continuous transformations of the symbol *a*_{k}.

### A. Trace-class property and Toeplitz operator

Let $ak\u2208ESkm$ so that by Lemma 2.1, $ad\u22121\u2208ESd\u22121m$ while $a=ad\u22121\u2212ixd\u2208ES\u0303m$. We denote by *H* = *H*_{d−1} = Op *a*_{d−1}. Let *U*(*H*) = *e*^{i2πφ(H)} with $\phi \u2208C\u221eS[0,1]$ while *W*(*H*) = *U*(*H*) − *I*.

We use the following notation for classes of symbols. These classes are also recalled in the Appendix. For *M* = *M*(*X*), a weight function, we denote by *S*^{0}(*M*) the space of symbols with components *b*(*X*) such that

for each multi-index *α*. Therefore, $S0(wkm)$ is a larger class of symbols than $Skm$ defined in (4). The pseudo-differential calculus provides the following results, as recalled in the Appendix:

If

*a*∈*S*^{0}(*M*_{1}) and*b*∈*S*^{0}(*M*_{2}), then Op*a*Op*b*= Op*c*with*c*∈*S*^{0}(*M*_{1}*M*_{2}) constructed explicitly in (A9) (with*h*= 1).If $\varphi \u2208Cc\u221e(R)$ and $H\u2208OpESd\u22121m$, then $\varphi (H)\u2208OpSd\u22121\u2212\u221e$.

If

*a*∈*S*(*M*) with $M\u2208L1(R2d)$, then Op*a*is trace-class and (A11) holds; namely, the trace of Op*a*is given by (2*π*)^{−d}times the integral of the trace of*a*over $R2d$, or equivalently by the integral of the trace of their Schwartz kernel along the diagonal; see (A7) (with*h*= 1).

This allows us to obtain the following results:

*Let* $ak\u2208ESkm$ *and* $P\u2208C\u221eS[0,1]$*. Let* $\varphi \u2208Cc\u221e(R)$ *and* $p,q\u2208N$*. Then* [*P*, *ϕ*(*H*)] *and* *H*^{p}[*P*, *H*^{q}]*ϕ*(*H*) *are trace-class operators with symbols in* $Sd\u2212\u221e$*. When* $P\u0303\u2208S[0,1]$ *is an orthogonal projector, then* $T\u2254P\u0303U(H)P\u0303RanP\u0303$ *is a Fredholm operator on* $RanP\u0303\u2282H\u03030$ *with index given by* $Tr[U(H),P\u0303]U*(H)=Tr[U(H),P]U*(H)$*. All the above operator traces may be computed by integrating the Schwartz kernel of the operator along the diagonal.*

By composition calculus (i), for any *A* = Op *a* with *a* ∈ *S*^{0}(*M*), then the decomposition $[A,P]=(1\u2212\chi \u0303(xd))A\chi \u0303(xd)\u2212\chi \u0303(xd)A(1\u2212\chi \u0303(xd))$ for $\chi \u0303(xd)$ a smooth function equal to 1 for *x* > 1 and 0 for *x* < −1 shows that [*A*, *P*] has symbol in $S0(M\u27e8xd\u27e9\u2212\u221e)$ (i.e., in $S0(M\u27e8xd\u27e9\u2212N)$ for each $N\u2208N$). By assumption on *a*_{k} and using the functional calculus result (ii) (see Lemma A.2), we obtain for $\varphi \u2208Cc\u221e(R)$ that $\varphi (H)\u2208OpS0(\u27e8xd\u22121\u2032,\xi \u27e9\u2212\u221e)$, which is larger that $OpSd\u22121\u2212\u221e$ (since *m* > 0). To simplify notation, we use the same notation for *S*^{0}(*M*) and $S0(M)\u2297M(n)$ for any *n*. Therefore, by composition calculus, [*ϕ*(*H*), *P*] and [*H*, *P*]*ϕ*(*H*) as well as *H*^{p}[*H*^{q}, *P*]*ϕ*(*H*) for $p,q\u2208N$ all have symbols in *S*^{0}(⟨*X*⟩^{−∞}). We use this with *ϕ* = *φ*′ and *ϕ* = *W*. With additional effort, we verify that all symbols are in $Sd\u2212\u221e$, although this is not necessary for the rest of the proof and so we leave the details to the reader.

Using (iii), we deduce that [*P*, *ϕ*(*H*)] and *H*^{p}[*P*, *H*^{q}]*ϕ*(*H*) are trace-class operators with traces given as the integral of their Schwartz kernel along the diagonal. Applying the latter directly yields that Tr[*ϕ*(*H*), *P*] = 0, for instance, since the Schwartz kernel of [*ϕ*(*H*), *P*] vanishes along the diagonal. In particular, [*P*, *W*(*H*)] and [*H*, *P*]*φ*′(*H*) are trace-class operators.

*W*(

*H*)

*χ*(

*x*

_{d}) and

*χ*(

*x*

_{d})

*W*(

*H*) are trace-class. Since multiplication by $P\u0303\u2212P$ is a bounded operator, we conclude that all operators on the above right-hand side are trace-class, and hence so is $[U(H),P\u0303]$. As a consequence, $[U(H),P\u0303]U*(H)$ is trace-class as well.

*AB*= Tr

*BA*when

*A*is trace-class and

*B*is bounded, we deduce

That $T=P\u0303U(H)P\u0303|RanP\u0303$ is a Fredholm operator with index given by $Tr[U(H),P\u0303]U*(H)=Tr(Q\u0303\u2212P\u0303)$ for $P\u0303$ and $Q\u0303=U(H)P\u0303U*(H)$ projectors is a non-trivial consequence of the trace-class nature of $[P\u0303,U(H)]$ and the Fedosov formula (see, e.g., Ref. 2, Proposition 2.4).

We now relate the Fredholm operator *T* and the calculation of its index as a trace with the line conductivity *σ*_{I} = *σ*_{I}(*H*) defined in (11). We obtain the following result:

*Under the assumptions of Lemma 3.1, we have* $2\pi \sigma I=Tr[U(H),P]U*(H)=Tr[U(H),P\u0303]U*(H)=IndexP\u0303U(H)P\u0303RanP\u0303$*.*

*χ*= 1 on the support of

*W*and

*g*. Let

*W*

_{p}be a sequence of polynomials chosen such that

*χ*(

*W*−

*W*

_{p}) and $\chi (W\u2032\u2212Wp\u2032)$ converge to 0 uniformly on $R$ as

*p*→ ∞. We find, with

*δW*

_{p}≔

*W*−

*W*

_{p}

*δW*

_{p}

*gχ*,

*P*] = 0 and that

*δW*

_{p}[

*P*,

*g*] is trace-class since the [

*P*,

*g*] has symbol in

*S*

^{0}(⟨

*X*⟩

^{−∞}). Since Tr

*AB*= Tr

*BA*when

*A*is a trace-class and

*B*is bounded, we find that Tr

*δW*

_{p}[

*P*,

*g*]

*χ*= Tr

*χδW*

_{p}[

*P*,

*g*] = Tr[

*P*,

*g*]

*χδW*

_{p}. Therefore,

*W*

_{p},

*P*]

*g*. We verify that

*AB*,

*C*] =

*A*[

*B*,

*C*] + [

*A*,

*C*]

*B*,

*H*

^{n}[

*H*,

*P*]

*g*is trace-class so that Tr

*H*

^{n}[

*H*,

*P*]

*gχ*= Tr

*H*

^{n}

*χ*[

*H*,

*P*]

*g*and for the last equality, an induction in

*n*≥ 1. This proves the result for

*W*

_{p}as well. Is remains to realize that $(Wp\u2032\u2212W\u2032)g$ is uniformly small as

*p*→ ∞ to obtain that

*H*,

*P*]

*W*′. Let $1=\psi 12+\psi 22$ a partition of unity with 0 ≤

*ψ*

_{j}≤ 1 for

*j*= 1, 2 and such that $\psi 1\u2208Cc\u221e(R)$ equals 1 on the support of

*W*. Then, using (13), with

*ψ*

_{j}=

*ψ*

_{j}(

*H*), we have $Tr[H,P]W\u2032=Tr[H,P]W\u2032\psi 12=Tr[W(H),P]\psi 12.$ Now, since [

*W*(

*H*),

*P*] is trace-class,

*W*(

*H*)

*ψ*

_{2}(

*H*) = 0. Since Tr[

*W*,

*P*] = 0, we have $Tr[H,P]W\u2032=Tr[W,P]\psi 12=Tr[W,P]=0$. This proves that Tr[

*U*,

*P*]

*U** = 2

*πi*Tr[

*H*,

*P*]

*φ*′(

*H*) since

*W*′

*U** =

*U*′

*U** = 2

*πiφ*′.

### B. Stability results

We next derive several results showing that the index of the Toeplitz operator $T=P\u0303U(H)P\u0303RanP\u0303$ is stable against a number of (continuous) deformations. We first need a technical result showing that the index can be computed by approximating the symbol of *H*_{k} by an *isotropic symbol*. All results so far have been obtained for symbols satisfying (4). The space of *isotropic* elliptic symbols $ESd\u22121m(gi)$ is defined by a similar constraint where ⟨*x*⟩^{|α|}⟨*ξ*⟩^{|β|} is replaced by ⟨*X*⟩^{|α|+|β|}. The class of elliptic isotropic symbols is thus smaller than $ESd\u22121m=ESd\u22121m(gs)$. It is also invariant under permutation of the variables *X*, which is not the case for $ESd\u22121m(gs)$. The invariance under such permutations will be needed in the proof of conservation of the topological charge in Sec. IV. We thus state the following approximate result:

*Let* $T=P\u0303U(H)P\u0303RanP\u0303$ *with* $H=Hd\u22121\u2208OpESd\u22121m(gs)$*. Then there is a sequence of operators* *H*_{ɛ} *for* 0 ≤ *ɛ* ≤ 1 *with symbol in* $ESd\u22121m(gi)$ *for all* *ɛ* > 0 *and such that the corresponding* $[0,1]\u220b\epsilon \u2192T\epsilon =P\u0303U(H\epsilon )P\u0303RanP\u0303$ *is continuous in the uniform sense and* *T*_{0} = *T**. Therefore,* Index *T*_{ɛ} *is defined as being independent of* *ɛ* *and equal to* Index *T**. Moreover, the symbols* *a*_{ɛ} *are chosen so that for any compact domain in* *X* = (*x*, *ξ*)*,* *a*_{ɛ} = *a*_{d−1} *on that domain for* *ɛ* *sufficiently small.*

*S*(

*M*,

*g*

^{s}) may be suitably approximated by symbols in

*S*(

*M*,

*g*

^{i}) as follows. Let $v(r):R+\u2192R+$ be a smooth non-increasing function such that

*v*(

*r*) = 1 on [0,1] and

*v*(

*r*) = 2/

*r*on [2, ∞). Let

*a*∈

*S*(

*M*,

*g*

^{s}). We then define the family of regularized symbols,

*a*

_{ɛ}(

*X*) =

*a*(

*X*) for

*ɛ*|

*X*| ≤ 1 while $a\epsilon (X)=a(X\epsilon |X|)$ homogeneous of degree 0 for

*ɛ*|

*X*| ≥ 2. Then Ref. 40, Lemma 19.3.3 (where the metrics

*g*

^{s}and

*g*

^{i}are called

*g*and

*G*, respectively) proves that

*a*

_{ɛ}(

*X*) ∈

*S*(

*M*,

*g*

^{s}) uniformly in 0 ≤

*ɛ*≤ 1 (i.e., every semi-norm defining the symbol space is bounded for

*a*

_{ɛ}uniformly in

*ɛ*). Moreover,

*a*

_{ɛ}(

*X*) ∈

*S*(

*M*,

*g*

^{i}) when

*ɛ*> 0 with a bound that now depends on

*ɛ*. Since

*a*

_{ɛ}(

*X*) =

*a*(

*X*) for

*ɛ*|

*X*| ≤ 1, we obtain that

*a*

_{ɛ}converges to

*a*in

*S*(

*M*,

*g*

^{s}) as

*ɛ*→ 0.

We now mimic the proof of Ref. 40, Theorem 19.3.1′ extending the index theorem (10) from the isotropic metric *g*^{i} to the metric *g*^{s}. For *a*_{ɛ}(*X*) = *a*_{d−1}(*v*(*ɛ*|*X*|)*x*, *v*(*ɛ*|*X*|)*ξ*), we find that $a\epsilon \u2208ESd\u22121m(gs)$ uniformly in *ɛ* and $a\epsilon \u2208ESd\u22121m(gi)$ for *ɛ* > 0. Let *H*_{ɛ} = Op *a*_{ɛ} and $T\u0303\epsilon =P\u0303U(H\epsilon )P\u0303+I\u2212P\u0303$. By uniformity of $a\epsilon (X)\u2208Sd\u22121m(gs)$ in *ɛ* and uniformity of bounds in Lemma A.2 (see Remark A.3), we obtain that *W*(*H*_{ɛ}) has symbol in $S\u03030(gs)$ uniform in *ɛ*.

We next observe that $T\u0303\epsilon *T\u0303\epsilon \u2212I=P\u0303[W*(H\epsilon ),P\u0303]U(H\epsilon )P\u0303$ and $T\u0303\epsilon T\u0303\epsilon *\u2212I$ are uniformly compact in *ɛ* and even uniformly trace-class from the results of Lemma 3.1. We then apply Ref. 40, Theorem 19.1.10 to obtain that the indices of $T\u0303\epsilon $ and $T\u0303\epsilon *$, and hence that of *T*_{ɛ}, are independent of 0 < *ɛ* ≤ 1. In the limit *ɛ* → 0, this is the index of *T*. Therefore, Index *T* = Index *T*_{ɛ} for *ɛ* > 0 but now for a symbol $a\epsilon \u2208ESd\u22121m(gi)$.

The above result shows that we can replace the symbol in *H*_{d−1} with that obtained at *ɛ* > 0. We also observe that (10) is independent of *ɛ* for *ɛ* small. We may, therefore, assume that $ad\u22121\u2208ESd\u22121m(gi)$ in the computation of Index *T*. The main advantage of the more constraining metric *g*^{i} is that the corresponding symbol classes are now invariant under suitable rotations and permutations of the phase space variables *X*. The following result is then used. Let *Y* = (*x*_{1}, …, *x*_{d−1}, *ξ*_{1}, …, *ξ*_{d−1}).

*Let* *g* = *g*^{i}*. Let* [0, 1] ∋ *t* → *L*_{t} *be a continuous family of linear invertible transformations in* $GL(2d\u22122,R)$ *in the* *Y* *variables leaving the variables* (*x*_{d}, *ξ*_{d}) *fixed. Let* $a(X)\u2208ESd\u22121m(gi)$*. Then* $a(t,X)=a(LtX)\u2208ESd\u22121m(gi)$*. Let* *T*_{t} *be the corresponding Toeplitz operator. Then* *T*_{t} *is Fredholm with index independent of* *t* ∈ [0, 1]*.*

*W*(see the Appendix), we compute

*s*∈ [0, 1] so that $(Ht\u2212Hs)(i\u2212Hs)\u22121=Op(at\u2212as)\u266frs$. Then, from Ref. 16, (18) and composition calculus, there exists a seminorm

*k*independent of

*t*such that

*L*

_{t}−

*L*

_{s})

*X*times derivatives of

*a*by chain rule. We thus obtain terms of the form $Xi\u2202Xja$ (with

*X*

_{j}∉ {

*x*

_{d},

*ξ*

_{d}}), which are operations that are stable from

*S*(

*M*

_{d−1},

*g*

^{i}) to itself provided that

*g*=

*g*

^{i}. Note that the vector field [implementing rotation in the variables (

*x*

_{j},

*ξ*

_{j})] $\xi j\u2202xj\u2212xj\u2202\xi j$ does not preserve

*S*(

*M*

_{d−1},

*g*

^{s}), and hence the importance of working with symbols in the smaller isotropic class.

Higher-order derivatives are bounded in the same way, allowing us to obtain that $C\Vert a(t,X)\u2212a(s,X)\Vert k;S(Md\u22121;g)$ is bounded by a constant times |*t* − *s*|. Therefore, *U*(*H*_{t}) − *U*(*H*_{s}) is small in the uniform sense for small (*t* − *s*), so that the index of *T*_{t} is continuous in *t* and hence independent of *t* ∈ [0, 1].

This result states, in particular, that for *a*_{d−1} ∈ *ES*_{d−1}(*g*^{i}), then the index of *T* is independent of any rescaling *Y*_{j} → *λY*_{j} for *λ* > 0 (leaving all other variables fixed) as well as any rotation in the phase space variables mapping (*Y*_{j}, *Y*_{k}) to (*Y*_{k}, − *Y*_{j}) (note the sign change to preserve orientation). We will use the above lemma only for such transformations (dilations and permutations).

We could show similarly that *σ*_{I} is independent of changes in *φ* although this property will automatically come from (14) proved in Sec. IV. However, we need the following straightforward result:

*Let* *g* ∈ {*g*^{s}, *g*^{i}}*. For* *t* ∈ [0, 1]*, let* $t\u21a6at\u2208ESd\u22121m(g)$ *be a continuous path of elliptic symbols. Then the indices of the corresponding Fredholm operators* *F*_{t} = Op (*a*_{t} − *im*_{d}) *and* $Tt=P\u0303U(Opat)P\u0303|RanP\u0303$ *are independent of* *t* ∈ [0, 1]*.*

By assumption and construction, Λ^{−m}*F*_{t} and *T*_{t} are continuous in *t* as operators from $H\u03030$ to itself. Their indices are, therefore, constant in *t*.

We apply the preceding lemma to $a1=a(xd\u22121\u2032,xd,\xi )$ and $a0=a(xd\u22121\u2032,0,\xi )$ while *a*_{t} = *ta*_{1} + (1 − *t*)*a*_{0}. The path of symbols belongs to $ESd\u22121m$ so that the indices of the respective operators are defined with clear continuity in *t*. This allows us to replace the *x*_{d}-dependent elliptic symbol *a*_{d−1} with an *x*_{d}-independent one, which is used below in the proof of topological charge conservation.

## IV. TOPOLOGICAL CHARGE CONSERVATION

We recall that *H*_{k} = Op *a*_{k} for an elliptic symbol $ak\u2208ESkm$ with 0 ≤ *k* ≤ *d* − 1 and *m* > 0. We constructed in Sec. II, with operator *H*_{d−1} = Op *a*_{d−1} confined in all but the last variable and a Fredholm operator *F* = Op *a* confined in all variables. Associated to *H*_{d−1} is the edge conductivity *σ*_{I} defined in (11). We associated two topological charges to *F* and *σ*_{I} in Theorems 2.3 and 3.2, respectively. This section shows that the two classifications are in fact equivalent and that we have the following conservation of the topological charge:

*Let*$a\u2208ES\u0303m$

*. Then we have*

As a first step, we continuously deform $ad\u22121(xd\u22121\u2032,xd,\xi )$ to $ad\u22121(xd\u22121\u2032,0,\xi )$ using Lemma 3.5 and the paragraph that follows it. Note that all terms in (14) are stable under this change of symbols (see Remark 2.5). We next use the approximation of a symbol in $ad\u22121\u2208ESd\u22121m=ESd\u22121m(gs)$ by $ad\u22121\u2208ESd\u22121m(gi)$ using Lemma 3.3. Note that again, all terms in (14) are stable under this change of symbols since both symbols agree on the support of $SR2d\u22121$.

To simplify the presentation, we change notation to $(y,x)=(xd\u22121\u2032,xd)$ and to (*ζ*, *ξ*) = *ξ* with the new $\zeta ,y\u2208Rd\u22121$ and $\xi ,x\u2208R$. Therefore, the symbol *a*_{d−1} = *a*_{d−1}(*y*, *ζ*, *ξ*).

We finally continuously deform *a*(*X*) to *a*_{d−1}(*y*, *ζ*, *ξ*) − *ix*. This does not change the definition of *σ*_{I}. Using the homotopy result of Remark 2.5, this does not change the integral over $SR2d\u22121$ either by continuously deforming the mass term $md(X)=\u27e8xd\u27e9m\u22121xd$ to *x*_{d} ≡ *x* (using $\u27e8xd\u27e9t(m\u22121)xd$ for *t* ∈ [0, 1]) on the ball of radius *R* while preserving the continuity of *a*^{−1} on $SR2d\u22121$.

For the same reason, we may replace *a* by *a*_{d−1}(*y*, *ζ*, *ξ*) + *α* − *ix* for any $\alpha \u2208R$ by continuity in *α* and for a fixed radius *R* for *α* in a compact domain.

*a*

_{d−1}does not depend on

*x*, we introduce the partial spectral decomposition $H\u0302=H\u0302[\xi ]$ such that

*Y*= (

*y*,

*ζ*) and a Weyl quantization in the variables

*Y*for each parameter $\xi \u2208R$. We thus obtain, using the trace-class properties of Lemma 3.1 providing traces as integrals of Schwartz kernels along diagonals, that

_{y}denotes the integration in all variables but

*x*=

*x*

_{d}(using Fubini), and

*H*(

*x*−

*x*′) is the dependence in (

*x*,

*x*′) of the Schwartz kernel of the operator

*H*[the above duality product is well defined since

*x*→

*φ*′(

*H*)(

*x*) is smooth] and we used

*x*=

*x*

_{d}with the imaginary part

*ω*. The dual variable

*ξ*is considered as another parameter, and pseudo-differential operators and semiclassical operators are now defined in the variables

*Y*= (

*y*,

*ζ*). We denote by

*σ*

_{z}(

*y*,

*ζ*,

*ξ*) =

*z*−

*a*

_{d−1}(

*y*,

*ζ*,

*ξ*) =

*λ*−

*a*(

*y*,

*ω*,

*ζ*,

*ξ*) the symbol of $z\u2212H\u0302[\xi ]$.

We now introduce the semiclassical parameter 0 < *h* ≤ 1 and the operator $H\u0302h$ with the symbol *a*_{d−1}(*y*, *hζ*, *ξ*). Using the semiclassical notation recalled in the Appendix (in the phase-space variable *Y*), we thus observe that $z\u2212H\u0302h=Oph\sigma z$. Using Lemma A.1, we know that $(z\u2212H\u0302h)\u22121=Ophrz$ is a PDO with the semiclassical symbol *r*_{z}(*y*, *ζ*, *ξ*; *h*). Note that the latter term has a complicated dependence on *h*. We know from Lemma 3.4 that *σ*_{I}(*H*_{h}) is independent of 0 < *h* ≤ 1. We may, therefore, compute it in the limit *h* → 0.

*h*

_{0}small enough] that $\phi \u2032(H\u0302)$ is a PDO and define

*s*such that $\u2202\xi H\u0302\phi \u2032(H\u0302)=Ops$. We, therefore, obtain from Lemma 3.1 that

*ζ*→

*hζ*with $\u2202\xi H\u0302h\phi \u2032(H\u0302h)=Ophs$ with now

*s*(

*y*,

*ζ*,

*ξ*;

*h*), we have

*ς*be such that $\phi \u2032(H\u0302h)=Oph\u03c2$ while $\u2202\xi H\u0302h=\u2212\u2202\xi \sigma z$. Then, by application of the semiclassical composition calculus and of the Helffer–Sjöstrand formula (A12), we have

*j*= 1, this is {

*a*,

*b*} the standard Poisson bracket. The term

*O*(

*h*

^{N+1}) in (16) is given explicitly in the proof of Ref. 58, Theorem 4.12 and from Ref. 58, Theorem 4.18 is bounded in

*S*

^{0}(

*M*

_{1}

*M*

_{2}) if

*a*∈

*S*

^{0}(

*M*

_{1}) and

*b*∈

*S*

^{0}(

*M*

_{2}).

*h*

_{0}small enough):

*h*≤

*h*

_{0}. Moreover, since $H\u0302h$ has a semiclassical symbol that does not depend on

*h*, we have from semiclassical calculus (see Ref. 25, Chap. 8, p. 102 and Ref. 58) the decomposition

*S*

^{0}(⟨

*Y*,

*ξ*⟩

^{−N}) for any

*N*(uniformly in 0 <

*h*≤

*h*

_{0}). We thus observe that

*s*

_{d−1}. In addition, from the functional calculus, we have

*ρ*

_{z}(

*Y*,

*ξ*;

*h*) is uniformly bounded in 0 <

*h*≤

*h*

_{0}and (

*Y*,

*ξ*) ∈ [−R,R]

^{2d−1}by

*C*|

*ω*|

^{−q}for some finite

*q*, we obtain from the property of the almost analytic extension $|\u2202\u0304\phi \u0303\u2032|\u2264C|\omega |q$ that the first equality in (18) indeed holds. The bound (18) is essentially derived in Ref. 25, Chap. 8, and is obtained as follows: from the expansion (16) in

*σ*

_{z}♯

_{h}

*r*

_{z}=

*I*and the estimate $|\u2202\xi ,Y\alpha rz|\u2264C\alpha |\omega |\u2212p\u2212|\alpha |$ for some finite

*p*uniformly in 0 <

*h*≤

*h*

_{0}and $(\xi ,Y)\u2208R2d\u22121$ coming from Ref. 25, Proposition 8.6 [see also (A10) in the Appendix], the error term

*ρ*

_{z}(

*Y*,

*ξ*;

*h*) in the Taylor expansion in

*h*of

*r*

_{z}is also bounded by

*C*|

*ω*|

^{−q}for some finite

*q*.

*σ*

_{z}♯

_{h}

*r*

_{z}=

*r*

_{z}♯

_{h}

*σ*

_{z}=

*I*, and using the expansion (16),

*ω*≠ 0. Then a higher-order equation can be solved iteratively for

*r*

_{zj}. The next two equations are, for instance,

*r*

_{zj}(

*Y*,

*ξ*) is a product of a maximum of 2

*j*+ 1 terms alternating a derivative (possibly of order 0) of

*σ*

_{z}with one (possibly of order 0) of $\sigma z\u22121$. The same property holds for

*τ*→

*ητ*for

*τ*one variable in (

*Y*,

*ξ*) → (

*Y*

_{η},

*ξ*

_{η}). By Lemma 3.4, the trace is independent of

*η*when $H\u0302$ now has symbol

*a*

_{d−1}(

*Y*

_{η},

*ξ*

_{η}). Let

*s*

_{d−1}(

*Y*,

*ξ*;

*η*) be the corresponding symbol appearing in the trace calculation. We thus have

*β*

_{j}∈ { −1, 1} and

*γ*is the number of derivatives in the variable

*τ*that appear in

*s*

_{d−1}. Integrating the latter expression over $R2d\u22121$ and changing variables (

*Y*

_{η},

*ξ*

_{η}) → (

*Y*,

*ξ*) shows that necessarily

*γ*= 1 in order for 2

*πσ*

_{I}to be independent of

*η*. This implies that exactly one differentiation in each of the variables (

*Y*,

*ξ*) appears in the terms that contribute to the integral of

*s*

_{d−1}. Therefore,

*∂*

_{ξ}

*σ*

_{z}is the only term involving a derivative in

*ξ*. In addition, any term in (19) with

*k*≥ 2 does not contribute to the trace defining

*σ*

_{I}. Therefore, only

*j*=

*d*− 1 with

*k*= 0 and

*j*=

*d*− 2 with

*k*= 1 remain in (19). We will see that the terms

*k*= 1 are in

*divergence form*and do not contribute eventually. However, they cannot be discarded purely by homogeneity. We thus obtain

*z*→

*τ*is analytic when

*ω*≠ 0 since

*z*→

*σ*

_{z}and $z\u2192\sigma z\u22121$ are analytic so that $\u2202\u0304\phi \u0303\u2032(z)tr\tau =\u2202\u0304(\phi \u0303\u2032(z)tr\tau )$. Using $\u2202\u0304=12(\u2202\lambda +i\u2202\omega )$ and the fact that $\phi \u0303\u2032(z)$ is compactly supported while $\phi \u0303\u2032(\lambda )=\phi \u2032(\lambda )$ on the real axis, we integrate by parts on

*ω*> 0 and

*ω*< 0 to obtain that

*τ*

_{1}is of the form {

*a*,

*b*} and hence is in divergence form. Therefore, after integration in

*Y*, we obtain a term supported on the boundary of [−R, R]

^{2d−1}, where by assumption $\sigma z\u22121(\xi ,Y)$ is defined and $\tau 1|\lambda \u2212i0\lambda +i0$ vanishes there.

*τ*

_{0}= −

*∂*

_{ξ}

*σ*

_{z}

*r*

_{z(d−1)}that involve exactly one differentiation in each variable

*Y*. Since, again, {·,·}

^{j}applies

*j*derivatives, only

*j*= 1 contributes to the trace integral, which we obtain iteratively,

*y*

_{k},

*ζ*

_{k}) appears at most once. Denote $cj=(i2)j$. We have for

*k*=

*d*− 1 the expression

*d*− 1}. The edge conductivity is given, using the cyclicity of the trace, by

*d*− 1)-form on $R2d$ [in the variables (

*ω*,

*ξ*,

*Y*)]. However, it is not quite $(\sigma \u22121d\sigma )2d\u22121$ yet. This is where we use the invariance of 2

*πσ*

_{I}against rotations of the variables

*Y*stated in Lemma 3.4.

*σ*=

*σ*(

*x*) in generic variables in

*n*dimensions, we have

*ρ*: (1, …,

*n*) → (

*ρ*

_{1}, …,

*ρ*

_{n}). By cyclicity of the trace, the term $\sigma z\u22121\u2202\xi \sigma z$ can always be brought to the left of the product. However, ${\sigma z\u22121,\sigma z}fd\u22121$ involves a summation over only specific permutations of the variables

*Y*. It is where having a symbol in an isotropic class with

*g*=

*g*

^{i}is used. From Lemma 3.4, any rotation in the variables

*Y*does not change

*σ*

_{I}, so that any permutation of the variables in

*Y*with a positive determinant leads to the same

*σ*

_{I}, and any permutation with a negative determinant leads to −

*σ*

_{I}.

*y*

_{j},

*ζ*

_{j}) permuted come with negative orientation. We thus find that

*d*− 2 and (−1)

^{ρ}is the signature of the permutation. Combining the permutations generating ${\u22c5,\u22c5}fd\u22121$, each term $\u220fj=12d\u22122\sigma z\u22121\u2202\rho j(Y)\sigma z(\u22121)\rho $ appears

*γ*

_{d}= 2

^{d−1}(

*d*− 1)! times, where 2

^{d−1}comes from the difference of products in each Poisson bracket and (

*d*− 1)! from the possible permutations of the variables.

*c*

_{d−1}2

^{d−1}=

*i*

^{d−1}yields

*λ*be fixed (in a compact interval since

*φ*′ has compact support). The form $tr(\sigma z\u22121d\sigma z)2d\u22121$ is closed since we verify that $d(tr(\sigma z\u22121d\sigma z)2d\u22121)=0$ as a 2

*d*-form in the variables (

*ω*,

*ξ*,

*Y*) (see, for instance, Ref. 40, p. 220). The integral in (22) at fixed

*λ*is over the closed 2

*d*− 1 surface

*S*= {0 +} × [−R, R]

^{2d−1}∪ {0 −} × [−R, R]

^{2d−1}. Note that on

*∂*[−R, R]

^{2d−1}, $\sigma z\u22121$ is defined so that $tr(\sigma z\u22121d\sigma z)2d\u22121\lambda \u2212i0\lambda +i0=0$ there for R large enough. By the Stokes theorem, the integral remains unchanged if the domain of integration is deformed from the surface

*S*to the sphere $SR$ of radius

*R*sufficiently large in the variables (

*ω*,

*ξ*,

*Y*), since $\sigma z\u22121$ is continuously defined on the volume with boundary given by $S\u222aSR$.

The integral on $SR$ is also independent of *λ* in a compact interval since *λ* → *σ*_{z} is continuous as an application of the result in Remark 2.5. Since $\u222bR\phi \u2032(\lambda )d\lambda =1$ and *σ*_{α+iω}(*ξ*, *Y*) = −*a*(*ω*, *ξ*, *Y*), the integrals in (14) and (22) agree modulo a sign. Upon inspection, we observe that the integral in (22) has been computed for the orientation *dx*_{d} ∧ *dξ*_{d} ∧ *dx*_{1} ∧ *dξ*_{1}…*dx*_{d−1} ∧ *dξ*_{d−1} > 0 with *ω* ≡ *x*_{d} the first variable defining the surface *S*. The latter orientation is (−1)^{d}*dξ*_{1} ∧ *dx*_{1} ∧ ⋯∧ *dξ*_{d} ∧ *dx*_{d}. With the latter choice of orientation, we obtain the topological charge conservation between the topological charge given by the index of *F* and the transport asymmetry given by the conductivity 2*πσ*_{I} stated in (14).

In dimension *d* = 2 and in the setting of bounded domain walls, the Fedosov–Hörmander formula may be interpreted as a difference of *bulk* quantities since the integral over the sphere $SR3$ may be deformed into the integral over two hyperplanes in the bulk phases where *x*_{1} = ±*R* constant. This also uses as in the above derivation that $d(tr(a\u22121da)\u22273)=0$ and the Stokes theorem (see Ref. 6). The topological charge conservation of Theorem 4.1 thus generalizes the two-dimensional bulk-interface correspondence to an arbitrary space dimension.

## V. GENERALIZED DIRAC OPERATORS AND DEGREE THEORY

We assume in this section that $ak\u2208ESkm$ has the following form:

where h^{k}(*X*) is a (*d* + *k*)-dimensional vector field on $X\u2208R2d$ and Γ_{k} is a collection of matrices in a representation of the Clifford algebra $Clnk(C)$; see, e.g., Ref. 47 for details on Clifford algebras and their central role in the analysis of topological insulators. The objective of this section is to show that the index of the Fredholm operator *F* associated with *H*_{k} = Op *a*_{k} may be computed as a topological degree associated with the map h^{k}. This simplifies the estimation of the integral (10).

For 0 ≤ *k* ≤ *d*, let $\kappa \u2254\kappa k=\u230ad+k2\u230b$ and $nk=2\kappa k$. The matrices $\Gamma k=(\gamma \kappa j)j$ for 1 ≤ *j* ≤ *d* + *k* are constructed to satisfy the commutation relations

These properties imply that $ak2=|hk|2Ink$ is proportional to identity. The matrices Γ_{k} may be defined explicitly as follows: the matrices $\gamma \kappa j$ at level *κ* are constructed starting from $\gamma 11,2,3=\sigma 1,2,3$ the standard Pauli matrices and then iteratively as

The last matrix plays the role of the chiral symmetry matrix in even dimension *d* + *k* = 2*κ*_{k}. The construction of the augmented Hamiltonians *H*_{j} for *k* < *j* ≤ *d* in Sec. II mimics the construction of the above matrices. When *d* + *k* is even, the chiral symmetry is implemented as

For *a*_{k} = h^{k} · Γ_{k}, we denote by h^{j} for *k* ≤ *j* ≤ *d* the vector fields of dimension *d* + *j* such that the augmented Hamiltonians constructed in Sec. II satisfy *H*_{j} = Op *a*_{j} with, as we verify, *a*_{j} = h^{j} · Γ_{j}.

Dirac operators are the prototypical example of operators in the form (23). In two dimensions, we have explicitly Γ_{0} = (*σ*_{1}, *σ*_{2}) while h^{0}(*X*) = (*ξ*_{1}, *ξ*_{2}) and Γ_{1} = (*σ*_{1}, *σ*_{2}, *σ*_{3}) while h^{1}(*X*) = (*ξ*_{1}, *ξ*_{2}, *x*_{1}). In dimension *d* = 3, we have Γ_{0} = (*σ*_{1}, *σ*_{2}, *σ*_{3}) while h^{0}(*X*) = (*ξ*_{1}, *ξ*_{2}, *ξ*_{3}), next Γ_{1} = (*σ*_{1} ⊗ *σ*_{1}, *σ*_{1} ⊗ *σ*_{2}, *σ*_{1} ⊗ *σ*_{3}, *σ*_{2} ⊗ *I*_{2}) while h^{1}(*X*) = (*ξ*_{1}, *ξ*_{2}, *ξ*_{3}, *x*_{1}), and finally Γ_{2} = (*σ*_{1} ⊗ *σ*_{1}, *σ*_{1} ⊗ *σ*_{2}, *σ*_{1} ⊗ *σ*_{3}, *σ*_{2} ⊗ *I*_{2}, *σ*_{3} ⊗ *I*_{2}) while h^{2}(*X*) = (*ξ*_{1}, *ξ*_{2}, *ξ*_{3}, *x*_{1}, *x*_{2}). When *d* = 3, then *κ*_{0} = 1 while *κ*_{1} = *κ*_{2} = 2 for a maximum of matrices satisfying (24) equal to 2*κ*_{2} + 1 = 5. Several other examples will be presented in Sec. VI.

### A. Topological charge computation

For elliptic operators that admit the Clifford representation (23), the explicit computation of the index in (10) significantly simplifies as does the computation of the degree of the map h^{k}.

We recall the definition of the degree of a map following Ref. 46, Chaps. 1.3 and 1.4; see also Ref. 28, Chaps. 13 and 14. Let *C* be an open set in $Rn$ with compact closure $C\u0304=C\u222a\u2202C$. Let $h:C\u0304\u2192Rn$ be a sufficiently smooth map such that |h(*ζ*)| > 0 for *ζ* ∈ *∂C*. There are regular values *y*_{0} of h arbitrarily close to 0 by Sard’s theorem that allow us to define the degree of h as

The above sum ranges over a finite set and is independent of the regular value *y*_{0} in an open vicinity of 0.

The definition of the index of a map from a manifold M to another manifold N depends on the chosen orientation on M. We consider two natural orientations in the context of topological insulators. Let $Bd\u2282R2d$ the ball of radius *R* given by {|*X*| ≤ *R*}. By ellipticity assumption, |h^{d}| > 0 on *∂B*_{d} for *R* large enough. We now define degrees for h^{d} with two possible orientations,

We observe that

The degree $deg\u0303$ is naturally related to Index *F*, while the degree deg is more naturally related to that of h^{k} as we now describe.

Using Lemma 3.5, we obtain that the index is unchanged if $hk(xk\u2032,xk\u2032\u2032,\xi )$ is replaced by $hk(xk\u2032,\xi )\u2254hk(xk\u2032,0,\xi )$ in the definition of the symbol. We may, therefore, interpret h^{k} as a map from $Rd+k$ to $Rd+k$ such that, thanks to the ellipticity constraint, |h^{k}| ≥ *h*_{0} > 0 for $|(xk\u2032,\xi )|\u2265R$. Let $Bk={|(xk\u2032,\xi )|<R}$. We define

The orientation of *B*_{k} is inherited from that of *B*_{d} as the subset $xk\u2032\u2032=0$. With these definitions, we obtain the main result of this section.

*We have*

In other words, Index *F* = deg(h^{k}) when *d* = 1, 2 mod 4 and Index *F* = −deg(h^{k}) when *d* = 3, 4 mod 4.

The rest of this section is devoted to the proof of the theorem. Its main steps are as follows: (i) write (10) in terms of *σ*_{d}; (ii) next in terms of h^{d}; (iii) identify (10) with the degree of h^{d} on the sphere $SR$; (iv) identify it with the degree $deg(hd,B\u0304d,0)$; (v) decompose $hd=(hk,h\u0303)$ with $h\u0303$ the augmentation map. The degree of h^{d} is then the product of the other two degrees. Now the degree of $h\u0303$ is one and this gives the result.

*We have*

*a*=

*a*

_{d−1}−

*im*

_{d}and

*a*

_{d}=

*σ*

_{+}⊗

*a*+

*σ*

_{−}⊗

*a**. Therefore, $ad\u22121dad=Diag(a\u2212*da*,a\u22121da)$ and hence $(ad\u22121dad)2d\u22121=Diag((a\u2212*da*)2d\u22121,(a\u22121da)2d\u22121)$. Therefore, with $\gamma d0=\sigma 3\u2297I$

*F*, while the integral of the first term gives Index

*F** = −Index

*F*since

*a** is the symbol of

*F**. This gives the result.

*We have,*

*k*≤

*j*≤

*d*is proportional to identity thanks to (24). Since $ad2=|hd|2$ is scalar, then $ad\u22121=wad$ for $w=ad\u22122$ scalar so that

*dwda*

_{d}

*a*

_{d}= −

*dwa*

_{d}

*da*

_{d}, we find $(ad\u22121dad)4=dw2ad(dad)3+w2(dad)4$ and more generally

*d*≥ 2),

*d*≥ 1 that

*dw*vanishing since the it involves the trace of a product of an even number of necessarily different (because of the product of exterior differentiations) gamma matrices. Such traces necessarily vanish for Clifford matrices as one verifies from their construction (25).□

*For*

*a*

_{d}= h

^{d}· Γ

_{d}

*, we have*

We recall that $Sn$ is the set of permutations of {1, …, *n*}. The proof of the lemma directly comes from the construction of the Clifford matrices in (25) (and their orientation) and generalizes that tr*σ*_{3}*σ*_{1}*σ*_{2} = 2*i*. The above three lemmas show that the index of *F* is related to an appropriate integral of h^{d}.

*We have,*

*X*=

*X*(

*u*) for $u\u2208R2d\u22121$. We introduce the 2

*d*× 2

*d*matrix

*L*(

*u*) constructed as follows (see Ref. 28, Corollary 14.2.1). The first row is $L1i(u)=hid\u25e6X(u)$ while the following rows are $Lj+1,i(u)=\u2202ujhid\u25e6X(u)$ for 1 ≤

*j*≤ 2

*d*− 1. We then observe that

_{u}h′ the Jacobian matrix, we have

*ρ*, we observe that (−1)

^{ρ}and Det∇

_{u}h′ change signs together so that denoting by $L\rho 1$ the matrix

*L*with the first row and the

*ρ*

_{1}column deleted, we have

*d*− 1)! is the number of permutations in $Sd\u22121$.

*γ*

_{2d−1}is the volume of the unit sphere $S2d\u22121$.

Let *f* be the Gauss map associated with h^{d} and given by *f*(*X*) = |h^{d}(*X*)|^{−1}h^{d}(*X*) for $X\u2208SR2d\u22121$. Then we recognize in the integration of the right-hand side of (31) over $SR2d\u22121$ the degree of *f* (Ref. 28, Corollary 14.21). Moreover, the degree of the Gauss map *f* is given in Ref. 28, Theorem 14.4.4 precisely by the sum in (26) and so equals $deg(hd;B\u0304d,0)$ when 0 is a regular value of h^{d}. When 0 is not a regular value, we apply the result of h^{d} − *y*_{0} for *y*_{0} small with the result independent of *y*_{0}; see also Ref. 46, Remark 1.5.10. With the chosen orientation to define $deg\u0303$ and Theorem 2.3, we thus obtain the result of the lemma.

*We have* deg(h^{d}) = deg(h^{k})*.*

*ζ*

_{j}↦ h

_{j}(

*ζ*

_{j}) for

*j*= 1, 2 be two smooth functions from $Rnj$ to itself with |h

_{j}(

*ζ*

_{j})| ≥

*c*

_{0}> 0 for |

*ζ*

_{j}| ≥

*R*

_{j}. Let

*B*

_{j}=

*B*

_{j}(0,

*R*

_{j}) be the centered balls of radius

*R*

_{j}for the Euclidean metric in $Rnj$ for

*j*= 1, 2. Let now (

*ζ*

_{1},

*ζ*

_{2}) =

*ζ*↦ h(

*ζ*) be the function from $Rn$ to itself with

*n*=

*n*

_{1}+

*n*

_{2}defined by

*ζ*)| ≥

*c*

_{0}> 0 for

*ζ*∈

*∂C*, where

*C*=

*B*

_{1}×

*B*

_{2}. Since 0 does not belong to the range of h or h

_{j}on the respective boundaries, we can define the degrees

*y*

_{0}be a regular value of h, i.e., a point in $h(C\u0304)\h(\u2202C)$ such that $h\u22121(y0)={\zeta \u2208B\u0304R;h(\zeta )=y0}$ are a finite number of isolated regular points (where ∇h is invertible). Note that

*y*

_{0}= (

*y*

_{1},

*y*

_{2}) with

*y*

_{j}∈

*B*

_{j}. By Sard’s theorem, such regular values exist. Then, independently of such a

*y*

_{0},

*J*

_{h}is the non-vanishing Jacobian of the map

*ζ*→ h(

*ζ*). Now, by construction,

*ζ*) =

*y*

_{0}means (h

_{1}(

*ζ*

_{1}), h

_{2}(

*ζ*

_{2})) = (

*y*

_{1},

*y*

_{2}) so that $h\u22121(y0)=h1\u22121(y1)\xd7h2\u22121(y2)$ and hence

*C*⊂

*B*

_{R}the ball of radius $R=R12+R22$. Since |h| ≥

*c*

_{0}> 0 on $B\u0304R\C$, invariance of the results with respect to (continuous) domain changes (see Ref. 46, Proposition 1.4.4) shows that

We now choose h_{1} = h^{k} and h_{2} the vector so that h^{d} = (h^{k}, h_{2}) with *n*_{1} = *d* + *k* and *n*_{2} = *d* − *k* where $R2d$ is oriented using *dξ*_{1}…*dξ*_{d}*dx*_{1}…*dx*_{d} > 0 and the subspaces $Rd+k$ (for h^{k}) and $Rd\u2212k$ (for h_{2}) with the induced orientation. We observe that the degree of $h2=h2(xk\u2032\u2032)=(mk+1,\u2026,md)(xk\u2032\u2032)$ equals 1 since the only point in $h2\u22121(0)=0$ and the Jacobian is identity there with the above orientation. Using (32) and the definitions (27) and (29) proves the result.□

The above lemmas together with the change of orientation relation (28) conclude the Proof of Theorem 5.1.

## VI. APPLICATIONS

The classification presented in Sec. II applies to Hamiltonians that are (a) *continuous* (with an open “Brillouin” zone $\xi \u2208Rd$), (b) defined on Euclidean space $Rd$, and (c) appropriately *elliptic* with a symbol that tends to infinity at infinity in the variables (*x*′, *ξ*). Besides these constraints, the Hamiltonians are general when *d* + *k* is odd and to Hamiltonians with a chiral symmetry when *d* + *k* is even. The Fedosov–Hörmander formula (10) shows that the index is controlled by the symbol *a* of *F*, and hence that of *H*_{k}, restricted to any sphere with a sufficiently large radius *R*. This implies that the topological charge is independent of the symbol *a* in the complement of that ball. The main assumptions to apply (10) are that: (i) the symbol *a* of *F* is uniformly invertible for |*X*| ≥ *R* for some *R* > 0, in which case: (ii) the topological charge solely depends on *a* restricted to the sphere |*X*| = *R*.

The theory of Sec. II applies only to operators whose symbols satisfy the ellipticity constraint (5), which combined with the growth condition (4) implies that the symbol *a* grows to infinity as |*X*| → ∞ with the same homogeneity in all phase-space variables. This should be contrasted to the two-dimensional results in Refs. 6 and 49, where the domain wall *m*(*x*_{1}) is assumed to be bounded and constant away from a compact domain.

For any symbol *a*_{k} such that (i) and (ii) hold, we allow for the following modifications of the symbol *a*_{k} in order to apply the theory of Sec. II. Let *ɛ* > 0 and *r* → ⟨*r*⟩_{ɛ} a smooth non-decreasing function from $R+$ to $R+$ such that

We use the same notation for the smooth function $Rp\u220by\u2192\u27e8y\u27e9\epsilon \u2254\u27e8|y|\u27e9\epsilon $. This function has the same leading asymptotic behavior as ⟨*ɛy*⟩ for |*y*| → ∞. We consider the above regularization for *y* being one or several of the variables in *X*. Such modifications of *a*_{k} preserve (i)-(ii) and allow us to satisfy (5) as well as (4) so that the theory of Sec. II applies.

Consider, for instance, the regularized “Dirac” operator *H*_{1} = *D*_{1}*σ*_{1} + *D*_{2}*σ*_{2} + (*μ* − *ηD* · *D*)*σ*_{3} with here $D\u22c5D=D12+D22$ the (positive) Laplacian and $R\u220b\eta \u22600$. The definition of a bulk invariant is ambiguous when *η* = 0, while it yields a Chern number $12(sign(\mu )+sign(\eta ))$ when *η* ≠ 0.^{3,11} As mentioned a number of times already in this paper, we do not consider bulk invariants but rather topological charges and interface invariants, which in two space dimensions may be related to bulk-difference (rather than bulk) invariants.^{6} To define a topology in the class of symbols analyzed in this paper and satisfy (5), we modify the above Hamiltonian as

where we assume that *μ*(*x*_{1}) equals *x*_{1}, say, outside of a compact set in $R$. We will verify below that the topological charge of *H*_{1} equals 1 and is independent of the regularization terms *ɛ* and *η* as expected since *η* affects the bulk invariants but not the bulk-difference invariant.^{3} Note that (i) and (ii) now hold with *m* = 1. Alternatively, we could introduce $H1=\u27e8D\u27e9\epsilon (D1\sigma 1+D2\sigma 2)+(\u27e8x1\u27e9\epsilon x1\u2212\eta D\u22c5D)\sigma 3$ satisfying (i) and (ii) with now *m* = 2.

We next consider several examples of topological insulators and superconductors in dimensions *d* = 1, 2, and 3^{11,48,51,52,56} where the theories of both Secs. II and V apply. We refer to Ref. 9 for an application to Floquet topological insulators, where a variation on Theorem 4.1 is used to compute invariants for operators that are not in the form (23).

While the theory leading to Theorems 4.1 and 5.1 applies to a large class of practical settings, as the rest of this section illustrates, it does not apply to situations where the ellipticity conditions are not met. This is the case for confinements generating flat bands, for instance in Schrödinger or Dirac equations with magnetic fields, which may be analyzed by other techniques.^{26,50} This is also the case for the 3 × 3 Hamiltonian (39) describing fluid waves before regularization. See (39) and the following paragraphs for a regularized version.

### A. Dirac operator

The first example is the Dirac operator with *H*_{0} = Op *a*_{0} for $a0(X)=h0(X)\u22c5\Gamma 0\u2208ES01(gs)$ in dimension *d*, where

and Γ_{0} are Clifford matrices acting on spinors in $C2\kappa 0$ with $\kappa 0=\u230ad2\u230b$. These generalize the cases *d* = 1, 2, and 3 considered in the introduction. We then observe from (26) that deg h^{0} = 1 since $(h0)\u22121(0)={0}$ and ∇h^{0}(0) = *I*_{d}, and that the topological charge of *H*_{0} is given by $IndexF=2\pi \sigma I(Hd\u22121)=(\u22121)12d(d+1)+1$, i.e., Index *F* = 1 in dimensions 1, 2 mod 4 and Index *F* = −1 in dimensions 3, 4 mod 4.

The topological charge is given by the degree of h^{0} or by that of h^{d} = (*ξ*_{1}, …, *ξ*_{d}, *x*_{1}, …, *x*_{d}) since the deg h^{0} = deg h^{d} = 1.

If *A* is a non-singular (constant) matrix in $Md(R)$ and we consider instead the operator *H*_{0} = Op (*A*h^{0})(*ξ*) · Γ_{0}, then we find that Index *F* = sign det *A* in dimensions 1, 2 mod 4 and Index *F* = −sign det *A* in dimensions 3, 4 mod 4.

The topological charge is also stable against large classes of smooth perturbations of arbitrary amplitude, so long as the perturbed symbol remains appropriately elliptic. Perturbations need to be smooth in order to apply the PDO techniques used in Secs. II and III. It is possible to use the stability of indices of Fredholm operators against compact perturbations and consider less regular perturbations as well, although we will not do so here.

For instance, we may consider $H0=Oph\u03030\u22c5\Gamma 0$ with $h\u0303j0(X)=bj(x)hj0(\xi )$ and *b*_{j}(*x*) smooth, bounded below and above by positive constants, and say equal to 1 outside of a compact set in $Rd$. Then we verify that the corresponding symbol $h\u03030\u22c5\Gamma 0\u2208ES01(gs)$ though not necessarily in $ES01(gi)$. Note that a more isotropic perturbation of the form $bj(X)hj0$ for *b*_{j}(*X*) smooth and equal to 1 outside of a compact set in $R2d$ would generate a perturbation in $ES01(gi)$ although one that is no longer a differential operator. This illustrates the reason why we considered the (reasonably large) classes $Skm(gs)$.

The model Hamiltonian in the presence of one domain wall is h^{1}(*X*) = (*ξ*_{1}, …, *ξ*_{d}, *x*_{1}). Based on Theorem 5.1, its topological charge is again given by deg h^{1} = 1. Domain walls of the form *b*_{j}(*x*)*x*_{1} even with *b*_{j}(*x*) = 1 outside of a compact domain no longer necessarily generate perturbations such that *a*_{1} remains in $S11$ and are, therefore, not allowed in the theoretical framework of this paper. We may, however, replace *x*_{1} by *m*(*x*_{1}) equal to *x*_{1} outside of a compact set. Inside that compact set, the level set *m*(*x*_{1}) = 0 is then arbitrary.

For a time-dependent picture of how wavepackets propagate along curved interfaces for two-dimensional Dirac equations, see also Refs. 5, 7, and 8.

For concreteness and illustration, we spell out some details of the calculations of the indices in Theorems 2.3 and 5.1 when *d* = 1. We then have *a*_{0} = *ξ* = h^{0} while *a* = *ξ* − *ix* and *a*_{1} = *ξσ*_{1} + *xσ*_{2}. We then observe that $a\u22121da=(\xi 2+x2)\u22121(\xi d\xi +xdx+i(\xi dx\u2212xd\xi ))$. In polar coordinates *ξ* = *r* cos *θ* and *x* = *r* sin *θ*, we observe that *a*^{−1}*da* = *r*^{−1}*dr* − *idθ* whose integral along the curve *r* = 1, gives −2*πi*, and hence Index Op *a* = 1 as a direct application of the Fedosov–Hörmander formula (10). We now observe that the index may be computed as in Lemma 5.2 from *a*_{1} = *ξσ*_{1} + *xσ*_{2} with $tr\sigma 3a1\u22121da1=a\u2212*da*\u2212a\u22121da$ and *a** = *ξ* + *ix* so that *a*^{−}**da** = *r*^{−1}*dr* + *idθ*. This shows that $tr\sigma 3a1\u22121da1=2id\theta $ whose appropriate integral gives the topological charge. Now, $a1\u22121=|h1|\u22122(h11\sigma 1+h21\sigma 2)$ for h^{1} = (*ξ*, *x*). Therefore, as in Lemma 5.4, $tr\sigma 3a1\u22121da1=|h1|\u22122tr\sigma 3(h11\sigma 1+h22\sigma 2)(dh11\sigma 1+dh21\sigma 2)=2i(h11dh21\u2212h21dh11)$. We recognize in the integral of the latter form over the circle an expression for the degree of h^{1} written as the degree of the Gauss map, which to $X\u2208S1$ associates h^{1}(*X*)/|h^{1}(*X*)|. Using the expression (26) of the degree over the unit disk *C* gives deg h^{1} = 1 since ∇h^{1} = *I*_{2} at the unique point *X* = 0 where h^{1} = 0.

The regularized “Dirac” operator considered at the beginning of this section (this is not quite a Dirac operator as $H12$ is not a second-order Laplace operator when *η* ≠ 0) is given by *H*_{1} = Op h^{1} · Γ_{1} with vector field $h1=(\xi 1,\xi 2,x1\u2212\eta \u27e8\xi \u27e9\epsilon \u22121|\xi |2)$ so that $h1\u22c5\Gamma 1\u2208ES11$. We observe that h^{1} = 0 only at the point (0, 0, 0), where the Jacobian is upper-triangular with diagonal entries equal to 1. Applying (26) and Theorem 5.1, we thus obtain that the topological charge of *H*_{1} equals 1 independent of *ɛ* and *η* as advertised.

The above orientation of the vector fields h^{k} is natural in the context of topological insulators or superconductors, which are typically first written for spatially-independent coefficients. A different orientation helps to better display the invariance of the indices of Dirac operators across spatial dimensions (see Ref. 40, Proposition 19.2.9 for a related construction). We start with $F1=D1\u2212ix1=\u2212ia1$ and then define iteratively

The above construction is an example of the more general structure,

where we verify that Index *f*♯*g* = Index *f* Index *g*. We apply it with *g* = *F*_{n−1} and *f* = *D*_{n} − *ix*_{n}. It is then straightforward to obtain that Index *F*_{n} = 1 for all *n* ≥ 1. We then observe that $Fn(1,0,\u2026,0)te\u221212|x|2=0$ with spinor (1, 0, …, 0)^{t} of dimension 2^{n−1}.

### B. Dirac operator with magnetic field

We now incorporate constant magnetic fields at infinity for magnetic potentials written in an appropriate gauge. Let us consider the case *d* = 2 for concreteness and the (minimal coupling) operator,

with *A* = (*A*_{1}, *A*_{2}) the magnetic vector potential and *V* a bounded scalar potential with compact support, say. The magnetic field is given by *B* = ∇ × *A* = *∂*_{1}*A*_{2} − *∂*_{2}*A*_{1}. We choose the Landau gauge such that $A2=B0x1+A\u03032$ and $A1=A\u03031$ for $A\u0303$ an arbitrary (smooth) compactly supported perturbation. In that gauge, we obtain that

is an operator *H*_{1} = Op *a*_{1} with $a1\u2208ES11$ for *n*_{1} = 2. Note that for *H*_{0} = Op *a*_{0}, we do not have that *a*_{0} belongs to $ES01$ because of the presence of the unbounded magnetic potential. We would also not have that *a*_{1} belongs to $ES11$ if $A=(\u221212B0x2,12B0x1)$ were chosen in the symmetric gauge, for instance. While physical phenomena have to be independent of the choice of a gauge, the appropriate functional setting to handle constant magnetic fields, and hence unbounded magnetic potentials, is not. With the above construction, we obtain that 2*πσ*_{I}(*H*_{1}) = Index *F* = 1 for *F* = *H*_{1} − *ix*_{2}, since the topological charge is given by

We could more generally consider a magnetic field with constant and opposite values as *x*_{1} → ±∞, for instance, with $A2=B02\pi arctan(x1)x1$. The topological charge of *H*_{1} remains equal to 1. The magnetic field, therefore, has no influence on the topological charge in this setting.

Such a result should be contrasted with the very different outcome we obtain in Ref. 50 for the same model of magnetic potential but with a bounded domain wall *μ*(*x*_{1}) [with *x*_{1}*σ*_{3} replaced by *μ*(*x*_{1})*σ*_{3} in the above definition of *H*_{1}]. In such a setting, both magnetic and mass confinements compete to generate asymmetric transport. Only when the bounded domain wall *μ*(*x*_{1}) converges to sufficiently large values *μ*_{±} as *x*_{1} → ±∞ (for a fixed magnetic field) do we retrieve that the asymmetric transport of 2*πσ*_{I}(*H*_{1}) equals 1 (see Ref. 50, Theorem 2.1). Whereas replacing a bounded domain wall with an unbounded one is practically irrelevant when mass terms are the only confining mechanism, this is no longer the case when several confining mechanisms are present in the system.

### C. Higher-order topological insulator

Let us consider the Weyl operator *D* · *σ* in dimension *d* = 3. As we mentioned in the introduction, the operator *H*_{2} = *σ*_{1} ⊗ *D* · *σ* + *σ*_{2} ⊗ *Ix*_{1} + *σ*_{3} ⊗ *Ix*_{2} generates a *hinge* in the third direction along which asymmetric transport is possible. With our choice of orientation, we have 2*πσ*_{I}(*H*_{2}) = −deg(*ξ*_{1}, *ξ*_{2}, *ξ*_{3}, *x*_{1}, *x*_{2}) = −1.

By implementing more general domain walls, an arbitrary number of asymmetric modes may be obtained. This is done by considering for $p\u2208Z$,

We thus deduce from Theorem 5.1 that

The last result is most easily obtained by identifying, as we did in the Proof of Lemma 5.5, the degree of h^{2} on the unit ball with the degree of the Gauss map $x\u2192h\u03022=h2/|h2|$ from the unit circle $S1$ to itself and then to the degree (winding number) of the map $x1+ix2\u2192(x1+ix2)p$ from the unit circle to itself, which equals *p*.

By an appropriate construction of the coefficients in the Hamiltonian *H*_{2} in (34) acting on $C4$, we thus obtain a low-energy model for a coaxial cable with an arbitrary number of asymmetric protected modes along the hinge (see, e.g., Ref. 52 for additional details on higher-order topological insulators).

### D. Topological superconductors

Several superconductors and superfluids^{11,56} are modeled by Hamiltonians of the form

with coupling term *H*_{Δ} = ∑_{i,j=1,2}Δ_{ij}(*X*)*σ*_{i} ⊗ *σ*_{j} for scalar operators Δ_{ij} and $\eta =(2m*)\u22121$ for a mass of the quasi-particle *m** > 0. For the above choice of the order parameter^{59} Δ, these Hamiltonians acting on $C4$ separate into two 2 × 2 Hamiltonians (acting on the first and fourth components, and the second and third components, respectively). We now consider several such examples in one and two space dimensions.

### E. One dimensional examples

For *d* = 1, an example with the order parameter Δ proportional to *D*_{x} gives

with $0\u2260\Delta \u2208C$. Let Δ = |Δ|*e*^{iθ} and $g=ei\theta 2\sigma 3$. We then verify that

and so we may assume Δ real-valued. Define $g2=ei\pi 4\sigma 2$ and $g1=ei\pi 4\sigma 1$. We verify that $g1g2\sigma 1,2,3(g1g2)*=\sigma 3,1,2$, so that

This is of the form *σ*_{−} ⊗ *F** + *σ*_{+} ⊗ *F* with $F=(\eta Dx2\u2212\mu )\u2212i\Delta Dx$.

In order for *F* to be a Fredholm operator, we need to introduce a domain wall. This may be achieved in two different ways: it may be implemented by either the chemical potential *μ* = *μ*(*x*) or by the order parameter Δ = Δ(*x*). As we mentioned in the introduction of this section, the symbol of *H*_{1} has to be asymptotically homogeneous for the ellipticity condition (5) to hold and the theories developed in the preceding sections to apply. We thus regularize the operator using $Rp\u220by\u2192\u27e8y\u27e9\epsilon =\u27e8|y|\u27e9\epsilon $ in (33). The regularization does not modify the symbol on compact domains in $R2d$ for 0 < *ɛ* sufficiently small and hence does not affect the computations of the index in (10) and (29).

When *η* > 0, we consider two regularized operators, one with a domain wall in the chemical potential,

and one with a domain wall in the order parameter

We observe that for *H*_{1} = Op *a*_{1}, then $a1\u2208ES1m$ is elliptic with *m* = 1 in the first example and *m* = 2 in the second example. Consider the second case (37). We wish to show that |h^{1}|^{2} ≥ *C*(|*X*| − 1)^{4}. This is clear for |*x*| ≤ 1 and for |*x*| ≥ 1, then |*x*| ≥ *C*⟨*x*⟩_{ɛ} for *C* > 0, so that $|h1|2\u2265(\eta \xi 2\u2212\mu \u27e8x\u27e9\epsilon 2)2+C\xi 2\u27e8x\u27e9\epsilon 2$ for *C* > 0. The latter expression is homogeneous in (*ξ*, ⟨*x*⟩_{ɛ}) and non-vanishing on the unit sphere in these variables. This shows that $a1\u2208ES12$. A similar computation shows that $a1=h1\u22c5\Gamma \u2208ES11$ in (36). Note that we could also have used the following regularization for the first example: h^{1} = (*ηξ*^{2} − *μx*⟨*x*⟩_{ɛ}, Δ*ξ*⟨*ξ*⟩_{ɛ}), in which case $h1\u22c5\Gamma 1\u2208ES12$.

We now compute the topological charges of the regularized operators starting with (36). We observe that h^{1}(*ξ*, *x*) vanishes only at *x* = *ξ* = 0. The Jacobian there has a determinant equal to *μ*Δ. The topological charge of (36) is, therefore, equal to Index *F* = deg h^{1} = sign(*μ*Δ) assuming *μ*Δ ≠ 0. Here and below, *F* is defined as usual by the relation *H*_{1} = *σ*_{−} ⊗ *F** + *σ*_{+} ⊗ *F*.

We next turn to (37), where h^{1}(*ξ*, *x*) vanishes when *ηξ*^{2} = *μ* and *x* = 0. When *μ* < 0, there is no real solution to this equation, and the topological charge vanishes. When *μ* > 0, we have two solutions $\xi =\xb1\mu /\eta $. At these points, the Jacobian matrix ∇h^{1} has components (2*ηξ*; 0; Δ*x*; Δ*ξ*) with a determinant equal to 2*η*Δ*ξ*^{2}. The topological charge of *H*_{1} in (37) is, therefore, equal to Index *F* = deg h^{1} = 2 sign(Δ).

Let us finally consider the asymptotic regime *η* = 0 for a mass term *m** → ∞ and a corresponding Hamiltonian *H*_{1} = −*μσ*_{1} + Δ*D*_{x}*σ*_{2}. A domain wall in the chemical potential is then modeled by *μ*(*x*) = *μx*. We then observe that *H*_{1} = Op *a*^{1} with $a1\u2208ES11$ and a topological charge equal to Index *F* = deg h^{1} = sign(*μ*Δ) as in the setting *η* > 0. A domain wall in the order parameter requires the following regularized Hamiltonian $H1=\u2212\mu \u27e8x\u27e9\epsilon 2\sigma 1+\Delta 12(Dxx+xDx)\sigma 2$, which is, however, gapped for *μ* ≠ 0 and hence topologically trivial.

### F. Two-dimensional examples

We now consider two-dimensional examples of the above superconductor models. The *p* + *ip* (or p-wave) model with an order parameter proportional to momentum, is of the form

We assume here that Δ_{1} and Δ_{2} are real-valued. The case *η* = 0 is a Dirac operator and was treated earlier. We thus assume *η* > 0. A domain wall in the chemical potential is then implemented as

The symbol of this operator is h^{1} · Γ_{1} with $h1(\xi 1,\xi 2,x1)=(\Delta 1\xi 1,\Delta 2\xi 2,\eta \u27e8\xi \u27e9\epsilon \u22121|\xi |2\u2212\mu x1)$. This regularization ensures that $a1\u2208ES11$ is elliptic. We could have defined a regularization in $ES12$ instead with $h1=(\Delta 1\xi 1\u27e8\xi 1\u27e9\epsilon ,\Delta 2\xi 2\u27e8\xi 2\u27e9\epsilon ,\eta |\xi |2\u2212\mu x1\u27e8x1\u27e9\epsilon )$.

It remains to compute the degree of h^{1}. We find that h^{1} = 0 when *ξ*_{1} = *ξ*_{2} = *x*_{1} = 0 and that the Jacobian determinant there is equal to −*μ*Δ_{1}Δ_{2}. The topological charge of the above operator is therefore 2*πσ*_{I}(*H*_{1}) = deg h^{1} = − sign(*μ*Δ_{1}Δ_{2}), which is consistent with Ref. 11.

In Ref. 56, Chap. 22, the domain wall is implemented in the order parameter Δ_{1}, which, after appropriate regularization, gives

The constants Δ_{1}, Δ_{2}, and *μ* are assumed not to vanish and *μ* > 0. The symbol $a1=h1\u22c5\Gamma 1\u2208ES12$ is given by

We have h^{1} = 0 when *ξ*_{2} = 0, *x*_{1} = 0, and $\eta \xi 12=\mu $. At each of the two solutions, the Jacobian of h^{1} is given by

Therefore, the topological charge of *H*_{1} is equal to 2*πσ*_{I}(*H*_{1}) = deg h^{1} = −2 sign(Δ_{1}Δ_{2}) as in Ref. 56, Chap. 22. When *μ* < 0, the find deg h^{1} = 0 again.

In Ref. 56, Chap. 22, a model for a d-wave superconductor is given as

Following Ref. 56, Chap. 22, we implement a domain wall in Δ_{1}(*x*_{1}) and a regularization that gives the operator

This generates a symbol $a1=h1\u22c5\Gamma 1\u2208ES12$ as may be verified. Then h^{1} = 0 when $\xi 12=\xi 22=\mu 2\eta $ while *x*_{1} = 0. At each of these four roots, we compute

The sign of the Jacobian is the same at each of the roots so that deg h^{1} = 4 sign(Δ_{1}Δ_{2}). Therefore, we obtain a topological charge 2*πσ*_{I}(*H*_{1}) = deg h^{1} = 4 sign(Δ_{1}Δ_{2}) when both *μ* > 0 and *η* > 0. When *μ* < 0, the operator is gapped and topologically trivial again.

### G. Three-dimensional example

Following Ref. 11, (17.24), we consider the time-reversal invariant superconductor (or superfluid) model

acting on $C4$. When *η* = 0 and *μ*(*X*) = *μx*_{1}, we obtain a standard Dirac operator with a topological charge sign(*μ*Δ), as may be verified (see also the following calculations). When *η* > 0, we conjugate the above operator by *g*_{1} ⊗ *I* (which maps *σ*_{3} to −*σ*_{2}), and after regularization and domain wall *μ*(*X*) = *μx*_{1}, we obtain the Hamiltonian

The operator has an elliptic symbol in $ES11$. We can then introduce as earlier *H*_{2} = *H*_{1} + *σ*_{3} ⊗ *Ix*_{2} and *F* = *H*_{2} − *ix*_{3}. Following Theorem 5.1, the topological charge of *H*_{1} is then defined as Index *F* = 2*πσ*_{I}(*H*_{2}) = −deg h^{1} with

We find h^{1} = 0 at the point *ξ* = 0 and *x*_{1} = 0. The Jacobian ∇h^{1} at this point has a determinant Δ^{3}*μ* so that the topological charge is given by Index *F* = − sign(*μ*Δ).

### H. Other Hamiltonians

The above examples all fit within the framework of operators with symbols *a*_{k} = h^{k} · Γ_{k} verifying that $ak2$ is a scalar operator resulting in two energy bands. The ellipticity requirement is that the energies tend to infinity as |*X*| goes to infinity with a prescribed power *m* > 0. In this setting, the topological charge can conveniently be computed as the degree of the field h^{k} as shown in the preceding examples.

The computations easily extend to operators of the form $Hk\u2295H\u0303k$ or more general direct sums of operators that are in the above form. More generally, the topological charge conservation result in Theorem 4.1 applies to operators beyond those of the form *a*_{k} = h^{k} · Γ_{k} provided that the symbol has eigenvalues appropriately converging to ∞ as |*X*| → ∞. For instance, the topological charge conservation in Theorem 4.1 applies to the sequence of effective Hamiltonians one obtains for continuous models of two-dimensional Floquet topological insulators. Such effective Hamiltonians are not in the form (23) and their asymmetric transport properties are most easily estimated by bulk-difference invariants related to the Fedosov–Hörmander formula (10) (see Ref. 9).

The theory presented in this paper does not apply for operators with flat bands for which the ellipticity condition cannot hold. A typical example is based on the shallow water wave (two-dimensional) Hamiltonian^{24,54}

where *f* = *f*(*x*_{1}) represents a (real-valued) Coriolis force. The symbol of that operator has two eigenvalues ±*λ*(*x*, *ξ*) with $\lambda (x,\xi )=\xi 12+\xi 22+f2(x1)$ similar to those of a Dirac operator and a third uniformly vanishing eigenvalue. Therefore, *H*_{0} + *α* is gapped for *α* ≠ 0, but with a gap independent of *ξ*. The presence of this flat band in the essential spectrum creates difficulties that are not only technical; the topological charge conservation (a bulk-interface correspondence in dimension *d* = 2) in Theorem 4.1 does not always hold, although it does for certain profiles *f*(*x*) (see Refs. 6, 37, and 55).

A properly regularized version of the above Hamiltonian, however, does fit into the framework of the current paper. We observe that the kernel of *a*_{0} is associated with the eigenvector $\psi 0=(\xi 12+\xi 22+f2)\u221212(if,\xi 2,\u2212\xi 1)t$. We define the projector $\Pi 0=\psi 0\psi 0*$ and for $0\u2260\mu \u2208R$ the regularized (pseudo-differential) Hamiltonian $H\mu =H0+\mu Op\lambda 2(1+\lambda 2)\u221212\Pi 0$. The symbol now has eigenvalues given by ±*λ* and $\mu \lambda 2(1+\lambda 2)\u221212$ (ensuring that the symbol of *H*_{μ} is smooth). If we choose *f*(*x*_{1}) = *νx*_{1} with *ν* ≠ 0 to generate a domain wall in the first variable, we observe that *H*_{μ} is elliptic with a symbol in $ES11$ (with *m* = 1). Following computations in, e.g., Refs. 6 and 49, which we do not reproduce here, we find that the topological charge of *H*_{μ} equals 2 sign(*ν*) independently of the choice of *μ* ≠ 0. This is the topological charge obtained when *μ* = 0 under smallness constraints in Ref. 6.

As we mentioned earlier, Hamiltonians with flat Landau levels do not satisfy the required ellipticity conditions. One such example is the ubiquitous two-dimensional scalar magnetic Schrödinger operator,

for instance, for *A* = (0, *Bx*_{1}), so that ∇ × *A* = *B* is a constant magnetic field. For the numerous applications of this model, both discrete and continuous, to the understanding of the integer quantum Hall effect, we refer the reader to, e.g., Refs. 1, 2, 10, 22, 26, 29, 39, and 53. The spectral decomposition of this operator gives rise to a countable number of infinitely degenerate flat bands, the Landau levels, which are incompatible with the elliptic structure we impose on the symbol of the Hamiltonian in this paper. See Refs. 26 and 50 for the analysis of the transport asymmetry for Schrödinger and Dirac operators in the presence of magnetic domain walls. Note that the integral in (10) vanishes for *a* scalar-valued in dimension *d* ≥ 2 since then $(a\u22121da)\u22272=0$. Therefore, (10) would not predict the asymmetric transport observed in Refs. 26 and 50 in the presence of magnetic domain walls. Beyond ellipticity constraints, we finally remark that the edge conductivity (11) for magnetic Schrödinger (40) or Dirac operators is *not* invariant under semiclassical scaling *D* → *hD*.

## ACKNOWLEDGMENTS

This work was funded in part by the NSF under Grant Nos. DMS-1908736 and EFMA-1641100.

## AUTHOR DECLARATIONS

### Conflict of Interest

The author has no conflicts to disclose.

### Author Contributions

**Guillaume Bal**: Writing – original draft (lead).

## DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

### APPENDIX: NOTATION, OPERATORS AND FUNCTIONAL CALCULUS

This appendix recalls the results summarized in Ref. 15 allowing us to characterize spaces of symbols *a*_{k} for *H*_{k} = Op *a*_{k} adapted to operators modeling unbounded domain walls, domains of definition for *H*_{k}, as well as functional calculus showing, in particular, that $(z\u2212Hd\u22121)\u22121$ and *φ*′(*H*_{d−1}) are pseudo-differential operators. We also recall results on semiclassical calculus and the Helffer–Sjöstrand formula following.^{25}

**Symbol spaces.**** ^{15} ** See also Ref. 40, Chaps. 18 and 19, Refs. 14, 16, and 17 and Ref. 58, Chap. 8.3.

On phase space $R2d$ in *d* spatial dimensions parameterized by *X* = (*x*, *ξ*) with $x\u2208Rd$ and $\xi \u2208(Rd)*\u2243Rd$, we define a Riemannian metric *g* in the Beals–Fefferman form by

We use the notation $\u27e8u\u27e9=1+|u|2$ for |·| the Euclidean norm applied to a vector *u*. For *u* = (*u*_{1}, *u*_{2}), we use the notation $\u27e8u1,u2\u27e9=1+|u1|2+|u2|2$. Associated to the above metric *g*, we define the Planck function *h*(*X*) and its inverse *λ*(*X*) by 1 ≤ *h*^{−1}(*X*) = *λ*(*X*) = Φ_{ξ}(*X*)Φ_{x}(*X*).

In this paper, we consider two metrics: *g*^{i} and *g*^{s}. The metric *g*^{i} is defined by $\Phi xi(X)=\Phi \xi i(X)=\u27e8X\u27e9\u22651$ with then *h*^{i} = ⟨*x*,*ξ*⟩^{−2}. The metric *g*^{s} is defined by $\Phi xs(X)=\u27e8x\u27e9\u22651$ and $\Phi \xi s(X)=\u27e8\xi \u27e9\u22651$ with Planck function *h*^{s} = ⟨*x*⟩^{−1}⟨*ξ*⟩^{−1}.

For 0 ≤ *k* ≤ *d*, we decompose $x=(xk\u2032,xk\u2032\u2032)$ with $xk\u2032\u2208Rk$ and $xk\u2032\u2032\u2208Rd\u2212k$. We define the weights

Following classical calculations,^{15} the weights *M*_{k} are admissible for *g* ∈ {*g*^{i}, *g*^{s}} in the sense of Ref. 15, Definition 2.3, and satisfy that *M*_{k} ≤ *Cλ*^{p} for some *C* > 0 and *p* < ∞ when *λ* ∈ {*λ*^{i}, *λ*^{s}}. This implies that *M*_{k}*h*^{N} goes to 0 as *X* → ∞ for *N* sufficiently large when *h* ∈ {*h*^{i}, *h*^{s}}.

For *g* = *g*^{s}, a symbol *b* ∈ *S*(*M*, *g*^{s}) when for each multi-index (*α*, *β*), we have

This is (4) when $M(X)=wkm(X)$. For *g* = *g*^{i}, a symbol *b* ∈ *S*(*M*, *g*^{i}) when instead

The metric *g*^{i} is referred to as the isotropic metric. We have that *S*(*M*, *g*^{i}) a subspace of *S*(*M*, *g*^{s}). Since they appear repeatedly in the derivations, we define for 0 ≤ *k* ≤ *d* the spaces,

We also define $Skm\u2254Skm(gs)$ and $S\u0303m\u2254S\u0303m(gs)$. Here, *n*_{k} and *n*_{d−1} are the dimensions of the spinors given in the introduction and in the construction of the augmented Hamiltonians in Sec. II, while $M(n)$ is the space of *n* × *n* matrices with complex coefficients.

Associated to the weights *M*_{k} are the Hilbert spaces $H(Mk,g)$ defined such that $Opau\u2208L2(Rd)$ whenever *a* ∈ *S*(*M*_{k}, *g*). These spaces are independent of *g* ∈ {*g*^{s}, *g*^{i}}^{15} and hence referred to as $H(Mk)$. We observe that $H(1,g)=L2(Rd)$. The Hilbert spaces associated with $Skm$ and $S\u0303m$ are denoted for 0 ≤ *k* ≤ *d* by

When $ak\u2208Skm(g)$, we thus obtain that Op *a*_{k} maps $Hkm$ to $Hk0=L2(Rd)\u2297M(nk)$. Note that $Hd\u221210=H\u03030=L2(Rd)\u2297M(nd\u22121)$. Pseudo-differential operators with Weyl quantization are defined in (2) with integrals defined as oscillatory integrals.

**Ellipticity (Ref.** **15 ****, Sec. 2.3.3).** We say that $a\u2208S(M,g)\u2297M(n)$ Hermitian valued is **elliptic** when

for some positive constants *C*_{1,2}. This is equivalent to imposing that each eigenvalue of *a*(*X*) is bounded away from 0 by at least *CM*(*X*) outside of a compact set. We then say that $a\u2208ES(M,g)\u2297M(n)$ and define the corresponding spaces of Hermitian elliptic symbols as $ESkm(g)$ for 0 ≤ *k* ≤ *d* and $ES\u0303m(g)$.

Since *M*(*X*) ≤ *Cλ*^{p}(*X*), ellipticity implies that $H=Opa\u2208OpES(M,g)\u2297M(n)$ is a self-adjoint operator with domain of definition $D(H)=H(M)\u2297M(n)$ and such that for some positive constant *C*,^{15}

For $ak\u2208ESkm$, we thus obtain that *H*_{k} = Op *a*_{k} is a self-adjoint operator from its domain of definition $D(Hk)=Hkm$ to $Hk0$. Similarly, for $a\u2208ES\u0303m$, then *F* = Op *a* is an unbounded operator from its domain of definition $D(F)=H\u0303m$ to $H\u03030$.

**Functional calculus (Ref.** **15 ****, Sec. 2.3).** For *H* = Op *a* and *a* elliptic, the above results show that the resolvent (*z* − *H*)^{−1} is an isomorphism from $L2(Rd)\u2297M(n)$ to $H(M)\u2297M(n)$ for $z\u2208C$ when Im(*z*) ≠ 0.

With the above assumptions, we have the *Wiener property*^{15} stating that: (i) *A* ∈ Op *S*(1, *g*) invertible in $L(L2)$ implies that *A*^{−1} ∈ Op *S*(1, *g*); and (ii) *A* ∈ Op *S*(*M*, *g*) bijection from $H(M1,g)$ to $H(M1/M,g)$, then *A*^{−1} ∈ Op *S*(*M*^{−1}, *g*).

This allows us to state the following result:

*Let* $H\u2208OpESd\u22121m$*. Then* $(\xb1i+H)\u22121\u2208OpESd\u22121\u2212m(g)$ *is an isomorphism from* $Hd\u221210$ *to* $Hd\u22121m$*.*

The proof follows.^{14,15} Associated to *H* is a resolvent operator *R*_{z} = (*z* − *H*)^{−1}, which is always defined as a bounded operator by spectral theory. When *H* is elliptic, then the domain $D(H)=Hd\u22121m$. Moreover, *R*_{z} is a bijection from $Hd\u221210=L2(Rn)\u2297M(nd\u22121)$ to that domain. We then apply above Wiener property^{14–17} to obtain that $Rz\u22121\u2208OpSd\u22121\u2212m(g)$.□

The above shows that $(I+H2)\u22121$ maps $Hd\u221210$ to $Hd\u221212m$ and has a symbol in $ESd\u22121\u22122m$. Moreover, using the Helffer–Sjöstrand formula as done in Ref. 15, Theorem 4 using *p* → −∞ in the notation there, we obtain the following result on the functional calculus:

*Let* $\varphi \u2208Cc\u221e(R)$ *and* $H\u2208OpESd\u22121m$*. Then* $\varphi (H)\u2208OpSd\u22121\u2212\u221e$*.*

*The above result means the following in terms of seminorms: for each* $N\u2208N$ *and each seminorm* *k* *defining the topology on the space of symbols, there is a seminorm* *l* *such that* *ϕ*(*H*) *is bounded for the seminorm* *k* *uniformly in the seminorm* *l* *applied to* *a**. For a sequence of operators* *H*(*ɛ*) = Op *a*(*ɛ*) *with* *a*(*ɛ*) *with seminorms of* $Sd\u22121m$ *uniformly bounded in* *ɛ**, this implies that the symbol of* *ϕ*(*H*_{ɛ}) *is bounded in any* $Sd\u22121\u2212N$ *uniformly in* *ɛ* *as well.*

**Semiclassical calculus.**** ^{25} ** The computation of several topological invariants, as in the proof of Ref. 40, Theorem 19.3.1, simplifies in the semiclassical regime. Let 0 <

*h*≤

*h*

_{0}≤ 1 be the semiclassical parameter. We define semiclassical operators in the Weyl quantization as

for *a*(*X*; *h*) a matrix-valued symbol in $M(n)$ for each $X\u2208R2d$ and *h* ∈ (0, *h*_{0}] and *ψ*(*x*) a spinor in $Cn$. The semi-classical symbol *a*(*X*; *h*) is related to the Schwartz kernel *K*(*x*, *y*; *h*) of *H*_{h} by

Note that Op *a*(*x*, *hξ*; *h*) = Op_{h}*a*(*x*, *ξ*; *h*). We define the classes of semi-classical symbols as S^{j}(*M*) constructed with the semi-classical metric in Beals–Fefferman form with Φ_{x}(*X*) = 1 and Φ_{ξ}(*X*) = *h*^{−1}, and for *M* an order function, i.e., in this context, a non-negative function on $R2d$ satisfying *M*(*x*, *ξ*) ≤ *C*(1 + |*x* − *y*| + |*ξ* − *ζ*|)^{N}*M*(*y*, *ζ*) uniformly in (*x*, *y*, *ξ*, *ζ*) for some *C*(*M*) and *N*(*M*). Then $a\u2208Sj(M)\u2297M(n)$ if for each component *b* of *a*, we have for each 2*d*-dimensional multi-index *α*, a constant *C*_{α} such that

We will mostly use the case *j* = 0. We also use the notation *b*(*X*) ∈ *S*^{0}(*M*) for symbols *b*(*X*) independent of *h*.

For two operators Op_{h}*a* and Op_{h}*b* with symbols *a* ∈ *S*^{0}(*M*_{1}) and *b* ∈ *S*^{0}(*M*_{2}), we then define the composition Op_{h}*c* = Op_{h}*a*Op_{h}*b* with symbol *c* ∈ *S*^{0}(*M*_{1}*M*_{2}) given by the (Moyal) product (Ref. 25, Theorem 7.9),

For *a* ∈ *S*^{0}(1), we obtain (Ref. 25, Theorem 7.11, Ref. 13, Proposition 1.4) that Op_{h} *a* is bounded as an operator in $L(L2(Rd)\u2297Cn)$ with bound uniform in 0 < *h* ≤ *h*_{0} so that *I* − *h*Op_{h} *a* is invertible on that space when *h* is sufficiently small.

An operator is said to be semiclassically elliptic when the symbol *a* = *a*(*x*, *ξ*; *h*) ∈ *S*^{0}(*M*) is invertible in $Mn$ for all $(x,\xi )\u2208R2d$ and *h* ∈ (0, *h*_{0}] with then *a*^{−1} ∈ *S*^{0}(*M*^{−1}).

Following Ref. 25 (see Ref. 6, Lemma 4.14), we obtain the following results on resolvent operators. Let *H*_{h} = Op_{h} *a* with *a* ∈ *S*^{0}(*M*). Let $z=\lambda +i\omega \u2208C$ with *ω* ≠ 0. Then $(z\u2212Hh)\u22121$ is a bounded operator and there exists an analytic function *z* → *r*_{z} = *r*_{z}(*y*, *ζ*; *h*) such that $(z\u2212Hh)\u22121=Ophrz$ (compare to Lemma A.1). Moreover, the symbol *r*_{z} ∈ *S*^{0}(1) satisfies

for all multi-indices *β* = (*β*_{y}, *β*_{ζ}) and a constant *C*_{β} independent of *z* ∈ *Z* a compact set in $C$ and 0 < *h* ≤ *h*_{0}.

**Trace-class criterion.** We have the following trace-class criterion (Ref. 25, Chap. 9). Assume that $M\u2208L1(R2d)$ and that |*∂*^{α}*a*(*X*)| ≤ *C*_{α}*M*(*X*) for all |*α*| ≤ 2*d* + 1. Then Op *a* is a trace-class operator and

In other words, all symbols in *S*^{0}(*M*) with *M* integrable generate trace-class operators.

**Helffer–Sjöstrand formula.**** ^{23,25}** Finally, we recall some results on spectral calculus and the Helffer–Sjöstrand formula following Refs. 23 and 25; see also Ref. 13 for the vectorial case. For any self-adjoint operator

*H*from its domain $D(H)$ to $L2(Rd)\u2297Cn$ and any bounded continuous function

*ϕ*on $R$, then

*ϕ*(

*H*) is uniquely defined as a bounded operator on $L2(Rd)\u2297Cn$ (Ref. 25, Chap. 4). Moreover, for

*ϕ*compactly supported, we have the following representation:

where, for *z* = *λ* + *iω*, *d*^{2}*z* ≔ *dλdω*, $\u2202\u0304=12\u2202\lambda +12\u2202\omega $, and where $\varphi \u0303(z)$ is an almost analytic extension of *ϕ*. The extension $\varphi \u0303$ may be chosen as compactly supported in $C$. Moreover, $\varphi \u0303(\lambda )=\varphi (\lambda )$ and $\u2202\u0304\varphi \u0303(\lambda )=0$ on the real axis. We can choose the almost analytic extension such that $|\u2202\u0304\varphi \u0303|\u2264CN|\omega |N$ for any $N\u2208N$ uniformly in (*λ*, *ω*). Several explicit expressions, which we do not need here, for such extensions are available in Refs. 23 and 25.

## REFERENCES

We use Δ for the order parameter as is customary in the superconductor literature. The (positive) Laplace operator is denoted by *D* · *D*.