Homological bulk-edge correspondence for Weyl semimetal

For a certain translation invariant tight-binding model of three-dimensional Weyl semimetals, we establish a bulk-edge correspondence as an equality of two relative homology classes, based on an idea of Mathai and Thiang: From spectral information on the edge Hamiltonian, we construct a relative homology class on the surface momentum space. This class agrees with the image under the surface projection of a homology class on the bulk momentum space relative to the Weyl points, constructed from the bulk Hamiltonian. Furthermore, the relative homology class on the surface momentum space can be represented by homology cycles whose images constitute the Fermi arc, the locus where the edge Hamiltonian admits zero spectrum.

detecting the corresponding edge states. Thus, from theoretical viewpoint, the justification of the bulk-edge correspondence is an important theme, and has been achieved in various situations as the coincidence of two quantities called the bulk index and the edge index (see e.g. [5,6,8,10,14,16,19,20,25]).
For example, in a tight-binding model which describes an insulator as a quantum system on the d-dimensional square lattice Z d with translation invariance, the information on the Hamiltonian H of the quantum system is encoded into a continuous familyĤ = {Ĥ(k)} k∈T d of invertible Hermitian matrices parametrized by the quasi-momentum k, which forms the d-dimensional torus T d = R d /2πZ d called the Brillouin zone. The eigenspaces ofĤ(k) with negative eigenvalues form a vector bundle EĤ on T d called the Bloch bundle. In the case that d = 2, we have the first Chern number of EĤ , which we denote by c 1 (Ĥ) ∈ Z. This is the bulk index. A typical Hamiltonian, called the Qi-Wu-Zhang (QWZ) model in [2], iŝ H QWZ (k x , k y ) = sin k x σk x + sin k y σ y + (u + cos k x + cos k y )σ z , where σ x , σ y and σ z are the Pauli matrices If we take the parameter u to be 0 < u < 2, thenĤ QWZ (k x , k y ) is invertible for all (k x , k y ) ∈ T 2 , and the Chern number is c 1 (Ĥ QWZ ) = 1.
To formulate the edge index, one introduces an edge to the lattice, and then impose the Dirichlet boundary condition to define a self-adjoint operator H ♯ , often called the edge Hamiltonian. Using the translation invariance in the direction along the edge, H ♯ induces a continuous family of self-adjoint operatorsH ♯ = {H ♯ (k)} k∈S 1 parametrized by the circle T 1 = S 1 = R/2πZ. It turns out thatH ♯ (k) is Fredholm, so that the spectral flow sf(H ♯ ) ∈ Z makes sense. The edge index is −sf(H ♯ ). Now, the bulk-edge correspondence for the 2-dimensional topological insulator (Chern insulator) is stated as the equality c 1 (Ĥ) = −sf(H ♯ ).
1.2. Bulk-edge correspondence for Weyl semimetals. The bulk-edge correspondence is also prominent in other topological states of matters. This is the case for the Weyl semimetal [1,27,28], which is experimentally confirmed by observing its characteristic edge states, known as Fermi arcs. However, mathematical study of the bulk-edge correspondence for Weyl semimetals, in particular one which explains topology of Fermi arcs, seem to be not fully developed yet. In this direction of research, a work of Mathai and Thiang [21,22] suggests a formulation of the bulk-edge correspondence. Based on their idea, we mathematically formulate and prove a bulk-edge correspondence for a particular model of Weyl semimetals in this paper. The model is more or less a protypical one, so that our result should be regarded as a first step toward the general case.
Based on the idea in [21,22], we here formulate the bulk-edge correspondence that we anticipate. LetĤ : T 3 → Herm(C 2 ) 0 be a continuous map from the 3-dimensional torus to the space Herm(C 2 ) 0 of 2 by 2 traceless Hermitian matrices. ForĤ to describe a Weyl semimetal, we tentatively require the following two properties.
(W2) At each point (k 0 x , k 0 y , k 0 z ) ∈ W , there is an approximation We will call W in (W1) the set of Weyl points ofĤ. Away from the Weyl points, H(k) is invertible. Thus, as in the case of topological insulators, we have the Bloch bundle over T 3 \W , and hence its first Chern class in the integral cohomology group This is essentially the same as the (bulk) topological invariant of the Weyl semimetal H studied in [21,22]. Now, suppose an edge on the lattice Z 3 . To be concrete, let Z 2 × {1} be the edge. Then, as in the case of topological insulators, we can get the edge Hamiltonian, namely, a continuous family of self-adjoint operators {H ♯ (k)} k∈T 2 parametrized by the 2-dimensional torus T 2 consisting of quasi-momenta on the boundary surface Z 2 × {1}. A definition of the Fermi arc ofĤ is the locus of k ∈ T 2 whereH ♯ (k) admits zero spectrum: where σ(H ♯ (k)) is the spectrum ofH ♯ (k). Note that π(W ) ⊂ F , where π : T 3 → T 2 is the projection π(k x , k y , k z ) = (k x , k y ). This is becauseH ♯ (k) at k ∈ π(W ) has 0 as an essential spectrum. What suggested in [21,22] as the bulk-edge correspondence is that the Fermi arc represents the homology class π * (PD(c 1 (EĤ ))) ∈ H 1 (T 3 , π(W )), where PD : H 2 (T 3 \W ) → H 1 (T 3 , W ) is the Poincaré duality map, and π * : H 1 (T 3 , W ) → H 1 (T 2 , π(W )) induced from the projection π : T 3 → T 2 . For example, if both W and π(W ) consist of distinct two points, then there is an isomorphism H 1 (T 2 , π(W )) ∼ = H 1 (T 2 ) ⊕ Z. A basis of the summand Z ∼ = H 1 (T 2 , π(W ))/H 1 (T 2 ) is the homology class of a path which connects the two points in π(W ). Thus, when F consists of (the images of) genuine paths, they represent a homology class in H 1 (T 2 , π(W )), and this homology class is expected to be π * (PD(c 1 (EĤ ))).
By definition, F is just a set. Hence F has no a priori parametrizations as paths. Taking this issue into account, we formulate the homological bulk-edge correspondence for Weyl semimetals as the following claim: Claim 1.1. For any Weyl semimetalĤ : T 3 → Herm(C 2 ) 0 (with appropriate assumptions), there would exist a homology class which has the following properties.
(3) If F = ∅, then F is recovered as the images of paths which represent F (H ♯ ).
In comparison with the bulk-edge correspondence for topological insulators, c 1 (EĤ ) plays the role of the bulk index, and F (H ♯ ) that of the edge index.
It is rather easy to define F (H ♯ ) satisfying (1) and (2) in a general setup. Actually, in this paper, we define F (H ♯ ) by the Poincaré dual of the spectral flow ofH ♯ | T 2 \π(W ) regarded as a first cohomology class (see § §3.3 for detail, cf. [26]). To the contrary, the verification of (3) for F (H ♯ ), defined as above for instance, seems somehow difficult. This is because the eigenvalues of operators may merge and branch, so that the set F may not have a simple description in general. Hence to find appropriate assumptions aboutĤ is the crucial part in showing Claim 1.1 Toward a general proof of Claim 1.1, some non-traditional descriptions of homology groups such as in [7,11] may be useful. In a recent work of Thiang [26], the local behaviour of zero loci of the edge Hamiltonian of a Weyl semimetal is generally analyzed. Though their global nature is not yet clarified, his result would be helpful to show Claim 1.1 in general.

Main result.
As is mentioned, this paper considers a certain model of a Weyl semimetal, and then prove the homological bulk-edge correspondence. To explain our model, we introduceĤ loc : It is easy to see thatĤ loc has two Weyl points (1, 0, π) and (−1, 0, 0). The associated edge Hamiltonian is a family of self-adjoint operators {H ♯ loc (a, b)} (a,b)∈R 2 , and the Fermi arc turns out to be the interval F = {0} × [−1, 1]. TakingĤ loc as the local model, we introduce our model of Weyl semimetals as follows.
Assumption 1.2. We suppose that a mapĤ : T 3 → Herm(C 2 ) 0 is expressed aŝ in terms of smooth maps a, b : T 2 → R with the following property: to the open subset a −1 ((−1, 1)) ⊂ T 2 , then 0 ∈ R is a regular value and the inverse image a −1 ((−1, 1))∩b −1 (0) consists of a finite number of connected components.
We notice that the characterization (W1) of a Weyl semimetal follows from (a), because π(W ) = a −1 ({±1}) ∩ b −1 (0). The characterization (W2) implies (b) when we understand the meaning of an "approximation" suitably ( Lemma 3.19). In this sense, our model is a generalization of Weyl semimetals. From (c), the Fermi arc but the projected Weyl points F \π(W ) = a −1 ((−1, 1)) ∩ b −1 (0) is expressed as the union of a finite number of 1-dimensional submanifolds. They are the images of smooth embeddings of intervals or circles, from which we get relative homology cycles. Then, using (b) and (d), we can verify the third property in Claim 1.1 to obtain our main theorem: We close the introduction by an example. It is well known that a Weyl semimetal arises in a phase transition of topological insulators (see [23] for example). In view of the phase transitions of the QWZ model, we substitute u = 2 + cos k z intoĤ QWZ to getĤ (k x , k y , k z ) = sin k x σ x + sin k y σ y + (2 + cos k x + cos k y + cos k z )σ z .
Note that one can get this from a toy model in [1]. Under the change of coordinates k x ↔ k z and the conjugation by an orthogonal matrix (concretely T in the proof of Lemma 2.7),Ĥ(k x , k y , k z ) agrees withĤ loc (a(k x , k y ), b(k x , k y ), k z ), in which a(k x , k y ) = 2 + cos k x + cos k y , b(k x , k y ) = sin k y .
This example satisfies Assumption 1.2. The set W consists of two Weyl points (π/2, π, 0) and (3π/2, π, π) on T 3 . The Fermi arc is the interval F = [π/2, 3π/2] × {π} connecting the projected Weyl points (π/2, π) and (3π/2, π) on T 2 . It turns out that the relative homology class This paper is organized as follows: In §2, we start with a summary of spectral data of the edge Hamiltonian associated to our local modelĤ loc . Based on this result, we also give a topological proof of the bulk-edge correspondence c 1 (Ĥ) = −sf(H ♯ ) for Chern insulators here. In §3, the main theorem is proved. For this aim, we begin with a review of relevant (co)homology groups. We next construct bases of these (co)homology groups, and define the relative homology class F (H ♯ ). Then, by using the bulk-edge correspondence for Chern insulators and the (co)homology basis, we prove Theorem 1.3. The proof appeals to basic techniques in differential topology. Examples are supplied at the end of this section. Finally, in §4, a postponed proof about spectral data in §2 is given.
Throughout the paper, some basic notions in algebraic topology (cohomology, homology and homotopy groups [15]) will be assumed.
Acknowledgements. The author's research is supported by JSPS KAKENHI Grant Numbers 20K03606 and JP17H06461.
Let L 2 (S 1 , C 2 ) be the space of L 2 -functions on S 1 = R/2πZ with values in C 2 , and L 2 + (S 1 , C 2 ) ⊂ L 2 (S 1 , C 2 ) the L 2 -completion of the subspace n>0 Ce inθ ⊗ C 2 . Put differently, L 2 + (S 1 , C 2 ) is the closed subspace of L 2 -functions whose Fourier expansions involve positive modes only. We denote byP : L 2 (S 1 , C 2 ) → L 2 + (S 1 , C 2 ) the orthogonal projection. Its adjointP * : to be the compression of the multiplication operator withĤ loc (a, b, ·) : S 1 → Herm(C 2 ) 0 , namely, the composition of the following operators ) is a self-adjoint bounded operator. Notice thatĤ ♯ loc (a, b) is essentially the Toeplitz operator [12] associated toĤ loc (a, b, ·). HenceĤ ♯ loc (a, b) is Fredholm if (a, b) = (±1, 0). Also, by using the continuity and symmetry of the model as well as the fact that essential spectrum is unchanged under compact perturbations, we can identify the essential spectrum ofĤ ♯ loc (a, b) with the union of the spectrum of the Hermitian matrices {H loc (a, b, θ)} θ∈S 1 .
• The case that a = 0 and E = ± √ b 2 + 1. In this case, the space ofψ ∈ L 2 • The case that |a| < 1 and E = b. In this case, the space ofψ ∈ L 2 The proof of this proposition, which is elementary, will be given in §4.

2.2.
Proof of bulk-edge correspondence for topological insulators. As an application of Proposition 2.1, we show the bulk-edge correspondence for topological insulators as mentioned in the introduction. This result will be used in the sequel. Our proof appeals to basic topology, while more analytic ones are available in [10,17], for instance.
LetĤ : T 2 → Herm(C r ) * be a continuous map from the 2-dimensional torus T 2 = R 2 /2πZ 2 to the space Herm(C r ) * of invertible r by r Hermitian matrices. We assume thatĤ(k) has both positive and negative eigenvalues at each k ∈ T 2 . Associated toĤ is the Bloch vector bundle EĤ → T 2 whose fiber at k ∈ T 2 consists of negative eigenvectors ofĤ(k) By evaluating the fundamental class [T 2 ] ∈ H 2 (T 2 ; Z) of T 2 by the first Chern class c 1 (EĤ ) ∈ H 2 (T 2 ; Z) of the Bloch bundle, we get the first Chern number of EĤ , which we denote by Let L 2 (S 1 , C r ) be the space of L 2 -functions on S 1 with values in C r , and L 2 + (S 1 , C r ) the closed subspace consisting of L 2 -functions whose Fourier expansions involve positive modes only. For k x ∈ S 1 , we defineĤ ♯ (k x ) : L 2 + (S 1 , C r ) → L 2 + (S 1 , C r ) to be the compression of the multiplication operator withĤ(k x , ·) : S 1 → Herm(C r ) * , namely, the composition of , whereP is the orthogonal projection. ThenĤ ♯ (k x ) is a self-adjoint bounded Fredholm operator. By the assumption thatĤ(k x , k y ) has both positive and negative eigenvalues,Ĥ ♯ (k x ) is neither essentially positive nor negative, in the sense of [4]. As a consequence, we have a continuous mapĤ ♯ : is the set of self-adjoint bounded Fredholm operators which are neither essentially positive nor negative, topologized by the operator norm. It is well-known that Fred 1 is a model of the classifying space of the odd Ktheory, and the set [S 1 , Fred 1 ] of homotopy classes of continuous maps S 1 → Fred 1 , which is the odd K-theory K 1 (S 1 ) of the circle, is identified with Z. This identification is induced by the spectral flow [3]. Intuitively, the spectral flow sf(A) ∈ Z of a continuous map A : S 1 → Fred 1 counts how many times the eigenvalues of A(k) changes from negative to positive as k ∈ S 1 goes around the circle. A rigours definition can be found in [24]. Proposition 2.3. IfĤ : T 2 → Herm(C r ) * is a continuous map such thatĤ(k) admits both positive and negative eigenvalues at each k ∈ T 2 , then c 1 (Ĥ) = −sf(Ĥ ♯ ).
The proposition will be shown based on some lemmas below.
Lemma 2.4. For any integer n ∈ Z, we defineĤ n : Proof. It is clear that detĤ n (k x , k y ) = 0 whenever u = 0, ±2. In the case that n = 1, the map H 1 is the QWZ model of Chern insulators [2], and its Chern number is c 1 (Ĥ 1 ) = 1 for 0 < u < 2. The map H n factors asĤ n =Ĥ 1 • f n , where f n : T 2 → T 2 is given by f n (k x , k y ) = (nk x , k y ). Since f n carries the fundamental class [T 2 ] ∈ H 2 (T 2 ; Z) ∼ = Z of the torus to n[T 2 ], it follows that c 1 (Ĥ n ) = n.
Tentatively, we write Herm p (C r ) for the space of invertible Hermitian matrices which have p negative eigenvalues and r − p positive eigenvalues.
Lemma 2.5. Herm p (C r ) is homotopy equivalent to the Grassmannian Gr p (C r ) consisting of p-dimensional linear subspaces in C r . Thus, if p = 0, r, then we have the following homotopy groups Proof. Given a matrixĤ ∈ Herm p (C r ), we have a self-adjoint involution η = H/|Ĥ|. A self-adjoint involution η corresponds bijectively to an orthogonal projection p = 1 − 2η, and an orthogonal projection to a linear subspace in C r . Hence Herm p (C r ) contains as a subspace the Grassmannian Gr p (C r ) consisting of p-dimensional linear subspaces in C r . The subspace is a deformation retract, sinceĤ and η are homotopic by the homotopyĤ/|Ĥ| t , (t ∈ [0, 1]).
Proof. Suppose that we have two continuous mapsĤ,Ĥ ′ : . We shall show thatĤ andĤ ′ are homotopic. For this aim, we notice that any continuous mapĤ : The reason is as follows: Because Herm p (C r ) is simply connected, there exists a homotopy between the restriction ofĤ to S 1 ∨ S 1 and the constant map c p . Apparently, the torus T 2 has the structure of a 2-dimensional CW complex whose 1-skeleton is S 1 ∨S 1 . Then, applying the homotopy extension property, we can extend the homotopy on S 1 ∨ S 1 to that on T 2 . As a consequence, we can assume that the restrictions ofĤ andĤ ′ to S 1 ∨ S 1 are the constant map c p . Then there is a bijective correspondence between continuous maps T 2 → Herm p (C r ) whose restrictions to S 1 ∨ S 1 are constant maps and continuous maps T 2 /(S 1 ∨ S 1 ) → Herm p (C r ), where T 2 /(S 1 ∨ S 1 ) is the space given by collapsing the subspace S 1 ∨ S 1 , which is homeomorphic to the sphere S 2 .
To summarize, we have the following bijections The last bijection to Z is induced from the evaluation of the first Chern class of the tautological (Bloch) vector bundle on Herm p (C r ) ≃ Gr p (C r ). Consequently, To complete the proof, we need to show that there isF n : T 2 → Herm p (C r ) such that c 1 (F n ) = n for any given integer n ∈ Z. Thanks to Lemma 2.4, this is clear in the case that p = 1 and r = 2. In the general case, an example ofF n can be constructed asF n (k) =Ĥ n (k) ⊕ (−1 C p−1 ) ⊕ 1 C r−p−1 , whereĤ n is given in Lemma 2.4 with the parameter 0 < u < 2, and 1 C k is the (constant map with its value) identity matrix on C k . Lemma 2.7. For n ∈ Z and 1 < u < 2, we have sf(Ĥ ♯ n ) = −n, whereĤ n is the map given in Lemma 2.4. Proof. Let T be the following matrix By direct computation, we see where a n (k x ) = u + cos nk x and b n (k x ) = sin nk x . It follows that sf(Ĥ ♯ n ) = sf(Ĥ ♯ ), where we setĤ(k x , k y ) =Ĥ loc (a n (k x ), b n (k x ), k y ) to suppress notations. Now, we apply Proposition 2.1. In the case of n = 0, the Fredholm operatorĤ ♯ (k x ) turns out to be invertible for all k x ∈ S 1 , so that sf(Ĥ ♯ ) = 0. In the case of n > 0, the Fredholm operatorĤ ♯ (k x ) has zero eigenvalues of multiplicity one at k x = π(2ℓ − 1)/n for ℓ = 1, 2, . . . , n. Around these values of k x , the eigenvalues of H ♯ (k x ) are given by E = sin nk x , so that sf(Ĥ ♯ ) = −n. In the case of n < 0, we find sf(Ĥ ♯ ) = −n in the same way.
We are now in the position to prove Proposition 2.3: Proof of Proposition 2.3. Because of Lemma 2.6, if a mapĤ : T 2 → Herm(C r ) * satisfying the hypothesis of the proposition has the Chern number c 1 (Ĥ) = n, then H is homotopic toF n constructed in the proof of Lemma 2.6. Then the associated families of self-adjoint Fredholm operatorsĤ ♯ : S 1 → Fred 1 andF ♯ n : S 1 → Fred 1 are also homotopic. Since the spectral flow is a homotopy invariant, it follows that sf(Ĥ ♯ ) = sf(F ♯ n ). By Lemma 2.7, we see sf( 3. Homological bulk-edge correspondence for Weyl semimetals 3.1. Relevant homology groups. We summarize here homology groups relevant to Weyl semimetals as given in [21,22]. All the cohomology and homology groups are those with integer coefficients. Let W ⊂ T 3 be a finite subset. For each w ∈ W , we choose an open ball D(w) ⊂ T 3 centered at w ∈ W so that their closure do not intersect with each other: D(w)∩D(w ′ ) = ∅ for w = w ′ . The ball D(w) and its boundary sphere ∂D(w) inherit orientations from T 3 . We use this orientation to identify H 2 (∂D(w)) ∼ = Z. We write i : T 3 \W → T 3 and i w : ∂D(w) → T 3 for the inclusions.
where the homomorphism Σ is given by (q j ) → j q j .
Proof. As shown in [21,22], the exact sequence is derived from the Mayer-Vietoris exact sequence associated to the open sets U = T 3 \W and V = w∈W D(w).
As a result, if W consists of n point, then we get H 2 (T 3 \W ) ∼ = Z n+2 abstractly, since the exact sequence in Lemma 3.1 admits a splitting. However, it should be noticed that there is no natural choice of a splitting.
Lemma 3.2. The projection π : T 3 → T 2 induces the following homomorphism of exact sequences.
Proof. The exact sequences are associated to the pairs (T 3 , W ) and (T 2 , π(W )). These exact sequences are natural, and hence π : T 3 → T 2 induces the natural homomorphisms π * of homology groups that constitute the homomorphism of exact sequences.
Proof. For later convenience, we account for the duality isomorphism via a manifold with boundary: The open submanifold T 3 \W ⊂ T 3 is homotopy equivalent to T 3 \D, where D = w∈W D(w) is the disjoint union of the open balls centered at the points in W . Hence the homotopy axiom gives the isomorphism To the compact oriented 3-dimensional manifold T 3 \D with its boundary ∂(T 3 \D) = ∂D = w∈W ∂D(w), we can apply the Poincaré(-Lefshetz) duality [15] H Again by the homotopy axiom, we get H 1 (T 3 \D, ∂D) ∼ = H 1 (T 3 , W ). In summary, we have an isomorphism H 2 (T 3 \W ) ∼ = H 1 (T 3 , W ). The same argument also leads to an isomorphism H 1 (T 2 \W ) ∼ = H 1 (T 2 , π(W )).

3.2.
Construction of (co)homology basis. This subsection is devoted to a construction of bases of relevant (co)homology groups. As before, let W ⊂ T 3 be a finite subset. Suppose that its image π(W ) ⊂ T 2 under the projection consists of n points. We express the points in π(W ) as w 0 , . . . , w n−1 . For i = 0, . . . , n − 1, let m i be the number of points in π −1 (w i ). By definition, the number of points in W is with its boundary identified suitably. In this description, it is clear that the choice of the smooth paths α i is possible. In the same way, T 3 is regarded as the cube [k 0 x , k 0 with its boundary identified suitably. In this description of T 3 , we can find α j 0 . We can also construct the smooth lifts α j i of α i for i = 1, . . . , n by choosing the z-component appropriately.
where the integers q x , q y and q z are given by In Proof. To see the lemma, we review the construction of the Poincaré dual ω . In view of the proof of Lemma 3.3, we can consider H 2 (T 3 \D) instead of H 2 (T 3 \W ). Furthermore, since the cohomology group is torsion free, we can regard cohomology classes in H 2 (T 3 \D) as represented by differential forms. Then we can apply the construction of the Poincaré dual in [9]. Choosing the open disks D(w) smaller if necessary, we truncate and rescale the domain of the smooth path α j i : [0, 1] → T 3 \W for each i, j to get a smooth path Because we can choose N j i to be disjoint to each other, we have ∂D(w j i ) ω j ′ i ′ = 0 whenever i = i ′ or j = j ′ . By construction, we also have ∂D(w j i ) ω j i = −1, where the sign is due to the difference of the orientations on ∂(T 3 \D) = ∂D induced from T 3 .
Since the tubular neighbourhoods of α x , α y , α z can be chosen so as to be disjoint to D, we have ∂D(w j i ) ω k = 0 for k = x, y, z. Thus, in view of Lemma 3.1, the coefficients q j i are given by q j i = − ∂D(w j i ) ω. Since the images of the paths α j i have no intersection with the subtori {k 0 i on these subtori are trivial. Note that ω x , ω y and ω z can be regarded as the injective image of a basis of H 2 (T 3 ) ∼ = Z 3 . Then they can be represented by 2-forms dk y dk z , −dk x dk z and dk x dk z , respectively. This description leads to the expressions of q x , q y and q z as stated.
3.3. Definition of the edge index. LetĤ : T 3 → Herm(C r ) be a continuous map, where r ≥ 1. DefineĤ ♯ (k x , k y ) : L 2 + (S 1 , C r ) → L 2 + (S 1 , C r ) to be the selfadjoint bounded operator obtained as the compression of the multiplication witĥ H(k x , k y , ·) by the projection L 2 (S 1 , C r ) → L 2 + (S 1 , C r ). The Fourier transform identifies L 2 (S 1 , C r ) with the space L 2 (Z, C r ) of square summarable functions on Z with values in C r L 2 (Z, C r ) = ψ = (ψ(n)) n∈Z ψ(n) ∈ C r , n∈Z ψ(n) 2 < +∞ , and the subspace L 2 + (S 1 , C r ) ⊂ L 2 (S 1 , C r ) with L 2 (N, C r ) ⊂ L 2 (Z, C r ). The oper-atorH ♯ (k x , k y ) : L 2 (N, C r ) → L 2 (N, C r ) mentioned in §1 is the one arising as the Fourier transform ofĤ ♯ (k x , k y ). SinceH ♯ (k x , k y ) andĤ ♯ (k x , k y ) have the same spectral data, we will work withĤ ♯ (k x , k y ).
Proof. In the same manner as in the proof of Proposition 3.7, we have for any ω ∈ H 1 (T 2 \π(W )). If ω = −Sf(Ĥ ♯ | T 2 \π(W ) ), then The same expressions hold for other coefficients of the base elements.
(3) If F = ∅, then F is recovered as the images of paths which represent F (H ♯ ).
The homology class F (Ĥ ♯ ) is of course the one as given in Definition 3.9. The proof of the theorem will be given, after some lemmas.
By definition,Ĥ(k x , k y , k z ) is expressed aŝ in terms of the local modelĤ loc . Hence the set of Weyl points W ⊂ T 3 , its image under the projection π : T 3 → T 2 and the Fermi arc F ⊂ T 2 are given by By Assumption 1.2 (a), the sets W and π(W ) consist of finite number of points.  ((−1, 1) By Assumption 1.2 (c), the smooth map b : a −1 ((−1, 1)) → R has 0 ∈ R as a regular value. By the so-called preimage theorem, a −1 ((−1, 1) ((−1, 1)) is a 1-dimensional submanifold in the open submanifold a −1 ((−1, 1)) ⊂ T 2 . This is supposed to have a finite number of connected components in Assumption 1.2 (c). By the classification of connected 1-dimensional manifolds, each of the connected components is either an embedded open interval or a circle S 1 .
Then the following holds true.
(a) c f (t) gives rise to a well-defined continuous map c f : (0, 1) → R such that c f (t) = 0 for all t. As a result, the following sign is also well-defined (b) We have the following formula for all t If f : S 1 → T 2 is one of the smooth embeddings f c 1 , . . . , f c ℓ c in Lemma 3.13, then we can similarly define c f : S 1 → R and ǫ(f ) ∈ {±1} with the above properties. Proof. To suppress notation, let us introduce vectors v, w ∈ R 2 as follows .
Recall that f : (0, 1) → T 2 corresponds to one of the connected components of the submanifold a −1 ((−1, 1) This means that v and w are orthogonal, or equivalently, v is parallel to w rotated by π/2. As is seen, v = 0. We also have w = 0, because f is an embedding. As a result, there uniquely exists a real number c = 0 such that when t is fixed. It is easy to see that c = c f (t), and (a) is proved. We omit the verification of (b), which is straightforward. Finally, the argument so far is local, and hence applicable to f = f c 1 , . . . , f c ℓ c as well.
where ǫ(f j ) and ǫ(f c j ) are given in Lemma 3.15, and Proof. In view of the construction of the Poincaré dual PD −1 (f j ) as reviewed in the proof of Lemma 3.3, one sees that the integer I(f j , γ) agrees with the so-called intersection number, namely, the number of positive intersection points of f j and γ minus that of negative intersection points. Here an intersection point p ∈ T 2 of f j and γ is positive (resp. negative) if the pair of tangent vectors (df j /dt, dγ/dt) form a basis of the tangent space at p ∈ T 2 compatible (resp. incompatible) with the orientation on T 2 . By assumption, the determinant of the Jacobian det J of (a, b) : T 2 → R 2 does not vanish at each intersection point p of f j and γ. Thus, near this point, (a, b) is a diffeomorphism. Because f j corresponds to a connected component of the submanifold a −1 ((−1, 1)) ∩ b −1 (0), the path (a, b) • f = (a • f, 0) lies on the interval (−1, 1) × {0} ⊂ R 2 . With this interval, the path (a, b) • γ intersects transversally. For s ∈ S 1 such that −1 < a(γ(s)) < 1, the unique eigenvalue of (Ĥ ♯ • γ)(s) with finite multiplicity is E = b(γ(s)) of multiplicity one. Therefore an intersection point p = γ(s) of f j and γ corresponds bijectively to the zero eigenvalue of (Ĥ ♯ • γ)(s). Near this s, the eigenvalue changes its sign once, and hence contributes to the spectral flow by ±1. To identify the sign of this contribution, we need to know the sign of det J(p), which determines whether (a, b) is orientation preserving or not. We also need the sign of d(a • f j )/dt. By direct inspection, the positive (resp. negative) intersection point contributes to the spectral flow by +1 (resp. −1) when the product of the sings of det J and d(a•f j )/dt is +1 (resp. −1). This product of signs agrees with ǫ(f j ), as shown in Lemma 3.15.
To summarize, the intersection points of f j and γ contribute to the spectral flow by ǫ(f j )I(f j , γ). Taking all the connected components of a −1 ((−1, 1)) ∩ b −1 (0) into account, we conclude the expression of the spectral flow.
We define a singular homol- where ǫ is the sign given in Lemma 3.15.
Note that, in the case of ℓ > 0, Lemma 3.14 implies π(W ) = 0, and we use the continuous extension f i : [0, 1] → T 2 in the above definition of f (Ĥ ♯ ). . . , f c ℓ c . By the same argument as in the proof of Proposition 3.7, we can express PD −1 ([f ]) ∈ H 1 (T 2 \π(W )) as where D i , (i = 0, 1, . . . , n−1) are small open disks centered at w i ∈ π(W ) such that D i ∩ D j = ∅ for i = j. Taking the Poincaré duality, we get the following expression of [f ] ∈ H 1 (T 2 , π(W )) where I(f, γ) = γ PD −1 ([f ]) for any smooth map γ : S 1 → T 2 \π(W ). To summarize, we have Because of Assumption 1.2 (d), we can regard α y as a smooth embedding α y : S 1 → T 2 \π(W ). By the transversality theorem [14], we can perturb α y to another smooth embedding γ so that γ intersects with the closed submanifold a −1 ((−1, 1))∩ b −1 (0) ⊂ T 2 \π(W ) transversally. This means that γ intersects with each of f = Then, by the transversality theorem again, the smooth embedding ∂D i : S 1 → U wi \{w i } can be perturbed to a smooth embedding γ : In summary, we get This is nothing but the expression of F (Ĥ ♯ ) given in Lemma 3.10.
Now, we are in the position to prove Theorem 3.12: Proof of Theorem 3.12. We let F (Ĥ ♯ ) ∈ H 1 (T 2 , π(W )) be the homology class given in Definition 3.9. Then the first two properties (1) and (2)   Proof. By the conjugation of the matrix T in the proof of Lemma 2.7 and the exchange of coordinates k y ↔ k z , which comprise a locally linear transformation of coordinates, we have x , k 0 z ) = 0 and k 0 y = 0. At this point, if we demand from (W2) that the Taylor expansion of TĤ loc (a(k x , k z ), b(k x , k z ))T * agrees with i=x,y,z (k i − k 0 i )σ i up to the first order, then Thus, det J(k 0 x , k 0 z ) = 0, and hence Assumption 1.2 (b) is fulfilled. At a Weyl point such that a(k 0 x , k 0 z ) = 1, b(k 0 x , k 0 z ) = 0 and k 0 y = π, a similar argument is valid.
The third example iŝ H(k x , k y , k z ) = sin k y − cos k x 2 + cos k x + cos k y + e ikz 2 + cos k x + cos k y + e −ikz − sin k y + cos k x where a(k x , k y ) = 2 + cos k x + cos k y and b(k x , k y ) = sin k y − cos k x . Note that the determinant of the Jacobian of (a, b) : T 2 → R 2 is det J(k x , k y ) = − sin k x (cos k y − sin k y ) = − √ 2 sin k x cos(k y + π/4).
Note that a 2 − 1 = 0 under the present assumption ∆ > 0, which is equivalent to a 2 = 1. Hence the case (i) cannot occur: if |λ + | = |a| > 1, then there is only the trivial L 2 -solution (z = 0). If |λ + | = |a| < 1 (the case (ii)), then we have non-trivial L 2 -solutions which form a 1-dimensional space. Now, we are in the position to complete the proof of Proposition 2.1: Proposition 4.6. The equation (H ♯ − E)ψ = 0 for ψ = (ψ(n)) n∈N admits a non-trivial L 2 -solution in the following cases.
• The case that a = 0 and E = ± √ b 2 + 1. In this case, the space of L 2solutions is infinite dimensional.
• The case that |a| < 1 and E = b. In this case, the space of L 2 -solutions is 1-dimensional.
Proof. By the lemmas so far, we have non-trivial L 2 -solutions in the following cases.
(3) a = 0, |a| < 1 and E = b = 0. (4) a = 0, |a| < 1 and E = b = 0. In the first case, the space of L 2 -solutions is infinite dimensional, whereas the space is 1-dimensional in each of the remaining cases. These remaining cases can be summarized as a single case where |a| < 1 and E = b.