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.

Topics
Superconductors, Topological insulator, 2D materials, Operator theory, Clifford algebra, Particle symmetry

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-binding11,43 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 obstruction4,47 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 discrete29,35,38,47,53 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 invariant6 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 Hk, where 0 ≤ k ≤ d − 1 refers to the number of confined dimensions. Therefore, H0 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(x1) > 0 and m(x1) < 0, respectively. We wish to test the topological properties of Hk by assessing its response to the addition of domain walls. When k ≤ d − 2, we first construct a Hamiltonian Hd−1 by appropriately adjoining d − k − 1 domain walls to it. The resulting Hamiltonian Hd−1 models a system that is confined in all but one dimension, say xd. We, therefore, expect asymmetric transport to occur along the corresponding dth dimension. We will introduce an edge conductivity σI(Hd−1) in (11) in Sec. III; this is a physical observable of the system associated with the asymmetric transport. We will show that 2πσI 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 Hk by implementing a final domain wall m(xd) in the dth dimension and introduce an operator F = Hd−1 − im(xd). We will show that F is a (non-Hermitian) Fredholm operator. The topological charge of Hk 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 Hk to F is local in the sense that F = γ1 ⊗ Hk + γ2 for γ1 a constant matrix and γ2 an explicit multiplication operator in the physical variables. The evaluation of (10) is thus directly related to the symbol of a given Hamiltonian Hk 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 Hk 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 Hk. Such degrees, which naturally take integral values, are often quite easy to compute in problems of interest. We will see that the construction of Hd−1 and F from Hk in arbitrary dimensions requires a structure of Clifford algebras in order to define appropriately orthogonal domain walls. When Hk itself admits a natural decomposition in a Clifford algebra, then we will associate with Hk a map hk 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 H0 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 Hk, Hd−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 Hd−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 σI associated with Hd−1 is given in (11) in Sec. III. We define a corresponding Fredholm operator in Toeplitz form and show in Theorem 3.2 that 2πσI = Index T. Several lemmas in that section also show the stability of 2πσI = Index T against a number of continuous deformations of the Hamiltonian Hk.

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 hk associated with Hamiltonians Hk 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.

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 

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 crossings11,27,32–34,51 and arguably the simplest models for topological phases of matter.

Consider first a one-dimensional material and asymmetric transport modeled by the Hamiltonian H0 = Dx with Dx = −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 σ(H0)=σac(H0)=R and displays asymmetric transport along the x axis in the sense that solutions of (Dt + Dx)u = 0 with initial condition u(0, x) = u0(t) are given by u(t, x) = u0(x − t).

To generate confinement in the vicinity of x = 0, we introduce the domain wall m(x) = x and the operator F=Dx−ix=−ia with a=∂x+x an annihilation operator. The operator F is now a Fredholm operator from its domain of definition D(F)={f∈L2(R);f′∈L2(R) and xf∈L2(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−12x2⁠. We define Index F as the topological charge of H0.

Consider next the two-dimensional version of the above example, where H0 = D1σ1 + D2σ2 with Dj = −i∂j for j = 1, 2 and σ1,2,3 are the standard Pauli matrices. The operator H0 appears as a generic low-energy description of energy band crossings and is ubiquitous in works on topological insulators. We classify H0 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 m1(x1) = x1 along the first variable by introducing H1 = H0 + x1σ3. This models includes insulating regions x1 > 0 and x1 < 0 while transport remains possible in the vicinity of the interface x1 = 0.

To obtain confinement in the second variable, we introduce the second domain wall m2(x2) = x2. Associated to H0 and H1 is then the operator F = H1 − ix2. This is again a Fredholm operator from its domain of definition to L2(R2)⊗C2 and we verify that Index F = 1, which defines the topological charge associated with H0 (and H1). The kernel of F is spanned by the spinor e−12|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 c1 = −1 here. In dimension d = 2, we observe that F*F=(D12+D22+x12+x22+c2)I2 with I2 the 2 × 2 identity matrix. The latter is obtained because (σ1, σ2, σ3, i) satisfy appropriate orthogonality properties. In particular, σ1,2,3 satisfy σiσj + σjσi = 2δijI2. 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 H0 = D1σ1 + D2σ2 + D3σ3. 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 H1 = σ1 ⊗ H0 + σ2 ⊗ I2x1 with a domain wall m1(x1) = x1 in the first direction but now acting on spinors in C4⁠. The operator H1 thus admits surface states concentrated in the vicinity of x1 = 0, as does the operator H0 in the two-dimensional setting. Its topology is then characterized by asymmetric transport in the third dimension after a second domain wall in the x2 direction is introduced: H2 = H1 + σ3 ⊗ I2x2.

Confinement in the last variable is imposed by the domain wall m3(x3) = x3. We introduce the operator

(1)

This is a Fredholm operator [from its domain of definition to L2(R3)⊗C4]. We verify (and will show in greater generality in Sec. VI) that the topological charge of H0 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 Hj as well as the orientation of the domain walls. The kernel of F* has for eigenfunction the spinor e−12|x|2(1,−1,−1,−1)t⁠. The topological charge of H0, of H1, and of H2 is defined as Index F.

Note that in the above three-dimensional model, H2 models higher-order topological insulators with transport occurring along a hinge, or co-dimension two manifold.52 The classification of H2 by domain walls is arguably simpler than one based on bulk phases that may be hard to identify.

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 ξ∈Rd the dual (Fourier) variable and X=(x,ξ)∈R2d the phase space variable. The algebras of pseudo-differential operators (PDOs) we consider are written in Weyl quantization as

(2)

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) = σ1 ⊗ (ξ1σ1 + ξ2σ2 + ξ2σ3) + x1σ2 ⊗ I2 + x2σ3 ⊗ I2 − ix3I4 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 Hk = Op ak with ak a given matrix-valued symbol in M(nk) interpreted as confining the first k variables. Our first aim is to construct the operators H ≔ Hd−1 = Op ad−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′,xk′′) with xk′∈Rk and xk′′∈Rd−k⁠. We use the notation ⟨y⟩=1+|y|2 and ⟨y1,y2⟩=1+|y1|2+|y2|2 and define the weights

(3)

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

(4)

Here, wkm=(wk)m⁠. We define the space of symbols S̃m as Sdm but acting on spinors of (lower) dimension nd−1 instead of nd. Here m is the order of the operator.

For the two-dimensional Dirac operator, we find m = 1, H0 = Op a0 for a0 = ξ1σ1 + ξ2σ2, while H1 = Op a1 for a1 = a0 + x1σ3 and F = Op a for a = a1 − ix2. For n0 = n1 = 2, we observe that aj∈Sj1 for j = 0, 1, while a∈S̃1⁠.

We impose a number of assumptions on Hk and ak. 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

(5)

In other words, all eigenvalues of the Hermitian matrix ak(X) are bounded away from 0 by at least Cwkm(X) for X outside of a compact set in R2d⁠. Since ak∈Skm⁠, all (positive and negative) eigenvalues of ak are of order wkm(X) away from a compact set.

We denote by ESkm the elliptic symbols in Skm and ES̃m the elliptic symbols in S̃m⁠.

Associated to the spaces of symbols Skm and S̃m are Hilbert spaces Hkm and H̃m defined in (A5) in the  Appendix. These spaces are constructed so that for ak∈ESkm⁠, we obtain that Hk = Op ak is an unbounded self-adjoint operator on Hk0=L2(Rd)⊗M(nk) with domain of definition D(Opak)=Hkm⁠, while for a∈ES̃m⁠, we obtain that F = Op a is an unbounded operator with domain of definition D(F)=H̃m⁠. For any ak∈ESkm⁠, the space Hkm is defined explicitly as Hkm=(Opak+i)−1Hk0⁠.

For the Dirac operators in dimension d = 3, we have, for instance, the Hilbert space Hk1={ψ∈L2(R3;Cnk);(Djψ)1≤j≤3∈L2(R3;Cnk) and (xjψ)1≤j≤k∈L2(R3;Cnk)} with n0 = 2 and n1 = n2 = n3 = 4.

We start from a given elliptic (self-adjoint) operator Hk = Op ak for ak∈ESkm with m > 0. By ellipticity assumption, Hk is an unbounded self-adjoint operator with domain D(Hk)=Hkm and acts of spinors in Cnk⁠. The ellipticity of Hk and the construction of the weight wk(X)=⟨xk′,ξ⟩ imply that the first k variables parameterized by xk′ are confined in the sense that each singular value of ak is large for |xk′| large.

To obtain a non-trivial topological classification, the operator Hk needs to satisfy a chiral symmetry when d + k is even (complex class AIII47). 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 σ1,2,3 are the Pauli matrices, a set of Hermitian 2 × 2 matrices such that σiσj + σjσi = 2δij and σ1σ2 = iσ3. Using the notation σ±=12(σ1±iσ2)⁠, the chiral symmetry takes in a suitable basis the following form:

(6)

We next introduce the domain walls

(7)

They are constructed to have the same asymptotic homogeneity of order m as the Hamiltonian Hk. We then define the new spinor dimension nk+1 = nk and the augmented Hamiltonian

(8)

This implements a domain wall in the variable xk+1.

Assume now d + k is odd. We define the new spinor dimension nk+1 = 2nk and the augmented Hamiltonian,

(9)

The operator Hk+1 satisfies the chiral symmetry of the form (6), as requested since d + k + 1 is now even.

We denote by ak+1 the symbol of Hk+1 = Op ak+1 and observe that ak+1 = ak + mk+1σ3 ⊗ I when d + k is even and ak+1 = σ1 ⊗ ak + mk+1σ2 ⊗ I when d + k is odd.

The procedure is iterated until Hd has been constructed. Note that ak+2(X)∈M(2nk) with dimension of the spinor space on which the matrices act that doubles every time k is raised to k + 2. Since 2d is even, Hd = σ− ⊗ F* + σ+ ⊗ F for an operator F = Fd = Hd−1 − imd =: Op a, or equivalently, a = ad−1 − imd.

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

Hk+l=γ0⊗Hk+μ⋅γ⊗Ink,

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

μ=(mk+1,…,mk+l),γ0=σ1⊗p,γ=(γ1,…,γl).

We now show that all operators Hl are elliptic and that Hd and F are Fredholm operators.

Lemma 2.1.

Let ak∈ESkm and Hk = Op ak satisfying the chiral symmetry (6) when d + k is even. Then aj∈ESjm for all k ≤ j ≤ d and a∈ES̃m.

Proof.

Let ak+l be the symbol of Hk+l = Op ak+l for 0 < l ≤ d − k. By construction and commutativity {γi, γj} = 0, we obtain that ak+l2=I⊗ak2+∑j=1lmk+l2⊗I with I identity matrices with appropriate dimensions. This shows that ak+l satisfies the ellipticity condition (5) for the weight wk+l(X). The decay properties for derivatives of ak+l in (4) with k replaced by k + l follow from the corresponding properties for ak. That a∈ES̃m comes from the corresponding result for ad and the construction of F.

Let Λ=−Δ+|x|2+1=Op⟨x,ξ⟩2 be an elliptic self-adjoint operator, which by construction, maps H(wd−1m) to H(1)=L2(Rd)⁠.45 

Lemma 2.2.

The operators Hd and F are Fredholm operators from Hdm to Hd0 and H̃m to H̃0, respectively. Equivalently, Λ−mHd and Λ−mF are Fredholm operators on Hd0 and H̃0, respectively.

Proof.

This is Ref. 40, Theorem 18.6.6; see also Ref. 45 since the Planck function hs(X) (see the  Appendix for notation) tends to 0 as |X| → ∞.

The construction of Hl implements l − k domain walls to test the topology of the operator Hk. When l = d, the operator Hd 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 Hd is self-adjoint, and so its index vanishes. However, it satisfies the chiral symmetry (6), and the corresponding operator F = Hd−1 − imd is also Fredholm. Its index may not vanish, and we define the topological charge56 of Hk as Index F.

The intermediate operator Hd−1 is physically relevant with d − 1 confined spatial variables (in the vicinity of xd−1′=0⁠) and transport allowed along the direction xd−1′′=xd⁠. As we show in Secs. III and IV, this transport is asymmetric and quantized by the topological charge of Hk.

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

Theorem 2.3.
For F = Op a, we have
(10)
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−1 is chosen so that dξ1 ∧ dx1 ∧ ⋯ ∧dξd ∧ dxd > 0.

Remark 2.4

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

Remark 2.5

(Ref. 40, Theorem 19.3.1′). Applies to Λ−mF and not F directly. However, the index of Λ−tmF 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 at of Λ−tmF are uniformly invertible for |X| ≥ R for R sufficiently large and smooth in t. The integral (10) is, therefore, the same for a = a0 and a = a1. This may be proved as follows:

Let M and N be smooth closed manifolds. Here, M=SR2d−1 and N=GL(nd−1;C)⁠. For t ∈ [0, 1], let it: M → M × [0, 1] defined by it(x) = (x, t). Let a0 and a1 be homotopic smooth maps from M to N, and let a: M × [0, 1] → N be the smooth homotopy map such that a0 = a◦i0 and a1 = a◦i1.

For each p ≥ 1, there exists by Ref. 42, Lemma 17.9, a homotopy operator h: Ωp(M × [0, 1]) to Ωp−1(M) such that for each p-form ω̃∈Ωp(M×[0,1])⁠, then
h(dω̃)+d(hω̃)=i1*ω̃−i0*ω̃.
Now let ω ∈ Ωp(N) be a closed p-form on N. In our application, this is ω=tr(A−1dA)∧(2d−1)⁠. Then we have that ω̃=a*ω is a closed p-form on M × [0, 1] since da*ω = a*dω = 0. Using the above homotopy operator, we thus have
a1*ω−a0*ω=i1*a*ω−i0*a*ω=d(ha*ω)+hda*ω=dha*ω,
is an exact p-form on M. By the Stokes theorem, this means that for a top-degree form, ∫Ma1*ω=∫Ma0*ω⁠. As a result, the right-hand side in (10) is the same for a the symbol of F or that of Λ−mF.

While systematic and explicit, the classification of Hk 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 Hd−1 introduced in Sec. II C.

The operator Hd−1 confines in d − 1 directions while allowing transport in the remaining dimension parameterized by xd. The following edge conductivity quantifies asymmetric transport in that direction. Let H ≔ Hd−1 and φ∈S[0,1] a smooth non-decreasing switch function and P=P(xd)∈S[0,1] a smooth spatial switch function.

Here and below, a function f:R→R is called a switch function f∈S[0,1] if f is bounded measurable and there are xL and xR in R such that f(x) = 0 for x < xL and f(x) = 1 for x > xR. We denote by C∞S[0,1] the subset of smooth switch functions.

We define the edge conductivity

(11)

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 σI 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−itHψ be a solution of the Schrödinger equation i∂tψ(t) = Hψ(t) with initial condition ψ, and let P be a Heaviside function defined as P(xd) = 1 for xd>x̃ while P(xd) = 0 for xd<x̃ for some x̃∈R⁠. Then ⟨P⟩(t)≔(ψ(t),Pψ(t)) is interpreted as the mass of ψ(t) in the half space on the right of the hyperplane xd=x̃⁠. Its derivative ddt⟨P⟩=(ψ(t),i[H,P]ψ(t))=Tri[H,P]ψ(t)ψ(t)* thus describes current crossing the hyperplane xd=x̃⁠. We formally replace the density ψ(t)ψ(t)* by the spectral density φ′(H). This heuristically gives the interpretation of σI as the rate of signal propagating from the left to the right of the hyperplane xd=x̃ 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∈ESkm⁠, then i[Hd−1, P]φ′(Hd−1) is indeed a trace-class operator so that σI(Hd−1) is well-defined. We next relate the edge conductivity to the index of the Toeplitz operator T≔P̃U(H)P̃RanP̃ for P̃ an orthogonal projector in S[0,1] and U(H) = e2πiφ(H). In particular, we show that 2πσI=IndexT∈Z so that σI is indeed quantized. Finally, we prove that σI and the index of T are stable against a number of continuous transformations of the symbol ak.

Let ak∈ESkm so that by Lemma 2.1, ad−1∈ESd−1m while a=ad−1−ixd∈ES̃m⁠. We denote by H = Hd−1 = Op ad−1. Let U(H) = ei2πφ(H) with φ∈C∞S[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 S0(M) the space of symbols with components b(X) such that

(12)

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 ∈ S0(M1) and b ∈ S0(M2), then Op a Op b = Op c with c ∈ S0(M1M2) constructed explicitly in (A9) (with h = 1).

  • If ϕ∈Cc∞(R) and H∈OpESd−1m⁠, then ϕ(H)∈OpSd−1−∞⁠.

  • If a ∈ S(M) with M∈L1(R2d)⁠, then Op a is trace-class and (A11) holds; namely, the trace of Op a is given by (2π)−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:

Lemma 3.1.

Let ak∈ESkm and P∈C∞S[0,1]. Let ϕ∈Cc∞(R) and p,q∈N. Then [P, ϕ(H)] and Hp[P, Hq]ϕ(H) are trace-class operators with symbols in Sd−∞. When P̃∈S[0,1] is an orthogonal projector, then T≔P̃U(H)P̃RanP̃ is a Fredholm operator on RanP̃⊂H̃0 with index given by Tr[U(H),P̃]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.

Proof.

By composition calculus (i), for any A = Op a with a ∈ S0(M), then the decomposition [A,P]=(1−χ̃(xd))Aχ̃(xd)−χ̃(xd)A(1−χ̃(xd)) for χ̃(xd) a smooth function equal to 1 for x > 1 and 0 for x < −1 shows that [A, P] has symbol in S0(M⟨xd⟩−∞) (i.e., in S0(M⟨xd⟩−N) for each N∈N⁠). By assumption on ak and using the functional calculus result (ii) (see Lemma A.2), we obtain for ϕ∈Cc∞(R) that ϕ(H)∈OpS0(⟨xd−1′,ξ⟩−∞)⁠, which is larger that OpSd−1−∞ (since m > 0). To simplify notation, we use the same notation for S0(M) and S0(M)⊗M(n) for any n. Therefore, by composition calculus, [ϕ(H), P] and [H, P]ϕ(H) as well as Hp[Hq, P]ϕ(H) for p,q∈N all have symbols in S0(⟨X⟩−∞). We use this with ϕ = φ′ and ϕ = W. With additional effort, we verify that all symbols are in Sd−∞⁠, 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 Hp[P, Hq]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.

Let now P̃∈S[0,1] a switch function that is not necessarily smooth (and so that, for instance, P̃2=P̃⁠). The above functional calculus no longer applies directly. However, P−P̃ is compactly supported by assumption. Let χ(xd)∈C0∞(R) be equal to 1 on the support of P−P̃⁠. Then
[U(H),P̃]=[U(H),P]+[U(H),P̃−P]=[W(H),P]+W(H)χ(P̃−P)−(P̃−P)χW(H).
Since from (ii), W(H)∈OpS0(⟨xd−1′,ξ⟩−∞) and χ∈OpS0(⟨xd⟩−∞)⁠, we deduce from (i) and (iii) that W(H)χ(xd) and χ(xd)W(H) are trace-class. Since multiplication by P̃−P is a bounded operator, we conclude that all operators on the above right-hand side are trace-class, and hence so is [U(H),P̃]⁠. As a consequence, [U(H),P̃]U*(H) is trace-class as well.
Using the cyclicity of the trace TrAB = TrBA when A is trace-class and B is bounded, we deduce
TrW(P̃−P)=TrW(P̃−P)χ=TrχW(P̃−P)=Tr(P̃−P)χW=Tr(P̃−P)W.
In other words, Tr[U,P̃−P]=0⁠. Consequently
Tr[U,P̃−P]U*=Tr[W,P̃−P]W*=TrW(P̃−P)W*−TrTr(P̃−P)WW*,
but since by cyclicity, TrW(P̃−P)W*=Tr(P̃−P)W*W⁠, we obtain that Tr[U,P̃]U*=Tr[U,P]U*⁠.

That T=P̃U(H)P̃|RanP̃ is a Fredholm operator with index given by Tr[U(H),P̃]U*(H)=Tr(Q̃−P̃) for P̃ and Q̃=U(H)P̃U*(H) projectors is a non-trivial consequence of the trace-class nature of [P̃,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 σI = σI(H) defined in (11). We obtain the following result:

Theorem 3.2.

Under the assumptions of Lemma 3.1, we have 2πσI=Tr[U(H),P]U*(H)=Tr[U(H),P̃]U*(H)=IndexP̃U(H)P̃RanP̃.

Proof.
This is essentially (Ref. 6, Proposition 4.3) with slightly different assumptions. The last two equalities of the theorem were proved in Lemma 3.1. We focus on the first one. Let g∈Cc∞(R) and χ∈Cc∞(R) with χ = 1 on the support of W and g. Let Wp be a sequence of polynomials chosen such that χ(W − Wp) and χ(W′−Wp′) converge to 0 uniformly on R as p → ∞. We find, with δWp ≔ W − Wp
[δWp,P]gχ=δWp[P,g]χ+δWpg[P,χ]+[δWpgχ,P].
We deduce from Lemma 3.1 that Tr[δWpgχ, P] = 0 and that δWp[P, g] is trace-class since the [P, g] has symbol in S0(⟨X⟩−∞). Since TrAB = TrBA when A is a trace-class and B is bounded, we find that TrδWp[P, g]χ = TrχδWp[P, g] = Tr[P, g]χδWp. Therefore,
Tr[W−Wp,P]g=Tr[P,g](W−Wp)χ+Tr(W−Wp)g[P,χ]→0 as p→∞.
It remains to analyze Tr[Wp, P]g. We verify that
Tr[Hn,P]g=Trn[H,P]Hn−1g,Tr[Wp,P]g=Tr[H,P]Wp′g.
Indeed, from [AB, C] = A[B, C] + [A, C]B,
Tr[Hn+1,P]g=TrHn[H,P]g+[Hn,P]Hg=TrHn[H,P]gχ+[Hn,P]Hg=Tr[H,P]Hngχ+[Hn,P]Hg=Tr[H,P]Hng+[Hn,P]Hg=Tr[H,P](n+1)Hng,
using that Hn[H, P]g is trace-class so that TrHn[H, P]gχ = TrHnχ[H, P]g and for the last equality, an induction in n ≥ 1. This proves the result for Wp as well. Is remains to realize that (Wp′−W′)g is uniformly small as p → ∞ to obtain that
(13)
We next compute
Tr[U,P]U*=Tr[W,P]+Tr[W,P]W*=Tr[H,P]W′W*=Tr[H,P]W′U*−Tr[H,P]W′.
We now show that 0 = Tr[H, P]W′. Let 1=ψ12+ψ22 a partition of unity with 0 ≤ ψj ≤ 1 for j = 1, 2 and such that ψ1∈Cc∞(R) equals 1 on the support of W. Then, using (13), with ψj = ψj(H), we have Tr[H,P]W′=Tr[H,P]W′ψ12=Tr[W(H),P]ψ12. Now, since [W(H), P] is trace-class,
Tr[W,P]ψ22=Trψ2[W,P]ψ2=0,
since W(H)ψ2(H) = 0. Since Tr[W, P] = 0, we have Tr[H,P]W′=Tr[W,P]ψ12=Tr[W,P]=0⁠. This proves that Tr[U, P]U* = 2πiTr[H, P]φ′(H) since W′U* = U′U* = 2πiφ′.

We next derive several results showing that the index of the Toeplitz operator T=P̃U(H)P̃RanP̃ 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 Hk by an isotropic symbol. All results so far have been obtained for symbols satisfying (4). The space of isotropic elliptic symbols ESd−1m(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−1m=ESd−1m(gs)⁠. It is also invariant under permutation of the variables X, which is not the case for ESd−1m(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:

Lemma 3.3.

Let T=P̃U(H)P̃RanP̃ with H=Hd−1∈OpESd−1m(gs). Then there is a sequence of operators Hɛ for 0 ≤ ɛ ≤ 1 with symbol in ESd−1m(gi) for all ɛ > 0 and such that the corresponding [0,1]∋ε→Tε=P̃U(Hε)P̃RanP̃ is continuous in the uniform sense and T0 = T. Therefore, Index Tɛ is defined as being independent of ɛ and equal to Index T. Moreover, the symbols aɛ are chosen so that for any compact domain in X = (x, ξ), aɛ = ad−1 on that domain for ɛ sufficiently small.

Proof.
The proof is based on Ref. 40, Lemma 19.3.3 showing that symbols in S(M, gs) may be suitably approximated by symbols in S(M, gi) as follows. Let v(r):R+→R+ 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, gs). We then define the family of regularized symbols,
aε(X)=a(v(ε|X|)x,v(ε|X|)ξ).
We observe that aɛ(X) = a(X) for ɛ|X| ≤ 1 while aε(X)=a(Xε|X|) homogeneous of degree 0 for ɛ|X| ≥ 2. Then Ref. 40, Lemma 19.3.3 (where the metrics gs and gi are called g and G, respectively) proves that aɛ(X) ∈ S(M, gs) uniformly in 0 ≤ ɛ ≤ 1 (i.e., every semi-norm defining the symbol space is bounded for aɛ uniformly in ɛ). Moreover, aɛ(X) ∈ S(M, gi) when ɛ > 0 with a bound that now depends on ɛ. Since aɛ(X) = a(X) for ɛ|X| ≤ 1, we obtain that aɛ converges to a in S(M, gs) as ɛ → 0.

We now mimic the proof of Ref. 40, Theorem 19.3.1′ extending the index theorem (10) from the isotropic metric gi to the metric gs. For aɛ(X) = ad−1(v(ɛ|X|)x, v(ɛ|X|)ξ), we find that aε∈ESd−1m(gs) uniformly in ɛ and aε∈ESd−1m(gi) for ɛ > 0. Let Hɛ = Op aɛ and T̃ε=P̃U(Hε)P̃+I−P̃⁠. By uniformity of aε(X)∈Sd−1m(gs) in ɛ and uniformity of bounds in Lemma A.2 (see Remark A.3), we obtain that W(Hɛ) has symbol in S̃0(gs) uniform in ɛ.

We next observe that T̃ε*T̃ε−I=P̃[W*(Hε),P̃]U(Hε)P̃ and T̃εT̃ε*−I 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 T̃ε and T̃ε*⁠, and hence that of Tɛ, are independent of 0 < ɛ ≤ 1. In the limit ɛ → 0, this is the index of T. Therefore, Index T = Index Tɛ for ɛ > 0 but now for a symbol aε∈ESd−1m(gi)⁠.

The above result shows that we can replace the symbol in Hd−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−1∈ESd−1m(gi) in the computation of Index T. The main advantage of the more constraining metric gi 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 = (x1, …, xd−1, ξ1, …, ξd−1).

Lemma 3.4.

Let g = gi. Let [0, 1] ∋ t → Lt be a continuous family of linear invertible transformations in GL(2d−2,R) in the Y variables leaving the variables (xd, ξd) fixed. Let a(X)∈ESd−1m(gi). Then a(t,X)=a(LtX)∈ESd−1m(gi). Let Tt be the corresponding Toeplitz operator. Then Tt is Fredholm with index independent of t ∈ [0, 1].

Proof.
Using the Helffer–Sjöstrand formula recalled in (A12) with W̃ an almost analytic extension of W (see the  Appendix), we compute
U(Ht)−U(Hs)=W(Ht)−W(Hs)=−1π∫C∂̄W̃(z)(z−Ht)−1(Ht−Hs)(z−Hs)−1d2z.
We use (z−Hs)−1=(i−Hs)−1(I+(i−z)(z−Hs)−1) and ‖(z−Hs)−1‖L(L2)≤|Imz|−1⁠. We have |∂̄W̃(z)|≲|Imz|2⁠. We know from Lemma A.1 that (i−Hs)−1=Oprs with rs∈S(Md−1−1,gi) uniformly in s ∈ [0, 1] so that (Ht−Hs)(i−Hs)−1=Op(at−as)♯rs⁠. Then, from Ref. 16, (18) and composition calculus, there exists a seminorm k independent of t such that
‖(Ht−Hs)(i−Hs)−1‖≲‖a(t,X)−a(s,X)‖k;S(Md−1,gi)‖rs‖k;S(Md−1−1,gi),
which is bounded by C‖a(t,X)−a(s,X)‖k;S(Md−1,g)⁠. This involves contributions in the form of powers of (Lt − Ls)X times derivatives of a by chain rule. We thus obtain terms of the form Xi∂Xja (with Xj ∉ {xd, ξd}), which are operations that are stable from S(Md−1, gi) to itself provided that g = gi. Note that the vector field [implementing rotation in the variables (xj, ξj)] ξj∂xj−xj∂ξj does not preserve S(Md−1, gs), 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‖a(t,X)−a(s,X)‖k;S(Md−1;g) is bounded by a constant times |t − s|. Therefore, U(Ht) − U(Hs) is small in the uniform sense for small (t − s), so that the index of Tt is continuous in t and hence independent of t ∈ [0, 1].

This result states, in particular, that for ad−1 ∈ ESd−1(gi), then the index of T is independent of any rescaling Yj → λYj for λ > 0 (leaving all other variables fixed) as well as any rotation in the phase space variables mapping (Yj, Yk) to (Yk, − Yj) (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 σI 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:

Lemma 3.5.

Let g ∈ {gs, gi}. For t ∈ [0, 1], let t↦at∈ESd−1m(g) be a continuous path of elliptic symbols. Then the indices of the corresponding Fredholm operators Ft = Op (at − imd) and Tt=P̃U(Opat)P̃|RanP̃ are independent of t ∈ [0, 1].

Proof.

By assumption and construction, Λ−mFt and Tt are continuous in t as operators from H̃0 to itself. Their indices are, therefore, constant in t.

We apply the preceding lemma to a1=a(xd−1′,xd,ξ) and a0=a(xd−1′,0,ξ) while at = ta1 + (1 − t)a0. The path of symbols belongs to ESd−1m so that the indices of the respective operators are defined with clear continuity in t. This allows us to replace the xd-dependent elliptic symbol ad−1 with an xd-independent one, which is used below in the proof of topological charge conservation.

We recall that Hk = Op ak for an elliptic symbol ak∈ESkm with 0 ≤ k ≤ d − 1 and m > 0. We constructed in Sec. II, with operator Hd−1 = Op ad−1 confined in all but the last variable and a Fredholm operator F = Op a confined in all variables. Associated to Hd−1 is the edge conductivity σI defined in (11). We associated two topological charges to F and σI 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:

Theorem 4.1.
Let a∈ES̃m. Then we have
(14)

Proof.

As a first step, we continuously deform ad−1(xd−1′,xd,ξ) to ad−1(xd−1′,0,ξ) 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−1∈ESd−1m=ESd−1m(gs) by ad−1∈ESd−1m(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−1⁠.

To simplify the presentation, we change notation to (y,x)=(xd−1′,xd) and to (ζ, ξ) = ξ with the new ζ,y∈Rd−1 and ξ,x∈R⁠. Therefore, the symbol ad−1 = ad−1(y, ζ, ξ).

We finally continuously deform a(X) to ad−1(y, ζ, ξ) − ix. This does not change the definition of σI. Using the homotopy result of Remark 2.5, this does not change the integral over SR2d−1 either by continuously deforming the mass term md(X)=⟨xd⟩m−1xd to xd ≡ x (using ⟨xd⟩t(m−1)xd for t ∈ [0, 1]) on the ball of radius R while preserving the continuity of a−1 on SR2d−1⁠.

For the same reason, we may replace a by ad−1(y, ζ, ξ) + α − ix for any α∈R by continuity in α and for a fixed radius R for α in a compact domain.

Since ad−1 does not depend on x, we introduce the partial spectral decomposition Ĥ=Ĥ[ξ] such that
H=Fξ→x−1∫R⊕Ĥ[ξ]dξFx→ξ,Ĥ[ξ]=Opad−1(Y,ξ),
with Y = (y, ζ) and a Weyl quantization in the variables Y for each parameter ξ∈R⁠. We thus obtain, using the trace-class properties of Lemma 3.1 providing traces as integrals of Schwartz kernels along diagonals, that
2πσI=2πTry∫Ri[H,P](x,x′)φ′(H)(x′,x)dx′dx,=2πiTry∫R−x′H(x′)φ′(H)(x′−x)dx′=Try∫R∂ξĤφ′(Ĥ[ξ])dξ.
Here, Try denotes the integration in all variables but x = xd (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
ϕ(x′)≔∫R(P(x−x′)−P(x))dx=−x′, since ∂x′ϕ(x′)=−∫RP′(x−x′)dx=−1.
We introduce the complex variable C∋z=λ+iω and identify the spatial variable x = xd 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 σz(y, ζ, ξ) = z − ad−1(y, ζ, ξ) = λ − a(y, ω, ζ, ξ) the symbol of z−Ĥ[ξ]⁠.

We now introduce the semiclassical parameter 0 < h ≤ 1 and the operator Ĥh with the symbol ad−1(y, hζ, ξ). Using the semiclassical notation recalled in the  Appendix (in the phase-space variable Y), we thus observe that z−Ĥh=Ophσz⁠. Using Lemma A.1, we know that (z−Ĥh)−1=Ophrz is a PDO with the semiclassical symbol rz(y, ζ, ξ; h). Note that the latter term has a complicated dependence on h. We know from Lemma 3.4 that σI(Hh) is independent of 0 < h ≤ 1. We may, therefore, compute it in the limit h → 0.

We know from Lemma A.2 [or from the semiclassical version of the functional calculus (Ref. 25, Theorem 8.7) for h0 small enough] that φ′(Ĥ) is a PDO and define s such that ∂ξĤφ′(Ĥ)=Ops⁠. We, therefore, obtain from Lemma 3.1 that
2πσI=Try∫R∂ξĤφ′(Ĥ)dξ=1(2π)d−1∫R2d−1trs(Y,ξ)dYdξ.
Passing to the semiclassical regime ζ → hζ with ∂ξĤhφ′(Ĥh)=Ophs with now s(y, ζ, ξ; h), we have
2πσI=Try∫R∂ξĤhφ′(Ĥh)dξ=1(2πh)d−1∫R2d−1trs(Y,ξ;h)dYdξ.
Let ς be such that φ′(Ĥh)=Ophς while ∂ξĤh=−∂ξσz⁠. Then, by application of the semiclassical composition calculus and of the Helffer–Sjöstrand formula (A12), we have
(15)
with (z−Ĥh)−1=Ophrz⁠. The semiclassical composition calculus (Moyal product) is given in generic variables by
(16)
We defined here
(17)
For j = 1, this is {a, b} the standard Poisson bracket. The term O(hN+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 S0(M1M2) if a ∈ S0(M1) and b ∈ S0(M2).
We have the following bound coming from the functional calculus Lemma A.2 (or Ref. 25, Theorem 8.7 for h0 small enough):
|∂βs(Y,ξ;h)|+|∂βς(Y,ξ;h)|≤CN,β⟨Y,ξ⟩−N,
uniformly in 0 < h ≤ h0. Moreover, since Ĥh 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(Y,ξ;h)=∑j=0Mhjsj(Y,ξ)+hM+1ρ̃M(Y,ξ;h),
with ρ̃M bounded in S0(⟨Y,ξ⟩−N) for any N (uniformly in 0 < h ≤ h0). We thus observe that
2πσI=1(2π)d−1∫R2d−1trsd−1(Y,ξ)dYdξ=limR→∞1(2π)d−1∫[−R,R]2d−1trsd−1(Y,ξ)dYdξ.
It remains to identify sd−1. In addition, from the functional calculus, we have
sd−1=(−∂ξσz♯hς)d−1=∑j=0d−1(−∂ξσz♯hςj)d−1−j.
We thus expect from (15) that
(18)
Assuming that ρz(Y, ξ; h) is uniformly bounded in 0 < h ≤ h0 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 |∂̄φ̃′|≤C|ω|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 σz♯hrz = I and the estimate |∂ξ,Yαrz|≤Cα|ω|−p−|α| for some finite p uniformly in 0 < h ≤ h0 and (ξ,Y)∈R2d−1 coming from Ref. 25, Proposition 8.6 [see also (A10) in the  Appendix], the error term ρz(Y, ξ; h) in the Taylor expansion in h of rz is also bounded by C|ω|−q for some finite q.
We thus obtained that
2πσI=limR→∞−2(2π)d∫[−R,R]2d−1×C∂̄φ̃′(z)trτdξdYd2z,τ:=∑j=0d−1(−∂ξσz♯hrjz)d−1−j.
Since σz♯hrz = rz♯hσz = I, and using the expansion (16),
I=∑j≥0hjrzj♯hσz=∑j,khj+k1k!−i2k{rzj,σz}k implies∑j+k=l1k!−i2k{rzj,σz}k=δl0,l≥0.
The leading equation is rz0=σz−1⁠, which is defined for ω ≠ 0. Then a higher-order equation can be solved iteratively for rzj. The next two equations are, for instance,
−i2{rz0,σz}+rz1σz=0,−18{rz0,σz}2−i2{rz1,σz}+rz2σz=0.
We thus observe iteratively that rzj(Y, ξ) is a product of a maximum of 2j + 1 terms alternating a derivative (possibly of order 0) of σz with one (possibly of order 0) of σz−1⁠. The same property holds for
(19)
We now show that most terms in (19) do not contribute to the trace. Let us rescale τ → ητ for τ one variable in (Y, ξ) → (Yη, ξη). By Lemma 3.4, the trace is independent of η when Ĥ now has symbol ad−1(Yη, ξη). Let sd−1(Y, ξ; η) be the corresponding symbol appearing in the trace calculation. We thus have
∫trsd−1(Y,ξ;η)dYdξ=η∫trsd−1(Y,ξ;1)dYdξ.
From the calculations leading to (19), we obtain
sd−1(Y,ξ;η)=∏j=1J∂αjσβj(Yη,ξη)=ηγ∏j=1J(∂αjσβj)(Yη,ξη),
where βj ∈ { −1, 1} and γ is the number of derivatives in the variable τ that appear in sd−1. Integrating the latter expression over R2d−1 and changing variables (Yη, ξη) → (Y, ξ) shows that necessarily γ = 1 in order for 2πσI 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 sd−1. Therefore, ∂ξσz 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 σI. 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
2πσI=limR→∞−2(2π)d∫[−R,R]2d−1×C∂̄φ̃′(z)tr(τ0+τ1)dξdYd2z,τk=1k!−i2k{−∂ξσz,rz(d−1−k)}k.
We next verify that z → τ is analytic when ω ≠ 0 since z → σz and z→σz−1 are analytic so that ∂̄φ̃′(z)trτ=∂̄(φ̃′(z)trτ)⁠. Using ∂̄=12(∂λ+i∂ω) and the fact that φ̃′(z) is compactly supported while φ̃′(λ)=φ′(λ) on the real axis, we integrate by parts on ω > 0 and ω < 0 to obtain that
2πσI=limR→∞i(2π)d∫[−R,R]2d−1×Rφ′(λ)tr(τ0+τ1)λ−i0λ+i0dλdξdY.
We observe that τ1 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 σz−1(ξ,Y) is defined and τ1|λ−i0λ+i0 vanishes there.
It, therefore, remains to identify the terms in τ0 = −∂ξσzrz(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,
rzj=i2{rz,j−1,σz}σz−1+non contributing terms to the trace integral.
Let r̃zk collect the contributing terms. We have
r̃zj=i2{r̃z,j−1,σz}σz−1=i22{{r̃z,j−2,σz}σz−1,σz}σz−1=i2j{σz−1,σz}fjσz−1,
where {σz−1,σz}fj is the subset of {σz−1,σz}j where differentiation in a pair (yk, ζk) appears at most once. Denote cj=(i2)j⁠. We have for k = d − 1 the expression
{σz−1,σz}fd−1=∑ρ∈Sd−1{σz−1,σz}ρ1…{σz−1,σz}ρd−1,
where the sum is over ρ∈Sd−1 the set of permutations of {1, …, d − 1}. The edge conductivity is given, using the cyclicity of the trace, by
(20)
Note that trσz−1∂ξσz{σz−1,σz}fd−1dYdξ is a (2d − 1)-form on R2d [in the variables (ω, ξ, Y)]. However, it is not quite (σ−1dσ)2d−1 yet. This is where we use the invariance of 2πσI against rotations of the variables Y stated in Lemma 3.4.
For σ = σ(x) in generic variables in n dimensions, we have
(σ−1dσ)n=∑ρ∈Sn(−1)ρσ−1∂ρ1σ…σ−1∂ρnσdx1∧⋯∧dxn,
where (−1)ρ=ϵρ1,…,ρn is the signature of the permutation ρ: (1, …, n) → (ρ1, …, ρn). By cyclicity of the trace, the term σz−1∂ξσz can always be brought to the left of the product. However, {σz−1,σz}fd−1 involves a summation over only specific permutations of the variables Y. It is where having a symbol in an isotropic class with g = gi is used. From Lemma 3.4, any rotation in the variables Y does not change σI, so that any permutation of the variables in Y with a positive determinant leads to the same σI, and any permutation with a negative determinant leads to −σI.
Note that
{σz−1,σz}j=−σz−1∂ζjσzσz−1∂yjσz+σz−1∂yjσzσz−1∂ζjσz.
Therefore, all terms of the form σz−1∂yjσzσz−1∂ζjσz come with positive orientation while the terms with (yj, ζj) permuted come with negative orientation. We thus find that
(21)
where summation is over all permutations of 1, …, 2d − 2 and (−1)ρ is the signature of the permutation. Combining the permutations generating {⋅,⋅}fd−1⁠, each term ∏j=12d−2σz−1∂ρj(Y)σz(−1)ρ appears γd = 2d−1(d − 1)! times, where 2d−1 comes from the difference of products in each Poisson bracket and (d − 1)! from the possible permutations of the variables.
Therefore, combining (20) and (21) with cd−12d−1 = id−1 yields
(22)
Let λ be fixed (in a compact interval since φ′ has compact support). The form tr(σz−1dσz)2d−1 is closed since we verify that d(tr(σz−1dσz)2d−1)=0 as a 2d-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 2d − 1 surface S = {0 +} × [−R, R]2d−1 ∪ {0 −} × [−R, R]2d−1. Note that on ∂[−R, R]2d−1, σz−1 is defined so that tr(σz−1dσz)2d−1λ−i0λ+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 σz−1 is continuously defined on the volume with boundary given by S∪SR⁠.

The integral on SR is also independent of λ in a compact interval since λ → σz is continuous as an application of the result in Remark 2.5. Since ∫Rφ′(λ)dλ=1 and σα+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 dxd ∧ dξd ∧ dx1 ∧ dξ1…dxd−1 ∧ dξd−1 > 0 with ω ≡ xd the first variable defining the surface S. The latter orientation is (−1)ddξ1 ∧ dx1 ∧ ⋯∧ dξd ∧ dxd. 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 2πσI 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 x1 = ±R constant. This also uses as in the above derivation that d(tr(a−1da)∧3)=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.

We assume in this section that ak∈ESkm has the following form:

(23)

where hk(X) is a (d + k)-dimensional vector field on X∈R2d and Γk 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 Hk = Op ak may be computed as a topological degree associated with the map hk. This simplifies the estimation of the integral (10).

For 0 ≤ k ≤ d, let κ≔κk=⌊d+k2⌋ and nk=2κk⁠. The matrices Γk=(γκj)j for 1 ≤ j ≤ d + k are constructed to satisfy the commutation relations

(24)

These properties imply that ak2=|hk|2Ink is proportional to identity. The matrices Γk may be defined explicitly as follows: the matrices γκj at level κ are constructed starting from γ11,2,3=σ1,2,3 the standard Pauli matrices and then iteratively as

(25)

The last matrix plays the role of the chiral symmetry matrix in even dimension d + k = 2κk. The construction of the augmented Hamiltonians Hj 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

γκ0ak+akγκ0=0,γκ0≔γκ2κ+1.

For ak = hk · Γk, we denote by hj for k ≤ j ≤ d the vector fields of dimension d + j such that the augmented Hamiltonians constructed in Sec. II satisfy Hj = Op aj with, as we verify, aj = hj · Γj.

Dirac operators are the prototypical example of operators in the form (23). In two dimensions, we have explicitly Γ0 = (σ1, σ2) while h0(X) = (ξ1, ξ2) and Γ1 = (σ1, σ2, σ3) while h1(X) = (ξ1, ξ2, x1). In dimension d = 3, we have Γ0 = (σ1, σ2, σ3) while h0(X) = (ξ1, ξ2, ξ3), next Γ1 = (σ1 ⊗ σ1, σ1 ⊗ σ2, σ1 ⊗ σ3, σ2 ⊗ I2) while h1(X) = (ξ1, ξ2, ξ3, x1), and finally Γ2 = (σ1 ⊗ σ1, σ1 ⊗ σ2, σ1 ⊗ σ3, σ2 ⊗ I2, σ3 ⊗ I2) while h2(X) = (ξ1, ξ2, ξ3, x1, x2). When d = 3, then κ0 = 1 while κ1 = κ2 = 2 for a maximum of matrices satisfying (24) equal to 2κ2 + 1 = 5. Several other examples will be presented in Sec. VI.

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 hk.

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 C̄=C∪∂C⁠. Let h:C̄→Rn be a sufficiently smooth map such that |h(ζ)| > 0 for ζ ∈ ∂C. There are regular values y0 of h arbitrarily close to 0 by Sard’s theorem that allow us to define the degree of h as

(26)

The above sum ranges over a finite set and is independent of the regular value y0 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⊂R2d the ball of radius R given by {|X| ≤ R}. By ellipticity assumption, |hd| > 0 on ∂Bd for R large enough. We now define degrees for hd with two possible orientations,

(27)

We observe that

(28)

The degree deg̃ is naturally related to Index F, while the degree deg is more naturally related to that of hk as we now describe.

Using Lemma 3.5, we obtain that the index is unchanged if hk(xk′,xk′′,ξ) is replaced by hk(xk′,ξ)≔hk(xk′,0,ξ) in the definition of the symbol. We may, therefore, interpret hk as a map from Rd+k to Rd+k such that, thanks to the ellipticity constraint, |hk| ≥ h0 > 0 for |(xk′,ξ)|≥R⁠. Let Bk={|(xk′,ξ)|<R}⁠. We define

(29)

The orientation of Bk is inherited from that of Bd as the subset xk′′=0⁠. With these definitions, we obtain the main result of this section.

Theorem 5.1.
We have
2πσI=IndexF=(−1)d−1deg̃(hd)=(−1)12d(d+1)+1deg(hk).

In other words, Index F = deg(hk) when d = 1, 2 mod 4 and Index F = −deg(hk) 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 σd; (ii) next in terms of hd; (iii) identify (10) with the degree of hd on the sphere SR⁠; (iv) identify it with the degree deg(hd,B̄d,0)⁠; (v) decompose hd=(hk,h̃) with h̃ the augmentation map. The degree of hd is then the product of the other two degrees. Now the degree of h̃ is one and this gives the result.

Lemma 5.2.
We have
IndexF=−12−(d−1)!(2πi)d(2d−1)!∫SR2d−1trγ0d(ad−1dad)2d−1.

Proof.
By construction, a = ad−1 − imd and ad = σ+ ⊗ a + σ− ⊗ a*. Therefore, ad−1dad=Diag(a−*da*,a−1da) and hence (ad−1dad)2d−1=Diag((a−*da*)2d−1,(a−1da)2d−1)⁠. Therefore, with γd0=σ3⊗I
trγd0(ad−1dad)2d−1=tr(a−*da*)2d−1−tr(a−1da)2d−1.
The above traces are not necessarily linearly dependent. However, by Theorem 2.3, the integral of the second term gives −Index F, while the integral of the first term gives Index F* = −Index F since a* is the symbol of F*. This gives the result.

Lemma 5.3.
We have,
trγd0(ad−1dad)2d−1=(−1)d−1|ad|−2dtrγd0ad(dad)2d−1.

Proof.
We first observe that aj2=|hj|2 for k ≤ j ≤ d is proportional to identity thanks to (24). Since ad2=|hd|2 is scalar, then ad−1=wad for w=ad−2 scalar so that
ad−1dad=−dad−1ad=−dwad2−wdadad,
and hence (ad−1dad)2=−dwaddad−w(dad)2. Using dwdad2=0 so that dwdadad = −dwaddad, we find (ad−1dad)4=dw2ad(dad)3+w2(dad)4 and more generally
(ad−1dad)2n=(−1)n[dwnad(dad)2n−1+w2(dad)2n],
as well as (for d ≥ 2),
(ad−1dad)2d−1=(−1)d−1[dwd−1(dad)2d−2+wdad(dad)2d−1].
Therefore, we obtain for d ≥ 1 that
trγd0(ad−1dad)2d−1=(−1)d−1|ad|−2dtrγd0ad(dad)2d−1,
with the term in 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).□

Lemma 5.4.
For ad = hd · Γd, we have
trγd0ad(dad)2d−1=(2i)d∑ρ∈S2d(−1)ρhρ1ddhρ2d∧⋯∧dhρ2dd,(2i)dϵi1,…,i2d=trγd0γdi1…γdi2d.

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 trσ3σ1σ2 = 2i. The above three lemmas show that the index of F is related to an appropriate integral of hd.

Lemma 5.5.
We have,
IndexF=(−1)d−1deg̃hd.

Proof.
Let Σ be a smooth hypersurface in R2d locally parameterized by X = X(u) for u∈R2d−1⁠. We introduce the 2d × 2d matrix L(u) constructed as follows (see Ref. 28, Corollary 14.2.1). The first row is L1i(u)=hid◦X(u) while the following rows are Lj+1,i(u)=∂ujhid◦X(u) for 1 ≤ j ≤ 2d − 1. We then observe that
(30)
Indeed, with h′ the vector (hρ2,…,hρ2d) and ∇uh′ the Jacobian matrix, we have
dhρ2d∧⋯∧dhρ2dd=Det∇uh′du1∧⋯∧du2d−1.
For any permutation ρ, we observe that (−1)ρ and Det∇uh′ change signs together so that denoting by Lρ1 the matrix L with the first row and the ρ1 column deleted, we have
∑ρ∈S2d(−1)ρhρ1ddhρ2d∧⋯∧dhρ2dd=(2d−1)!∑ρ1=12d(−1)ρ1DetLρ1du1∧⋯∧du2d−1,
which gives (30), noting that (2d − 1)! is the number of permutations in Sd−1⁠.
Collecting the results of the above three lemmas, we obtained that
(31)
Therefore,
−(d−1)!(2πi)d(2d−1)!tr(a−1da)2d−1=(−1)d−1γ2d−1|hd|−2dDetLdu1∧⋯∧du2d−1,γ2d−1=2πd(d−1)!,
where γ2d−1 is the volume of the unit sphere S2d−1⁠.

Let f be the Gauss map associated with hd and given by f(X) = |hd(X)|−1hd(X) for X∈SR2d−1⁠. Then we recognize in the integration of the right-hand side of (31) over SR2d−1 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;B̄d,0) when 0 is a regular value of hd. When 0 is not a regular value, we apply the result of hd − y0 for y0 small with the result independent of y0; see also Ref. 46, Remark 1.5.10. With the chosen orientation to define deg̃ and Theorem 2.3, we thus obtain the result of the lemma.

Lemma 5.6.

We have deg(hd) = deg(hk).

Proof.
Let ζj ↦ hj(ζj) for j = 1, 2 be two smooth functions from Rnj to itself with |hj(ζj)| ≥ c0 > 0 for |ζj| ≥ Rj. Let Bj = Bj(0, Rj) be the centered balls of radius Rj for the Euclidean metric in Rnj for j = 1, 2. Let now (ζ1, ζ2) = ζ ↦ h(ζ) be the function from Rn to itself with n = n1 + n2 defined by
h(ζ)=(h1(ζ1),h2(ζ2)).
We find that |h(ζ)| ≥ c0 > 0 for ζ ∈ ∂C, where C = B1 × B2. Since 0 does not belong to the range of h or hj on the respective boundaries, we can define the degrees
deg(h,C̄,0) and deg(hj,B̄j,0),j=1,2.
Let y0 be a regular value of h, i.e., a point in h(C̄)\h(∂C) such that h−1(y0)={ζ∈B̄R;h(ζ)=y0} are a finite number of isolated regular points (where ∇h is invertible). Note that y0 = (y1, y2) with yj ∈ Bj. By Sard’s theorem, such regular values exist. Then, independently of such a y0,
deg(h,C̄,0)=∑ζ∈h−1(y0)sgndetJh(ζ),
where Jh is the non-vanishing Jacobian of the map ζ → h(ζ). Now, by construction,
detJh(ζ)=detJh1(ζ1)detJh2(ζ2).
Moreover, h(ζ) = y0 means (h1(ζ1), h2(ζ2)) = (y1, y2) so that h−1(y0)=h1−1(y1)×h2−1(y2) and hence
∑ζ∈h−1(y0)sgn det Jh(ζ)=∑ζ1∈h1−1(y1)sgn det Jh1(ζ1)∑ζ2∈h2−1(y2)sgn det Jh2(ζ2).
We recognize the product of degrees for the regular values yj∈hj(B̄j)\hj(∂Bj)⁠. Since degrees are independent of such regular values locally, we obtain that
deg(h,C̄,0)=∏j=12deg(hj,B̄j,0).
We observe that C ⊂ BR the ball of radius R=R12+R22⁠. Since |h| ≥ c0 > 0 on B̄R\C⁠, invariance of the results with respect to (continuous) domain changes (see Ref. 46, Proposition 1.4.4) shows that
(32)

We now choose h1 = hk and h2 the vector so that hd = (hk, h2) with n1 = d + k and n2 = d − k where R2d is oriented using dξ1…dξddx1…dxd > 0 and the subspaces Rd+k (for hk) and Rd−k (for h2) with the induced orientation. We observe that the degree of h2=h2(xk′′)=(mk+1,…,md)(xk′′) equals 1 since the only point in h2−1(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.

The classification presented in Sec. II applies to Hamiltonians that are (a) continuous (with an open “Brillouin” zone ξ∈Rd⁠), (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 Hk, 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(x1) is assumed to be bounded and constant away from a compact domain.

For any symbol ak such that (i) and (ii) hold, we allow for the following modifications of the symbol ak 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

(33)

We use the same notation for the smooth function Rp∋y→⟨y⟩ε≔⟨|y|⟩ε⁠. 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 ak 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 H1 = D1σ1 + D2σ2 + (μ − ηD · D)σ3 with here D⋅D=D12+D22 the (positive) Laplacian and R∋η≠0⁠. The definition of a bulk invariant is ambiguous when η = 0, while it yields a Chern number 12(sign(μ)+sign(η)) 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

H1=D1σ1+D2σ2+(μ(x1)−η⟨D⟩ε−1D⋅D)σ3≔Opξ1σ1+ξ2σ2+(μ(x1)−η⟨ξ⟩ε−1|ξ|2)σ3,

where we assume that μ(x1) equals x1, say, outside of a compact set in R⁠. We will verify below that the topological charge of H1 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=⟨D⟩ε(D1σ1+D2σ2)+(⟨x1⟩εx1−ηD⋅D)σ3 satisfying (i) and (ii) with now m = 2.

We next consider several examples of topological insulators and superconductors in dimensions d = 1, 2, and 311,48,51,52,56 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.

The first example is the Dirac operator with H0 = Op a0 for a0(X)=h0(X)⋅Γ0∈ES01(gs) in dimension d, where

h0(X)=(ξ1,…,ξd),

and Γ0 are Clifford matrices acting on spinors in C2κ0 with κ0=⌊d2⌋⁠. These generalize the cases d = 1, 2, and 3 considered in the introduction. We then observe from (26) that deg h0 = 1 since (h0)−1(0)={0} and ∇h0(0) = Id, and that the topological charge of H0 is given by IndexF=2πσI(Hd−1)=(−1)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 h0 or by that of hd = (ξ1, …, ξd, x1, …, xd) since the deg h0 = deg hd = 1.

If A is a non-singular (constant) matrix in Md(R) and we consider instead the operator H0 = Op (Ah0)(ξ) · Γ0, 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=Oph̃0⋅Γ0 with h̃j0(X)=bj(x)hj0(ξ) and bj(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 h̃0⋅Γ0∈ES01(gs) though not necessarily in ES01(gi)⁠. Note that a more isotropic perturbation of the form bj(X)hj0 for bj(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 h1(X) = (ξ1, …, ξd, x1). Based on Theorem 5.1, its topological charge is again given by deg h1 = 1. Domain walls of the form bj(x)x1 even with bj(x) = 1 outside of a compact domain no longer necessarily generate perturbations such that a1 remains in S11 and are, therefore, not allowed in the theoretical framework of this paper. We may, however, replace x1 by m(x1) equal to x1 outside of a compact set. Inside that compact set, the level set m(x1) = 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 a0 = ξ = h0 while a = ξ − ix and a1 = ξσ1 + xσ2. We then observe that a−1da=(ξ2+x2)−1(ξdξ+xdx+i(ξdx−xdξ))⁠. In polar coordinates ξ = r cos θ and x = r sin θ, we observe that a−1da = r−1dr − idθ whose integral along the curve r = 1, gives −2πi, 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 a1 = ξσ1 + xσ2 with trσ3a1−1da1=a−*da*−a−1da and a* = ξ + ix so that a−*da* = r−1dr + idθ. This shows that trσ3a1−1da1=2idθ whose appropriate integral gives the topological charge. Now, a1−1=|h1|−2(h11σ1+h21σ2) for h1 = (ξ, x). Therefore, as in Lemma 5.4, trσ3a1−1da1=|h1|−2trσ3(h11σ1+h22σ2)(dh11σ1+dh21σ2)=2i(h11dh21−h21dh11)⁠. We recognize in the integral of the latter form over the circle an expression for the degree of h1 written as the degree of the Gauss map, which to X∈S1 associates h1(X)/|h1(X)|. Using the expression (26) of the degree over the unit disk C gives deg h1 = 1 since ∇h1 = I2 at the unique point X = 0 where h1 = 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 H1 = Op h1 · Γ1 with vector field h1=(ξ1,ξ2,x1−η⟨ξ⟩ε−1|ξ|2) so that h1⋅Γ1∈ES11⁠. We observe that h1 = 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 H1 equals 1 independent of ɛ and η as advertised.

The above orientation of the vector fields hk 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−ix1=−ia1 and then define iteratively

Fn+1=σ−⊗Fn+σ+⊗Fn*+σ3⊗Dn+1−ixn+1.

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

f♯g≔f⊗II⊗g*I⊗g−f*⊗I,

where we verify that Index f♯g = Index f Index g. We apply it with g = Fn−1 and f = Dn − ixn. It is then straightforward to obtain that Index Fn = 1 for all n ≥ 1. We then observe that Fn(1,0,…,0)te−12|x|2=0 with spinor (1, 0, …, 0)t of dimension 2n−1.

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,

H0=(D1−A1)σ1+(D2−A2)σ2+V,

with A = (A1, A2) the magnetic vector potential and V a bounded scalar potential with compact support, say. The magnetic field is given by B = ∇ × A = ∂1A2 − ∂2A1. We choose the Landau gauge such that A2=B0x1+Ã2 and A1=Ã1 for à an arbitrary (smooth) compactly supported perturbation. In that gauge, we obtain that

H1=(D1−A1)σ1+(D2−A2)σ2+V+x1σ3,

is an operator H1 = Op a1 with a1∈ES11 for n1 = 2. Note that for H0 = Op a0, we do not have that a0 belongs to ES01 because of the presence of the unbounded magnetic potential. We would also not have that a1 belongs to ES11 if A=(−12B0x2,12B0x1) 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 2πσI(H1) = Index F = 1 for F = H1 − ix2, since the topological charge is given by

deg(ξ1,ξ2−B0x1,x1,x2)=deg(ξ1,ξ2,x1,x2)=1.

We could more generally consider a magnetic field with constant and opposite values as x1 → ±∞, for instance, with A2=B02πarctan(x1)x1⁠. The topological charge of H1 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(x1) [with x1σ3 replaced by μ(x1)σ3 in the above definition of H1]. In such a setting, both magnetic and mass confinements compete to generate asymmetric transport. Only when the bounded domain wall μ(x1) converges to sufficiently large values μ± as x1 → ±∞ (for a fixed magnetic field) do we retrieve that the asymmetric transport of 2πσI(H1) 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.

Let us consider the Weyl operator D · σ in dimension d = 3. As we mentioned in the introduction, the operator H2 = σ1 ⊗ D · σ + σ2 ⊗ Ix1 + σ3 ⊗ Ix2 generates a hinge in the third direction along which asymmetric transport is possible. With our choice of orientation, we have 2πσI(H2) = −deg(ξ1, ξ2, ξ3, x1, x2) = −1.

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

(34)

We thus deduce from Theorem 5.1 that

(35)

The last result is most easily obtained by identifying, as we did in the Proof of Lemma 5.5, the degree of h2 on the unit ball with the degree of the Gauss map x→ĥ2=h2/|h2| from the unit circle S1 to itself and then to the degree (winding number) of the map x1+ix2→(x1+ix2)p from the unit circle to itself, which equals p.

By an appropriate construction of the coefficients in the Hamiltonian H2 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).

Several superconductors and superfluids11,56 are modeled by Hamiltonians of the form

H1=ηD⋅D−μσ3⊗I2+HΔ,

with coupling term HΔ = ∑i,j=1,2Δij(X)σi ⊗ σj for scalar operators Δij and η=(2m*)−1 for a mass of the quasi-particle m* > 0. For the above choice of the order parameter59 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.

For d = 1, an example with the order parameter Δ proportional to Dx gives

H̃1=(ηDx2−μ)σ3+DxReΔσ1−DxImΔσ2,

with 0≠Δ∈C⁠. Let Δ = |Δ|eiθ and g=eiθ2σ3⁠. We then verify that

gH̃1g*=(ηDx2−μ)σ3+|Δ|Dxσ1,

and so we may assume Δ real-valued. Define g2=eiπ4σ2 and g1=eiπ4σ1⁠. We verify that g1g2σ1,2,3(g1g2)*=σ3,1,2⁠, so that

g1g2gH̃1(g1g2g)*=(ηDx2−μ)σ1+ΔDxσ2=h̃1⋅Γ1,h̃1=(ηξ2−μ,Δξ).

This is of the form σ− ⊗ F* + σ+ ⊗ F with F=(ηDx2−μ)−iΔDx⁠.

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 H1 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∋y→⟨y⟩ε=⟨|y|⟩ε 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,

(36)

and one with a domain wall in the order parameter

(37)

We observe that for H1 = Op a1, then a1∈ES1m 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 |h1|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≥(ηξ2−μ⟨x⟩ε2)2+Cξ2⟨x⟩ε2 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∈ES12⁠. A similar computation shows that a1=h1⋅Γ∈ES11 in (36). Note that we could also have used the following regularization for the first example: h1 = (ηξ2 − μx⟨x⟩ɛ, Δξ⟨ξ⟩ɛ), in which case h1⋅Γ1∈ES12⁠.

We now compute the topological charges of the regularized operators starting with (36). We observe that h1(ξ, x) vanishes only at x = ξ = 0. The Jacobian there has a determinant equal to μΔ. The topological charge of (36) is, therefore, equal to Index F = deg h1 = sign(μΔ) assuming μΔ ≠ 0. Here and below, F is defined as usual by the relation H1 = σ− ⊗ F* + σ+ ⊗ F.

We next turn to (37), where h1(ξ, x) vanishes when ηξ2 = 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 ξ=±μ/η⁠. At these points, the Jacobian matrix ∇h1 has components (2ηξ; 0; Δx; Δξ) with a determinant equal to 2ηΔξ2. The topological charge of H1 in (37) is, therefore, equal to Index F = deg h1 = 2 sign(Δ).

Let us finally consider the asymptotic regime η = 0 for a mass term m* → ∞ and a corresponding Hamiltonian H1 = −μσ1 + ΔDxσ2. A domain wall in the chemical potential is then modeled by μ(x) = μx. We then observe that H1 = Op a1 with a1∈ES11 and a topological charge equal to Index F = deg h1 = sign(μΔ) as in the setting η > 0. A domain wall in the order parameter requires the following regularized Hamiltonian H1=−μ⟨x⟩ε2σ1+Δ12(Dxx+xDx)σ2⁠, which is, however, gapped for μ ≠ 0 and hence topologically trivial.

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

H1=Δ1D1σ1+Δ2D2σ2+(ηD⋅D−μ)σ3.

We assume here that Δ1 and Δ2 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

(38)

The symbol of this operator is h1 · Γ1 with h1(ξ1,ξ2,x1)=(Δ1ξ1,Δ2ξ2,η⟨ξ⟩ε−1|ξ|2−μx1)⁠. This regularization ensures that a1∈ES11 is elliptic. We could have defined a regularization in ES12 instead with h1=(Δ1ξ1⟨ξ1⟩ε,Δ2ξ2⟨ξ2⟩ε,η|ξ|2−μx1⟨x1⟩ε)⁠.

It remains to compute the degree of h1. We find that h1 = 0 when ξ1 = ξ2 = x1 = 0 and that the Jacobian determinant there is equal to −μΔ1Δ2. The topological charge of the above operator is therefore 2πσI(H1) = deg h1 = − sign(μΔ1Δ2), which is consistent with Ref. 11.

In Ref. 56, Chap. 22, the domain wall is implemented in the order parameter Δ1, which, after appropriate regularization, gives

H1=Δ1x1D1+D1x12σ1+Δ2D2⟨D2⟩εσ2+(ηD⋅D−μ⟨x1⟩ε2)σ3.

The constants Δ1, Δ2, and μ are assumed not to vanish and μ > 0. The symbol a1=h1⋅Γ1∈ES12 is given by

h1(ξ1,ξ2,x1)=(Δ1ξ1x1,Δ2ξ2⟨ξ2⟩ε,η|ξ|2−μ⟨x1⟩ε2).

We have h1 = 0 when ξ2 = 0, x1 = 0, and ηξ12=μ⁠. At each of the two solutions, the Jacobian of h1 is given by

∇h1=00Δ1ξ10Δ202ηξ100,|∇h1|=−2ηΔ1Δ2ξ12.

Therefore, the topological charge of H1 is equal to 2πσI(H1) = deg h1 = −2 sign(Δ1Δ2) as in Ref. 56, Chap. 22. When μ < 0, the find deg h1 = 0 again.

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

H1=Δ1D1D2σ1+Δ2(D12−D22)σ2+(ηD⋅D−μ)σ3.

Following Ref. 56, Chap. 22, we implement a domain wall in Δ1(x1) and a regularization that gives the operator

H1=Oph1⋅Γ1,h1=Δ1x1⟨x1⟩εξ1ξ2,Δ2(ξ12−ξ22),η|ξ|2−μ⟨x1⟩ε2.

This generates a symbol a1=h1⋅Γ1∈ES12 as may be verified. Then h1 = 0 when ξ12=ξ22=μ2η while x1 = 0. At each of these four roots, we compute

∇h1=00Δ1ξ1ξ22Δ2ξ1−2Δ2ξ202ηξ12ηξ20,|∇h1|=4ηΔ1Δ2ξ12ξ22.

The sign of the Jacobian is the same at each of the roots so that deg h1 = 4 sign(Δ1Δ2). Therefore, we obtain a topological charge 2πσI(H1) = deg h1 = 4 sign(Δ1Δ2) when both μ > 0 and η > 0. When μ < 0, the operator is gapped and topologically trivial again.

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

H1=Δσ1⊗(D⋅σ)+(ηD⋅D−μ)σ3⊗I,

acting on C4⁠. When η = 0 and μ(X) = μx1, we obtain a standard Dirac operator with a topological charge sign(μΔ), as may be verified (see also the following calculations). When η > 0, we conjugate the above operator by g1 ⊗ I (which maps σ3 to −σ2), and after regularization and domain wall μ(X) = μx1, we obtain the Hamiltonian

H1=Δσ1⊗(D⋅σ)+(μx1−η⟨D⟩ε−1D⋅D)σ2⊗I.

The operator has an elliptic symbol in ES11⁠. We can then introduce as earlier H2 = H1 + σ3 ⊗ Ix2 and F = H2 − ix3. Following Theorem 5.1, the topological charge of H1 is then defined as Index F = 2πσI(H2) = −deg h1 with

h1=(Δ(ξ1,ξ2,ξ3),μx1−η⟨ξ⟩ε−1|ξ|2).

We find h1 = 0 at the point ξ = 0 and x1 = 0. The Jacobian ∇h1 at this point has a determinant Δ3μ so that the topological charge is given by Index F = − sign(μΔ).

The above examples all fit within the framework of operators with symbols ak = hk · Γk 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 hk as shown in the preceding examples.

The computations easily extend to operators of the form Hk⊕H̃k 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 ak = hk · Γk 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) Hamiltonian24,54

(39)

where f = f(x1) represents a (real-valued) Coriolis force. The symbol of that operator has two eigenvalues ±λ(x, ξ) with λ(x,ξ)=ξ12+ξ22+f2(x1) similar to those of a Dirac operator and a third uniformly vanishing eigenvalue. Therefore, H0 + 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 a0 is associated with the eigenvector ψ0=(ξ12+ξ22+f2)−12(if,ξ2,−ξ1)t⁠. We define the projector Π0=ψ0ψ0* and for 0≠μ∈R the regularized (pseudo-differential) Hamiltonian Hμ=H0+μOpλ2(1+λ2)−12Π0⁠. The symbol now has eigenvalues given by ±λ and μλ2(1+λ2)−12 (ensuring that the symbol of Hμ is smooth). If we choose f(x1) = νx1 with ν ≠ 0 to generate a domain wall in the first variable, we observe that Hμ 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 Hμ 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,

(40)

for instance, for A = (0, Bx1), 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−1da)∧2=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.

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

The author has no conflicts to disclose.

Guillaume Bal: Writing – original draft (lead).

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

This appendix recalls the results summarized in Ref. 15 allowing us to characterize spaces of symbols ak for Hk = Op ak adapted to operators modeling unbounded domain walls, domains of definition for Hk, as well as functional calculus showing, in particular, that (z−Hd−1)−1 and φ′(Hd−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∈Rd and ξ∈(Rd)*≃Rd⁠, we define a Riemannian metric g in the Beals–Fefferman form by

gX(dx,dξ)=Φx−2(X)dx2+Φξ−2(X)dξ2.

We use the notation ⟨u⟩=1+|u|2 for |·| the Euclidean norm applied to a vector u. For u = (u1, u2), we use the notation ⟨u1,u2⟩=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) = Φξ(X)Φx(X).

In this paper, we consider two metrics: gi and gs. The metric gi is defined by Φxi(X)=Φξi(X)=⟨X⟩≥1 with then hi = ⟨x,ξ⟩−2. The metric gs is defined by Φxs(X)=⟨x⟩≥1 and Φξs(X)=⟨ξ⟩≥1 with Planck function hs = ⟨x⟩−1⟨ξ⟩−1.

For 0 ≤ k ≤ d, we decompose x=(xk′,xk′′) with xk′∈Rk and xk′′∈Rd−k⁠. We define the weights

(A1)

Following classical calculations,15 the weights Mk are admissible for g ∈ {gi, gs} in the sense of Ref. 15, Definition 2.3, and satisfy that Mk ≤ Cλp for some C > 0 and p < ∞ when λ ∈ {λi, λs}. This implies that MkhN goes to 0 as X → ∞ for N sufficiently large when h ∈ {hi, hs}.

For g = gs, a symbol b ∈ S(M, gs) when for each multi-index (α, β), we have

(A2)

This is (4) when M(X)=wkm(X)⁠. For g = gi, a symbol b ∈ S(M, gi) when instead

(A3)

The metric gi is referred to as the isotropic metric. We have that S(M, gi) a subspace of S(M, gs). Since they appear repeatedly in the derivations, we define for 0 ≤ k ≤ d the spaces,

(A4)

We also define Skm≔Skm(gs) and S̃m≔S̃m(gs)⁠. Here, nk and nd−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.

Hilbert Spaces (Ref. 15 , Sec. 2.2.3).16  See also Refs. 14, 17, and 45 and Ref. 58, Chap. 8.3.

Associated to the weights Mk are the Hilbert spaces H(Mk,g) defined such that Opau∈L2(Rd) whenever a ∈ S(Mk, g). These spaces are independent of g ∈ {gs, gi}15 and hence referred to as H(Mk)⁠. We observe that H(1,g)=L2(Rd)⁠. The Hilbert spaces associated with Skm and S̃m are denoted for 0 ≤ k ≤ d by

(A5)

When ak∈Skm(g)⁠, we thus obtain that Op ak maps Hkm to Hk0=L2(Rd)⊗M(nk)⁠. Note that Hd−10=H̃0=L2(Rd)⊗M(nd−1)⁠. 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∈S(M,g)⊗M(n) Hermitian valued is elliptic when

|deta(X)|1n≥C1M(X)−C2,

for some positive constants C1,2. 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∈ES(M,g)⊗M(n) and define the corresponding spaces of Hermitian elliptic symbols as ESkm(g) for 0 ≤ k ≤ d and ES̃m(g)⁠.

Since M(X) ≤ Cλp(X), ellipticity implies that H=Opa∈OpES(M,g)⊗M(n) is a self-adjoint operator with domain of definition D(H)=H(M)⊗M(n) and such that for some positive constant C,15 

‖Hψ‖L2(Rd)⊗M(n)≥C(‖ψ‖H(M)⊗M(n)−‖ψ‖L2(Rd)⊗M(n)).

For ak∈ESkm⁠, we thus obtain that Hk = Op ak is a self-adjoint operator from its domain of definition D(Hk)=Hkm to Hk0⁠. Similarly, for a∈ES̃m⁠, then F = Op a is an unbounded operator from its domain of definition D(F)=H̃m to H̃0⁠.

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)⊗M(n) to H(M)⊗M(n) for z∈C when Im(z) ≠ 0.

With the above assumptions, we have the Wiener property15 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:

Lemma A.1.

Let H∈OpESd−1m. Then (±i+H)−1∈OpESd−1−m(g) is an isomorphism from Hd−10 to Hd−1m.

Proof.

The proof follows.14,15 Associated to H is a resolvent operator Rz = (z − H)−1, which is always defined as a bounded operator by spectral theory. When H is elliptic, then the domain D(H)=Hd−1m⁠. Moreover, Rz is a bijection from Hd−10=L2(Rn)⊗M(nd−1) to that domain. We then apply above Wiener property14–17 to obtain that Rz−1∈OpSd−1−m(g)⁠.□

The above shows that (I+H2)−1 maps Hd−10 to Hd−12m and has a symbol in ESd−1−2m⁠. 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:

Lemma A.2.

Let ϕ∈Cc∞(R) and H∈OpESd−1m. Then ϕ(H)∈OpSd−1−∞.

Remark A.3.

The above result means the following in terms of seminorms: for each N∈N 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−1m uniformly bounded in ɛ, this implies that the symbol of ϕ(Hɛ) is bounded in any Sd−1−N 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 ≤ h0 ≤ 1 be the semiclassical parameter. We define semiclassical operators in the Weyl quantization as

(A6)

for a(X; h) a matrix-valued symbol in M(n) for each X∈R2d and h ∈ (0, h0] 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 Hh by

(A7)

Note that Op a(x, hξ; h) = Opha(x, ξ; h). We define the classes of semi-classical symbols as Sj(M) constructed with the semi-classical metric in Beals–Fefferman form with Φx(X) = 1 and Φξ(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|)NM(y, ζ) uniformly in (x, y, ξ, ζ) for some C(M) and N(M). Then a∈Sj(M)⊗M(n) if for each component b of a, we have for each 2d-dimensional multi-index α, a constant Cα such that

(A8)

We will mostly use the case j = 0. We also use the notation b(X) ∈ S0(M) for symbols b(X) independent of h.

For two operators Opha and Ophb with symbols a ∈ S0(M1) and b ∈ S0(M2), we then define the composition Ophc = OphaOphb with symbol c ∈ S0(M1M2) given by the (Moyal) product (Ref. 25, Theorem 7.9),

(A9)

For a ∈ S0(1), we obtain (Ref. 25, Theorem 7.11, Ref. 13, Proposition 1.4) that Oph a is bounded as an operator in L(L2(Rd)⊗Cn) with bound uniform in 0 < h ≤ h0 so that I − hOph 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) ∈ S0(M) is invertible in Mn for all (x,ξ)∈R2d and h ∈ (0, h0] with then a−1 ∈ S0(M−1).

Following Ref. 25 (see Ref. 6, Lemma 4.14), we obtain the following results on resolvent operators. Let Hh = Oph a with a ∈ S0(M). Let z=λ+iω∈C with ω ≠ 0. Then (z−Hh)−1 is a bounded operator and there exists an analytic function z → rz = rz(y, ζ; h) such that (z−Hh)−1=Ophrz (compare to Lemma A.1). Moreover, the symbol rz ∈ S0(1) satisfies

(A10)

for all multi-indices β = (βy, βζ) and a constant Cβ independent of z ∈ Z a compact set in C and 0 < h ≤ h0.

Trace-class criterion. We have the following trace-class criterion (Ref. 25, Chap. 9). Assume that M∈L1(R2d) and that |∂αa(X)| ≤ CαM(X) for all |α| ≤ 2d + 1. Then Op a is a trace-class operator and

(A11)

In other words, all symbols in S0(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)⊗Cn and any bounded continuous function ϕ on R⁠, then ϕ(H) is uniquely defined as a bounded operator on L2(Rd)⊗Cn (Ref. 25, Chap. 4). Moreover, for ϕ compactly supported, we have the following representation:

(A12)

where, for z = λ + iω, d2z ≔ dλdω, ∂̄=12∂λ+12∂ω⁠, and where ϕ̃(z) is an almost analytic extension of ϕ. The extension ϕ̃ may be chosen as compactly supported in C⁠. Moreover, ϕ̃(λ)=ϕ(λ) and ∂̄ϕ̃(λ)=0 on the real axis. We can choose the almost analytic extension such that |∂̄ϕ̃|≤CN|ω|N for any N∈N uniformly in (λ, ω). Several explicit expressions, which we do not need here, for such extensions are available in Refs. 23 and 25.

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

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2023
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