Classification of topological invariants related to corner states

We discuss some bulk-surfaces gapped Hamiltonians on a lattice with corners and propose a periodic table for topological invariants related to corner states aimed at studies of higher-order topological insulators. Our table is based on four things: (1) the definition of topological invariants, (2) a proof of their relation with corner states (3) computations of K-groups and (4) a construction of explicit examples.


Introduction
Recent developments in condensed matter physics have greatly generalized the bulk-boundary correspondence for topological insulators to include corner states. Topological insulators have a gapped bulk, which incorporates some topology that do not change unless the spectral gap of the bulk Hamiltonian closes under deformations. Examples include the TKNN number for quantum Hall systems [65] and the Kane-Mele Z 2 index for quantum spin Hall systems [36]. It is known that, corresponding to these bulk invariants, gapless edge states appear, which is called the bulk-boundary correspondence [31]. After Schnyder-Ryu-Furusaki-Ludwig's classification of topological insulators [60] for ten Altland-Zirnbauer classes [2], Kitaev noted the role of K-theory and Bott periodicity in the classification problem and obtained the famous periodic table [42]. Recently, some (at least bulk) gapped systems possessing in-gap or gapless states localized around a higher codimensional part of the boundary (corners or hinges) are studied [30,12,41], which are called higherorder topological insulators (HOTIs) [59]. For example, for second-order topological insulators, not only is the bulk gapped but also the codimension-one boundaries (edges, surfaces), and an in-gap or a gapless state appears around codimensiontwo corners or hinges. In this framework, conventional topological insulators are regarded as first-order topological insulators. HOTIs are now actively studied and the classification of HOTIs has also been proposed [26,40,52]. Generalizing the bulk-boundary correspondence, relations between some gapped topology and corner states are much discussed [66,5,63].
Initiated by Bellissard, K-theory and index theory are known to provide a powerful tool to study topological insulators. Bellissard-van Elst-Schulz-Baldes studied quantum Hall effects by means of noncommutative geometry [10,11], and Kellendonk-Richter-Schulz-Baldes went on to prove the bulk-boundary correspondence by using index theory for Toeplitz operators [39]. The study of topological insulators, especially regarding its classification and the bulk-boundary correspondence for each of the ten Altland-Zirnbauer classes by using K-theory and index theory has been much developed [39,24,64,16,29,48,57,64,17,38,44,1]. In [32], three-dimensional (3-D) class A bulk periodic systems are studied on one piece of a lattice cut by two specific hyperplanes, which is a model for systems with corners. Based on the index theory for quarter-plane Toeplitz operators [62,23,54], a topological invariant is defined assuming the spectral gap both on the bulk Hamiltonian and two half-space compressions of it. This gapped topological invariant is topological in the sense that it does not change under continuous deformation of the bulk Hamiltonians unless the spectral gap of one of the two surfaces closes. It is proved that, corresponding to this topology gapless corner states appear. A construction of nontrivial examples by using two first-order topological insulators (of 2-D class A and 1-D class AIII) is also proposed. Class AIII codimension-two systems are also studied through this method in [33] and, as an application to HOTIs, the appearance of topological corner states in Benalcazar-Bernevig-Hughes' 2-D model [12] is explained based on the chiral symmetry. The construction of examples in [33] leads to a proposal of second-order semimetallic phase protected by the chiral symmetry [51].
The purpose of this paper is to expand the results in [32] to all Altland-Zirnbauer classes and systems with corners of arbitrary codimension. Since class A and class AIII systems (with codimension-two corners) were already discussed in [32,33] by using complex K-theory, we focus on the remaining eight cases, for which we use real K-theory. For our expansion, a basic scheme has already been well developed in the above previous studies, which we mainly follow: some gapped Hamiltonian defines an element of a KO-group of a real C * -algebra, and its relation with corner states are clarified by using index theory [39,24,16,29,44,64,17,38]. Although many techniques have already been developed in studies of topological insulators, in our higher-codimensional cases, we still lack some basic results at the level of K-theory and index theory; hence, the first half of this paper is devoted to these K-theoretic preliminaries, that is, the computation of KO-groups for real C *algebras associated with the quarter-plane Toeplitz extension and the computation of boundary maps for the 24-term exact sequence of KO-theory associated with it, which are carried out in Sect. 3. Since the quarter-plane Toeplitz extension [54] is a key tool in our study of codimension-two corners, such a variant for Toeplitz operators associated with higher-codimensional corners should be clarified, which are carried out in Sect. 4. These variants of Toeplitz operators were discussed in [23,22], and the contents in Sect. 4 will be well-known to experts. Since the author could not find an appropriate reference, especially concerning Theorem 4.1 which will play a key role in Sect. 5, the results are included for completeness. Note that the idea there to use tensor products of the ordinary Toeplitz extension for the study of these variants is based on the work of Douglas-Howe [23], where these higher-codimensional generalizations are briefly mentioned. The study of some gapped phases for systems with corners in Altland-Zirnbauer's classification is carried out in Sect. 5. In the framework of the one-particle approximation, we consider n-D systems with a codimension k corner and take compressions of the bulk Hamiltonian onto infinite lattices with codimension k − 1 corners 1 whose intersection makes the codimension k corner. We assume that they are gapped. Note that, under this assumption, bulk, surfaces and corners up to codimension k − 1 which constitute the codimension k corner are also gapped. For such a system, we define two topological invariants as elements of some KO-groups: one is defined for these gapped Hamiltonians while the other one is related to in-gap or gapless codimension k corner states. We then show a relation between these two which states that topologically protected corner states appear reflecting some gapped topology of the system. We first study codimension-two cases (Sect. 5.1 to Sect. 5.4) and then discuss higher-codimensional cases (Sect. 5.5). This distinction is made because many detailed results have been obtained for codimension-two cases by virtue of previous studies of quarter-plane Toeplitz operators [54,35] (the shape of the corner we discuss is more general than in higher-codimensional cases, and a relation between convex and concave corners is also obtained in [33]). Based on these results, we propose a classification table for topological invariants related to corner states ( Table 1). Note that the codimension-one case of Table 1 is Kitaev's table [42] and Table 1 is also periodic by the Bott periodicity. In order further to clarify a relation between our invariants and corner states, in Sect. 5.6, we introduce Z or Z 2 -valued numerical corner invariants when the dimension of the corner is zero or one. They are defined by (roughly speaking) counting the number of corner states. A construction of examples is discussed in Sect. 5.7. As in [32], this is given by using pairs of Hamiltonians of two lower-order topological insulators. In the real classes, there are 64 pairs of them and the results are collected in Table 12. By using this method, we can construct nontrivial examples of each entry of Table 1, starting from Hamiltonians of first-order topological insulators. The corner invariant for the constructed Hamiltonian is expressed by corner (or edge) invariants of constituent two Hamiltonians. This is given by using an exterior product of some KO-groups in general, though, as in [32,33], the formula for numerical invariants introduced in Sect. 5.6 is also included. For computations of KO-groups and classification of such gapped systems, we employ Boersema-Loring's unitary picture for KO-theory [14] whose definitions are collected in Sect. 2. Basic results for some Toeplitz operators are also included there. In Appendix A, we revisit Atiyah-Singer's study 1 In standard terminologies, they will be called edges, surfaces, hinges or edge of edges depending on n and k. In this paper, we may simply call them corners but state its codimensions. Table 1. Classification of (strong) topological invariants related to corner state in Altland-Zirnbauer (AZ) classification. In this table, n is the dimension of the bulk, and k is the codimension of the corner.
of spaces of skew-adjoint Fredholm operators [9] and collect necessary results from the viewpoint of Boersema-Loring's K-theory. Definitions of some Z 2 -spaces, maps between them, expression of boundary maps of 24-term exact sequences used in this paper are collected there. Finally, let us point out a relation with our results and the current rapidly developing studies on HOTIs. In [26], the HOTIs are divided into two classes: intrinsic HOTIs, which basically originate from the bulk topology protected by a point group symmetry, and others extrinsic HOTIs. Our study will be for extrinsic HOTIs since no point group symmetry is assumed and our classification table (Table 1) is consistent with that of Table 1 in [26].

Preliminaries
In this section, we collect the necessary results and notations.
2.1. Boersema-Loring's KO-Groups via Unitary Elements. In this subsection, we collect Boersema-Loring's definition of KO-groups by using unitaries satisfying some symmetries [14]. The basics of real C * -algebras and KO-theory can be found in [28,61], for example.
A C * ,τ -algebra is a pair (A, τ ) consisting of (complex) C * -algebra A and an antiautomorphism 2 τ of A satisfying τ 2 = 1. We call τ the transposition and write a τ for τ (a). There is a category equivalence between the category of C * ,τ -algebras and the category of real C * -algebras: for a C * ,τ -algebra (A, τ ), the corresponding real C *algebra is A τ = {a ∈ A | a τ = a * }, and its inverse is given by the complexification. A real structure on a (complex) C * -algebra A is an antilinear * -automorphism r satisfying r 2 = 1. For a real structure r, there is an associated transposition τ given by τ (a) = r(a * ), which gives a one-to-one correspondence between transpositions 2 i.e., a complex linear automorphism of A that preserves * and satisfies τ (ab) = τ (b)τ (a). and real structures on the C * -algebra 3 . We extend the transposition τ on A to the transposition (for which we simply write τ ) on the matrix algebra M n (A) by (a ij ) τ = (a τ ji ) where a ij ∈ A and 1 ≤ i, j ≤ n. This induces a transposition τ K on K ⊗ A where K = K(V) is the C * -algebra of compact operators on a separable complex Hilbert space V. Let ♯ ⊗ τ be a transposition on M 2 (A) defined by 4 a 11 a 12 a 21 a 22 .
If we identify the quaternions H with C 2 by x+yj → (x, y), the left multiplication by j corresponds to j(x, y) = (−ȳ,x). Then, we have ♯ ⊗ id = Ad j • * where * denotes the operation of taking conjugation of matrices and the C * ,τ -algebra (M 2 (C), ♯⊗id) corresponds to the real C * -algebra H of quaternions. We extend this transposition , we also consider a transposition ♯ ⊗ τ defined by where c ij ∈ M n (A). For an m × m matrix X, we write X n for the mn × mn block diagonal matrix diag(X, . . . , X). For example, we write 1 n for the n × n diagonal matrix diag(1, . . . , 1).
Definition 2.1 (Boersema-Loring [14]). Let (A, τ ) be a unital C * ,τ -algebra. For i = −1, 0, . . . , 6, let n i be a positive integer, R i be a relation and I (i) be a matrix, as indicated in Table 2. Let U k (A, τ ), we consider the equivalence relation ∼ i generated by homotopy and stabilization given by I (i) . We define KO i (A, τ ) = U (i) ∞ (A, τ )/ ∼ i which is a group by the binary operation given by [u] For a nonunital C * ,τ -algebra (A, τ ), the i-th KO-group KO i (A, τ ) is defined as the kernel of λ * : KO i (Ã, τ ) → KO i (C, id), whereÃ is the unitization of A and λ :Ã → C is the natural projection. In [14], they also describe the boundary maps of the 24-term exact sequence for KO-theory associated with a short exact sequence of C * ,τ -algebras. In Appendix A.3, we discuss an alternative description for some of them through exponentials.

Toeplitz Operators.
In this subsection, we collect the definitions and basic results for some Toeplitz operators used in this paper [22,54].
Let T be the unit circle in the complex plane C, and let c be the complex conjugation on C, that is, c(z) =z. Let n be a positive integer. On the n-dimensional torus T n , we consider an involution ζ defined as the n-fold product of c. This induces a transposition τ T on C(T n ) by (τ T f )(t) = f (ζ(t)). Let Z ≥0 be the set of nonnegative integers and P n be the orthogonal projection of l 2 (Z n ) onto l 2 ((Z ≥0 ) n ). For a continuous function f : T n → C, let M f be the multiplication operator on l 2 (Z n ) generated by f . We consider the operator P n M f P n on l 2 ((Z ≥0 ) n ), which is 3 Boersema-Loring called τ the real structure in [14]. In this paper, we distinguish these two since the antilinear structure naturally appears in our application. We call τ the transposition following [38]. 4 For notations of the transpositions introduced here, we follow [14].
the Toeplitz operator associated with the subsemigroup (Z ≥0 ) n of Z n of symbol f . We write T n for the C * -subalgebra of B(l 2 ((Z ≥0 ) n )) generated by these Toeplitz operators. The algebra T 1 is the ordinary Toeplitz algebra and we simply write T . Note that the algebra T n is isomorphic to the n-fold tensor product of T . The complex conjugation c on C induces an antiunitary operator 5 of order two on the Hilbert space l 2 (Z n ) by the pointwise operation, for which we also write c. This induces a real structure c on B(l 2 ((Z ≥0 ) n )) by c(a) = Ad c (a) = cac * . We write τ T for the transposition on T n given by its restriction onto T n . We next focus on the case of n = 2. We consider the Hilbert space l 2 (Z 2 ) and take its orthonormal basis {δ m,n | (m, n) ∈ Z 2 }, where δ m,n is the characteristic function of the point (m, n) on Z 2 . When f ∈ C(T 2 ) is given by f (z 1 , z 2 ) = z m 1 z n 2 , we write M m,n for the multiplication operator M f . Let α < β be real numbers, and let H α , H β ,Ĥ α,β andȞ α,β be closed subspaces of l 2 (Z 2 ) spanned by {δ m,n |−αm+n ≥ 0}, {δ m,n |−βm+n ≤ 0}, {δ m,n |−αm+n ≥ 0 and −βm+n ≤ 0}, and {δ m,n |−αm+n ≥ 0 or − βm + n ≤ 0}, respectively. In the following, we may take α = −∞ or β = ∞, but not both. Let P α , P β ,P α,β andP α,β be the orthogonal projection of l 2 (Z 2 ) onto H α , H β ,Ĥ α,β andȞ α,β , respectively. For f ∈ C(T 2 ), the operators P α M f P α on H α and P β M f P β on H β are called half-plane Toeplitz operators. The operatorP α,β M fP α,β onĤ α,β is called the quarter-plane Toeplitz operator, anď P α,β M fP α,β onȞ α,β is its concave corner analogue. We write T α and T β for C *algebras generated by these half-plane Toeplitz operators andT α,β andŤ α,β for C * -algebras generated by the quarter-plane and concave corner Toeplitz operators, respectively. There are * -homomorphisms σ α : T α → C(T 2 ) and σ β : T β → C(T 2 ), which map P α M f P α and P β M f P β to the symbol f , respectively. We define the C * -algebra S α,β as a pullback C * -algebra of these two * -homomorphisms. The real structure c on H = l 2 (Z 2 ) induces real structures c on T α , T β ,T α,β ,Ť α,β , and S α,β and thus induces transpositions τ α , τ β ,τ α,β ,τ α,β and τ S on T α , T β ,T α,β , T α,β and S α,β , respectively. For transpositions, we may simply write τ when it is clear from the context. The maps σ α and σ β preserve the real structures and we 5 An operator A on a complex Hilbert space V is called the antiunitary operator if A is an antilinear bijection on V satisfying Av, Aw = v, w for any v and w in V.

KO-Groups of C * -algebras Associated with Half-Plane and Quarter-Plane Toeplitz Operators
In this section, the KO-theory for half-plane and quarter-plane Toeplitz operators are discussed. In Sect. 3.1, KO-groups of the half-plane Toeplitz algebra is computed. Quarter-plane Toeplitz operators are discussed in the following sections, and the KO-groups of the C * ,τ -algebra (S α,β , τ S ) are computed in Sect. 3.2. In Sect 3.3, the boundary maps of the 24-term exact sequence for KO-theory associated with the sequence (2.2) are discussed and the KO-groups of the quarter-plane Toeplitz algebra (T α,β ,τ α,β ) are computed.
We first consider the case when α is a rational number or −∞. When α ∈ Q, we write α = p q where p and q are relatively prime integers and q is positive. Let m and n be integers such that −pm + qn = 1 and let Then, the action of Γ on Z 2 induces the Hilbert space isomorphism H α ∼ = H 0 and an isomorphism of C * ,τ -algebras (T α , τ α ) ∼ = (T 0 , τ 0 ). Thus, the C * ,τ -algebra (T α , τ α ) is isomorphic to (T , τ T ) ⊗ (C(T), τ T ), and its KO-groups are computed . For the first isomorphism, see Proposition 1.5.1 of [61]. Generators of the group KO i (C(T), τ T ) are obtained in Example 9.2 of [14], and the unital * -homomorphism ι : C → T induces an isomorphism (id ⊗ ι) * : . Combined with them, KO-group KO i (T α , τ α ) and its generators are given as follows.
The case of α = −∞ is computed similarly, and its generators are given by replacing p and q above with −1 and 0, respectively. We next consider the cases of irrational α. In this case, complex K-groups of T α are computed by Ji-Kaminker and Xia in [34,69].
where the isomorphism is given by λ α * .
We consider a split * -homomorphism of C * ,τ -algebras λ α : (T α , τ α ) → (C, id) given by the composition of σ α : (T α , τ α ) → (C(T 2 ), τ T ) and the pull-back onto a fixed point of the involution ζ on T 2 . Let T α 0 = Ker λ α . By the six-term exact sequence associated with the extension 0 → T α 3.2. KO-Groups of (S α,β , τ S ). In this subsection, we compute the KO-groups of the C * ,τ -algebra (S α,β , τ S ). The basic tool is the following Mayer-Vietoris exact sequence associated with the pull-back diagram (2.1) (see Theorem 1.4.15 of [61], for example): As in [54], the computation of the group KO * (S α,β , τ S ) is divided into three cases corresponding to whether α and β are rational (or ±∞) or irrational. As in Sect. 3, we have a unital * -homomorphism λ α • p α : (S α,β , τ S ) → (C, id) which splits. Correspondingly, the KO-group KO * (S α,β , τ S ) have a direct summand corresponding to KO * (C, id). Noting this, these KO-groups are computed by Lemma 3.1 and the sequence (3.2) when at least one of α and β is irrational. The results are collected in Tables 4 and 5. In the rest of this subsection, we focus on the cases when both α and β are rational (or ±∞).
When α, β ∈ Q, we write α = p q and β = r s by using mutually prime integers where q and s are positive. In the following discussion, the case of α = −∞ corresponds to the case where p = −1 and q = 0, and the case of β = +∞ corresponds to the case where r = 1 and s = 0. By using the action of Γ ∈ SL(2, Z) in (3.1) on Z 2 , there are isomorphisms (T α , τ α ) ∼ = (T 0 , τ 0 ) and (T β , τ β ) ∼ = (T γ , τ γ ), where γ = t u for u = ns − mr and t = −ps + qr. Note that t is positive since α < β. We have the following commutative diagram: where the vertical isomorphisms are induced by the action of Γ. In the following, we discuss the lower part of the diagram, which is enough for our purpose since the isomorphism KO i (S α,β , τ S ) ∼ = KO i (S 0,γ , τ S ) is also induced. We write ϕ i for the above map σ γ * − σ 0 * . By the exact sequence (3.2), we have the following short exact sequence.
Combined with the above computation and the exact sequence (3.3), KO-group KO i (S α,β , τ S ) are computed, though some complication appears when i = 2, 3. We discuss these two cases in the following subsections.
3.2.1. The Group KO 2 (S α,β , τ S ). We compute the group KO 2 (S 0,γ , τ S ), which is isomorphic to KO 2 (S α,β , τ S ). The computation is divided into two cases depending on whether t is even or odd. Note that u is odd when t is even since r and s are mutually prime.
When t is odd, Ker(ϕ 2 ) ∼ = Z 2 is generated by ([−I (2) ], [−I (2) ]) and the sequence (3.3) splits. Therefore, We next discuss the cases of even t. In this case, both of the kernel and the cokernel of Then, the sequence (3.3) reduces to the following extension: In the following, we show that this sequence (3.4) splits. We find a lift of the generator of Z 2 in KO 2 (S 0,γ , τ S ) and show this lift has order two. For (m, n) ∈ Z 2 and κ = 0 and γ, we write T κ m,n for P κ M m,n P κ , and let Q be the projection T γ u,0 T γ −u,0 . Note that 1 − Q is the projection onto the closed subspace spanned by {δ m,n | 0 ≤ γm − n < t }. For j = 1, . . . , t, let P j be a projection in T γ , defined inductively as follows: Specifically, P j is the orthogonal projection of H γ onto the closed subspace spanned by {δ n,tn−j+1 } n∈Z . Note that t j=1 P j = 1 − Q. For odd j = 1, 3, . . . , t − 1, let We first consider the following elements: where the double-sign corresponds. Elements a and b ± are self-adjoint unitaries satisfying a τ = −a and b τ ± = −b ± , and pairs (a, b ± ) are elements of M 2 (S 0,γ ); therefore, they define the elements of KO 2 (S 0,γ , τ S ).
Proof. We first show that [(a, b + )] = 0. For j = 1, 3, . . . , t−1, let r j = P j M 0,1 P j+1 + P j+1 M 0,−1 P j and r = t−1 j=1,odd r j . The element r satisfies ( i ) r * = r, (ii) r τ = r, Figure 1. The case of u = 1 and t = 4. 1 − Q is the projection onto the closed subspace corresponding to lattice points in between two lines (lattice points on the line y = γx are included, while that on y = γ(x − 1) are not). P j is the projection onto the closed subspace spanned by {δ n,4n−j+1 | n ∈ Z}. s j interchanges two points in a pair up to the sign.
1 (T γ , τ γ ). We further discuss b π 2 . Let consider lattice points (m, n) ∈ Z 2 satisfying 0 ≤ γm − n < t, as indicated in Figure 1 for the case where u = 1 and t = 4. As in Figure 1, we divide these points to t 2 pairs of lattice points: for n ∈ Z and odd j = 1, 3, . . . , t − 1, a pair consists of {(n, tn − j), (n, tn − j + 1)}. The action of b π 2 is closed on each pair of lattice points and is expressed by a 4 × 4 matrix (acting on C 2 ⊗ C 2 ; one C 2 corresponds to a pair of lattice points, and the other C 2 corresponds to the 2 × 2 matrix we consider). Let V be the following matrix.
Then V ∈ SO(4) and satisfies where the left matrix inside the conjugation is the restriction of b π 2 onto the closed subspace spanned by generating functions of these two lattice points tensor C 2 and the right matrix is that of d (note that Q = 0 on these lattice points). Let W be the unitary on H γ ⊗ C 2 defined by applying V to these pair of lattice points satisfying 0 ≤ γm − n < t and the identity on the lattice points satisfying t ≤ γm − n, then we have W b π 2 W * = d. Since SO(4) is path-connected, there is a path of self-adjoint unitaries from b π 2 to d preserving the relation of the KO 2 -group. Summarizing, we have a path in U 1 (T γ , τ γ ) from b + to d. By its construction, the pair of the constant path at a ∈ M 2 (T 0 ) and this path gives a path in U We next discuss the class Let consider the following elements: Proof. For 0 ≤ θ ≤ π, let consider the following element in M 4 (T γ ): 2 (S 0,γ , τ S ) and gives a path from (v 2 ⊕ a, w + ⊕ b + ) to (v 2 ⊕ a, w − ⊕ b − ). By using Lemma 3.2, we obtain the following equality in KO 2 (S 0,γ , τ S ): Then, A 0 θ and A γ θ are self-adjoint unitaries satisfying . Therefore, by Lemma 3.3, the following equality holds in Proposition 3.5. When α and β are rational numbers and t = −ps + qr is even, Proof. Since u is odd when t is even, the pair ( is nontrivial and has order two by Lemma 3.4. This element belongs to KO 2 (S 0,γ , τ S ) and, by mapping 1 ∈ Z 2 to [(v 2 , w + )], we obtain a splitting of the the sequence (3.4). Therefore, KO 2 (S 0,γ , τ S ) ∼ = (Z 2 ) 3 and the result follows.
We next discuss the cases of even t. In this case, the extension (3.3) is of the following form: As in Sect. 3.2.1, we show that this sequence splits by finding a lift of the generator of the right Z 2 in KO 3 (S 0,γ , τ S ) of order two. Let consider the following elements: where the double-sign in the second equality corresponds. Pairs (v 3 , z ± ) are uni- Table 3. KO * (S α,β , τ S ) when both α and β are rational (or ±∞). Table 4. KO * (S α,β , τ S ) when one of α and β is rational (or ±∞) and the other is irrational. Table 5. KO * (S α,β , τ S ) when both α and β are irrational.
are contained in M 4 (C), where they coincide. Since this unitary satisfies the symmetry of the KO 3 -group, this is an element of the quaternionic ) to (1 S ) 4 . By using Lemma 3.6, we obtain the following equality in KO 3 (S 0,γ , τ S ): Proposition 3.8. When α and β are rational numbers and t = −ps + qr is even, ] is nontrivial and has order two by Lemma 3.7. We thus obtain a splitting of the sequence (3.5) and the group KO 3 (S 0,γ , τ S ) is isomorphic to The results in this subsection are summarized in Tables 3, 4 and 5. 3.3. Boundary Maps Associated with Quarter-Plane Toeplitz Extensions and KO-Groups of (T α,β ,τ α,β ). We next consider the boundary maps 7 of the 24-term exact sequence for KO-theory associated with the sequence (2.2): Proof. When i = −1, 0, 4, 6, the group KO i−1 (K(Ĥ α,β ), τ K ) is trivial and the statement is obvious. We discuss the other cases. The proof is given by constructing explicit elements of the group KO i (S α,β , τ S ), which maps to a generator of the group KO i−1 (K(Ĥ α,β ), τ K ). As in [35], by using the action of SL(2, Z) on Z 2 , we assume 0 < α ≤ 1 2 and 1 ≤ β < +∞ without loss of generality. Let P m,n =P α,β M m,nP α,β M −m,−nP α,β . As in [35], we consider the following element inT α,β : The operatorÂ is Fredholm whose kernel is trivial and has one dimensional cokernel [35]. We also have the following.
•Â is a real operator, that is r(Â) =Â, andÂ τ = r(Â * ) =Â * holds. From these preliminaries, the proof of Proposition 3.9 is parallel to the computation in Example 9.4 of [14]. We summarize the results here.
In this section, Toeplitz operators associated with the subsemigroup (Z ≥0 ) n of Z n for n ≥ 3 are discussed. They are an n-variable generalization of the ordinary Toeplitz and quarter-plane Toeplitz operators and are briefly discussed in [23,22], where a necessary and sufficient condition for these Toeplitz operators to be Fredholm is obtained. We revisit these operators since, in our application to condensed matter physics, models of higher-codimensional corners are given by using these n-variable generalizations. Since the Toeplitz extension (4.1) and the quarter-plane Toeplitz extension (2.3) provide a framework for these applications, we seek this extension for our n-variable cases (Theorem 4.1). Note that we consider corners of arbitrary codimension, though of a specific shape compared to the codimension-two case [54]. In this section, let n be a positive integer bigger than or equals to three.
To study such Toeplitz operators, we follow Douglas-Howe's idea [23] to use the tensor product of the Toeplitz extension, There is a linear splitting of the * -homomorphism γ given by the compression onto be the * -homomorphism induced by γ. Specifically, π D R = a 1 ⊗ · · · ⊗ a n , where a i is id C(T) when i ∈ D, is γ when i ∈ R \ D and is id T otherwise. Note that π D R is a surjection and π ∅ ∅ = id. In the following, we use a subset A of {1, . . . , n} as a label to distinguish C * -algebras and the morphisms between them, which we may abbreviate brackets {·} in our notation. For example, we write T n 1,2 for T n {1,2} , π i for π ∅ {i} and π 1 1,2 for π {1} {1,2} . For each A ⊂ {1, . . . , n}, the map π A has a linear splitting ρ A : T n A → T n given by the compression onto l 2 ((Z ≥0 ) n ). By these preliminaries, we consider the following C * -subalgebra of T n 1 ⊕ · · · ⊕ T n n . .
Let (T 1 , . . . , T n ) ∈ S n . For a nonempty subset A ⊂ {1, . . . , n}, we take i ∈ A and consider the element π i A (T i ) ∈ T n A . This element does not depend on the choice of i ∈ A, and we write T A = π i A (T i ). Let ρ ′ : S n → T n be a linear map defined by for (T 1 , . . . , T n ) ∈ S n , where the second summation is taken over all subsets A ⊂ {1, . . . , n} consisting of k elements. Let K n = K(l 2 ((Z ≥0 ) n )), and let γ n : T n → S n be an * -homomorphism given by γ n (T ) = (π 1 (T ), . . . , π n (T )). Let ι n be the n-fold tensor product of ι.
Theorem 4.1. There is the following short exact sequence of C * -algebras: where the map γ n has a linear splitting given by ρ ′ .
and since (−1) n π 1 • ρ 2,...,n (T 2,...,n ) + (−1) n+1 π 1 • ρ 1,...,n (T 1,...,n ) = 0,  As in Sect. 2.2, the real structure c on l 2 (Z n ) induces real structures on T n i and S n . We write τ S for the transposition on S n associated with this real structure. The map γ n preserves the real structure, and we obtain the following exact sequence of C * ,τ -algebras: We next compute the K-groups of the C * -algebra S n and KO-groups of the C * ,τ -algebra (S n , τ S ).
. The result follows from the six-term exact sequence of K-theory associated with the sequence (4.2) in Theorem 4.1.
The results are collected in Table 9.
Proof. Note that KO i (T n , τ T ) ∼ = KO i (C, id). The result follows from the 24-term exact sequence of KO-theory associated with the sequence (4.4).
A Fredholm Toeplitz operator associated with a codimension-n corner whose Fredholm index is one is constructed as follows.
Example 4.5 (A Fredholm Operator of Index One). Let T z be the Toeplitz operator whose symbol z : T → C is the inclusion. Its adjoint T * z is a Fredholm operator on l 2 (Z ≥0 ) of index one. Let p = T z T * z and q = 1 T − p, then p, q ∈ T and are projections onto l 2 (Z ≥1 ) and Cδ 0 , respectively, where δ 0 is the characteristic function of the point 0 ∈ Z. For a subset A ⊂ {1, . . . , n}, let P n A = r 1 ⊗ · · · ⊗ r n , where r i is p when i ∈ A and is q otherwise. The operator P n A is a projection which satisfies A P n A = 1 T n . LetT = T * z ⊗ q ⊗ · · · ⊗ q and consider the following element in T n : where the sum is taken over all subsets of {1, . . . , n} except {1}. Then, we can see that Ker(G) ∼ = C and Coker(G) = 0, that is, G is a Fredholm Toeplitz operator associated with codimension-n corners whose Fredholm index is one.
This example leads to the following result.
Proposition 4.6. The boundary maps of the six-term exact sequence for K-theory associated with (4.2) are surjective. Moreover, the boundary maps of the 24-term exact sequence for KO-theory associated with (4.4) are surjective.
Proof. The result for complex K-theory is immediate from Example 4.5. For KOtheory, since the operator G in Example 4.5 satisfies G τ = G * , the result follows as in Proposition 3.9.
Since the Fredholm index of G is one, this gives a generator of K 1 (S n ) ∼ = Z. As in the proof of Proposition 3.9, generators of the KO-groups KO i (S n , τ S ) are also obtained by using G.
Remark 4.7. Let 1 ≤ j ≤ n. We have the following * -homomorphisms: where the first map maps (T 1 , . . . , T n ) to T j . We write σ n,n−1 for the composite of the above maps which induces the map σ n,n−1 *

Topological invariants and corner states in Altland-Zirnbauer classification
In this section, some gapped Hamiltonians on a lattice with corners are discussed in each of the Altland-Zirnbauer classes. Since two of them (class A and AIII) are already studied in [32,33], we consider the remaining cases here. The codimension of the corner will be arbitrary, though we mainly discuss codimension-two cases, with many detailed results being obtained by [54,35,33] and the results in Sect. 3. Higher-codimensional cases are discussed in a similar way, whose results are collected in Sect. 5.5. 5.1. Setup. Let V be a finite rank Hermitian vector space of complex rank N . Let n be a positive integer. Let Θ and Ξ be antiunitary operators on V whose squares are +1 or −1. Let Π be a unitary operator on V whose square is one. These operators Θ, Ξ and Π are naturally extended to the operator on l 2 (Z n ; V ) by the fiberwise operation; we also denote them as Θ, Ξ and Π, respectively. Let Herm(V ) be the space of Hermitian operators on V . We consider a continuous map T n → Herm(V ), t → H(t), where t = (t 1 , t 2 , . . . , t n ) in T n . Through the Fourier transform L 2 (T n ; V ) ∼ = l 2 (Z n ; V ), the multiplication operator generated by this continuous map defines a bounded linear self-adjoint operator H on the Hilbert space l 2 (Z n ; V ). We consider the lattice Z n as a model of the bulk and call H the bulk Hamiltonian. The Hamiltonian is said to preserve time-reversal symmetry (TRS) if it commutes with Θ (i.e., ΘHΘ * = H), particle-hole symmetry (PHS) if it anticommutes with Ξ (i.e., ΞHΞ * = −H) and chiral symmetry if it anticommutes with Π (i.e., ΠHΠ * = −H). Furthermore, TRS or PHS is called even (resp. odd) if Θ 2 = 1 or Ξ 2 = 1 (resp. Θ 2 = −1 or Ξ 2 = −1). Hamiltonians may preserve both TRS and PHS. In that case, Θ and Ξ are assumed to commute, and Π is identified with ΘΞ or iΘΞ such that Π 2 = 1 is satisfied.
By taking the partial Fourier transform in the variables t 1 and t 2 , we obtain a continuous family of bounded linear self-adjoint operators {H(t)} t∈T n−2 on H ⊗ V . By taking a compression onto H α ⊗ V , H β ⊗ V andĤ α,β ⊗ V , we obtain a family of operators H α (t), H β (t) andĤ α,β (t) parametrized by t = (t 3 , . . . , t n ) ∈ T n−2 . H α (t) and H β (t) are our models for two surfaces (codimension-one boundaries), andĤ α,β (t) is our model of the corner (codimension-two corner). We assume the following spectral gap condition. Under this assumption, the bulk Hamiltonian H is also invertible since, when we take a basis of V and identify V with C N , there is a unital * -homomorphism M N (S α,β ⊗ C(T n−2 )) → M N (C(T n )) that maps (H α , H β ) to H. In classes AI and AII, we further assume that the spectrum of H is not contained in either R >0 or R <0 . Note that in other classes where Hamiltonians preserve PHS or chiral symmetry, this condition follows from Assumption 5.1. Let h be the pair (H α , H β ). Under Assumption 5.1, we set When the bulk Hamiltonian H satisfies TRS, PHS or chiral symmetry, the operators H α , H β ,Ĥ α,β and sign(h) also satisfy the symmetry, that is, commutes with Θ or anticommutes with Ξ or Π.

Gapped Topological Invariants.
In the following, starting from a Hamiltonian satisfying Assumption 5.1 in each class AI, BDI, D, DIII, AII, CII, C and CI, we construct a unitary and see that this unitary satisfies the relation R i in Table 2. By using this unitary, we define a topological invariant as an element of some KO-group.
In class AI, the Hamiltonian has even TRS. We take an orthonormal basis of V to identify V with C N and express Θ as C = diag(c, . . . , c). Under our spectral gap condition, let This u is a self-adjoint unitary satisfying u τ = Ad C⊕C (u * ) = u * by the TRS. In class BDI, the Hamiltonian has both even TRS and even PHS. Note that the chiral symmetry is given by Π = ΘΞ and commutes with Θ and Ξ. For a Hamiltonian satisfying chiral symmetry and Assumption 5.1 to exist, the even/odd decomposition V ∼ = V 0 ⊕ V 1 with respect to Π should satisfy rank C V 0 = rank C V 1 , and we assume that. Then, there is an orthonormal basis of V to identify V with C N such that Π and Θ are expressed as follows: where C = diag(c, . . . , c). Since the Hamiltonian H anticommutes with Π, the operator sign(h) in (5.1) is written in the following off-diagonal form: where u is a unitary. By the TRS, we have u τ = Cu * C * = u * . In class D, the Hamiltonian has even PHS. We take an orthonormal basis of V to identify V ∼ = C N and express Ξ as C = diag(c, . . . , c). Let u = sign(h), then we have u τ = ΞuΞ * = −u by the PHS.
In class DIII, the Hamiltonian has both odd TRS and even PHS. Note that the chiral symmetry is given by Π = iΘΞ and anticommutes with Θ and Ξ. For such a Hamiltonian H satisfying Assumption 5.1 to exist, the complex rank of V must be a multiple of 4 since sign(H(t)), iΠ, i and Θ provides a Cl 1,1 ⊗Cl 2,0 ∼ = H(2)-module structure on V . We assume rank C V = 4M for some positive integer M .
Lemma 5.2. If a Hamiltonian H satisfying Assumption 5.1 exists, there is an orthonormal basis of V such that Π and Θ are expressed as follows.
We write J = diag(j, . . . , j), where j is the quaternionic structure on H.
We take this basis on V and express Π and Θ as above. By the chiral symmetry, we take u in (5.4). By the TRS, we have u ♯⊗τ = J u * J * = u.
In class AII, the Hamiltonian has odd TRS. The space V has a quaternionic structure given by Θ, and the complex rank of V is even, for which we write 2M . There is an orthonormal basis of V for identifying V with C 2M ∼ = H M and Table 10. i(♠) and c(♠) for each of the Altland-Zirnbauer classes ♠.
expressing Θ as J = diag(j, . . . , j). Let u be a self-adjoint unitary in (5.2). By the TRS, we have u ♯⊗τ = Ad J ⊕J (u * ) = u * . In class CII, the Hamiltonian has both odd TRS and odd PHS. The chiral symmetry is given by Π = ΘΞ and commutes with Θ and Ξ. As in the class BDI case, we take an orthonormal basis of V to identify V with C N and express Π and Θ as where J = diag(j, . . . , j). By the chiral symmetry, we take u in (5.4). By the TRS, we have u ♯⊗τ = J u * J * = u * . In class C, the Hamiltonian has odd PHS. Since Ξ provides a quaternionic structure on V , its complex rank is even, for which we write 2M . We take an orthonormal basis of V to identify V with C 2M ∼ = H M and express Ξ as J = diag(j, . . . , j). Let u = sign(h), then we have u ♯⊗τ = J u * J * = −u by the PHS.
In class CI, the Hamiltonian has both even TRS and odd PHS. The chiral symmetry is given by Π = iΘΞ and anticommutes with Θ and Ξ. As in Lemma 5.2, we take an orthonormal basis of V to express, where C = diag(c, . . . , c). By the chiral symmetry, we take u in (5.4). By the TRS, we have u τ = Cu * C * = u.  Table 10. We denote its class 9 [u] in the KO-group KO i(♠) (S α,β ⊗ C(T n−2 ), τ ) by I n,2,♠ Gapped (H). The groups KO * (S α,β ⊗ C(T n−2 ), τ ) are computed by results in Sect. 3.2.
Remark 5.4. We expressed the symmetry operators in a specific way, though we may choose another one. In class DIII, for example, the operator Θ can also be expressed as 0 −C C 0 , where C = diag(c, . . . , c). Then, we obtain unitaries satisfying u τ = −u, which are treated in [38].

Gapless Topological Invariants.
We next define another topological invariant by using our model for the cornerĤ α,β . By Assumption 5.1 and Theorem 2.6 in [54], {Ĥ α,β (t)} t∈T n−2 is a continuous family of self-adjoint Fredholm operators. Corresponding to its Altland-Zirnbauer classes, this family provides a Z 2 -map from (T n−2 , ζ) to some Z 2 -spaces of self-adjoint or skew-adjoint Fredholm operators introduced in Appendix A as follows.
• Class D, Z 2 -map iĤ α,β : (T n−2 , ζ) → (Fred (1,0) Here, we write r Θ = Ad Θ when Θ 2 = 1 and q Θ = Ad Θ when Θ 2 = −1. Involutions r Ξ and q Ξ are defined as Ad Ξ in the same way. By Corollary A.9, the Z 2 -homotopy classes of Z 2 -maps from (T n−2 , ζ) to the above Z 2 -space of self-adjoint or skewadjoint Fredholm operators is isomorphic to the KO-group KO i (C(T n−2 ), τ T ) of some degree i.  Tables 10 and 11. For a Hamiltonian H in class ♠ satisfying Assumption 5.1, we denote I n,2,♠ Gapless (H) for the class [c(♠)Ĥ α,β ] in KO i(♠)−1 (C(T n−2 ), τ T ). We call I n,2,♠ Gapless (H) the gapless corner invariant. If the gapless corner invariant is nontrivial, zero is contained in the spectrum of H α,β . In Sect. 5.6, we discuss more refined relations between the gapless corner invariant and corner states when k = n − 1 and n. Table 11. In each Altland-Zirnbauer class ♠, gapped invariants and gapless invariants are defined as elements of some K-and KO-groups of some degree, as indicated in this table. Classifying spaces for topological K-and KR-groups through self-adjoint or skew-adjoint Fredholm operators and unitaries are also included. (Z 2 -)spaces Fred ♠ and U ♠ are introduced in Appendix A.
5.4. Correspondence. By taking a tensor product of the extension (2.2) and (C(T n−2 ), τ T ), we have the following short exact sequence of C * ,τ -algebras, Let consider the following diagram containing the boundary map of 24-term exact sequence for KO-theory associated with this sequence: where F ♠ is the Z 2 -subspace of Fred ♠ as in Appendix A, whose inclusion F ♠ ֒→ Fred ♠ is the Z 2 -homotopy equivalence. Maps L and exp are as follows. , we use its expressions through exponentials (see [14] for even i(♠) and Appendix A.3 for odd i(♠)) and the diagram commutes. Note that, by Proposition 3.9, the boundary map∂ i(♠) is surjective. The following is the main result of this section. Proof. The operatorĤ α,β is a self-adjoint lift of (H α , H β ) and preserves the symmetries of the class ♠. Therefore, we have L(I n,2,♠ Gapped (H)) = [c(♠)Ĥ α,β ] and the results follows from the commutativity of the above diagram.
Remark 5.7 (Relation with bulk weak invariants). Under Assumption 5.1, the bulk Hamiltonian H is also invertible. When we take H in place of h = (H α , H β ) and define the unitary u ′ as in Sect. 5.2, this unitary defines an element [u ′ ] in KO i(♠) (C(T n ), τ T ), which classifies bulk invariants in class ♠. A relation between our gapped invariants I n,2,♠ Gapped (H) and these bulk invariants can be discussed through the map (σ ⊗ 1) * : which maps [u] to [u ′ ] and we briefly mention its consequences here. Our gapped invariant I n,2,♠ Gapped (H) has no relation with bulk invariants in the sense that, under Assumption 5.1, bulk invariants are trivial except for a component corresponding to KO-groups of a point 10 and for the cases when α, β are both rational (or ±∞) and t = −ps + qr is even. By Remark 3.10, when α and β takes these values, some bulk weak invariants can be nontrivial, though they have no relation with I n,2,♠ Gapless (H), which can be seen by comparing the above map (σ ⊗ 1) * and the boundary map ∂ i(♠) .
Remark 5.8 (Convex and concave corners). When we fix α and β, there exist two models of corners: convex and concave corners (Ĥ α,β andȞ α,β ). We have discussed convex corners though, as in [33], similar results also hold for concave corners by using (2.3) in place of (2.2) in our discussion. By Remark 3.11, the gapless invariants of these two are related by the factor −1.

5.5.
Higher-Codimensional Cases. Let n and k be positive integers satisfying 3 ≤ k ≤ n. In this subsection, we consider n-D system with a codimension-k corner. Let d = n − k. We consider a continuous map T n → Herm(V ) and the bounded linear self-adjoint operator H on l 2 (Z n ) generated by this map, which is our model of the bulk. We next introduce models of corners of codimension k − 1 whose intersection makes a codimension k corner. For this, we choose d variables t j1 , . . . , t j d in t 1 , t 2 , . . . , t n and consider the partial Fourier transform in these d variables to obtain a continuous family of self-adjoint operators {H(t)} t∈T d on l 2 (Z k ; V ). On the Hilbert space l 2 (Z k ; V ) ∼ = (l 2 (Z)⊗· · ·⊗l 2 (Z))⊗V , we consider projections P k = (P ≥0 ⊗ · · · ⊗ P ≥0 ) ⊗ 1 V , and P k,i = (P ≥0 ⊗ · · · ⊗ 1 ⊗ · · · ⊗ P ≥0 ) ⊗ 1 V for 1 ≤ i ≤ k where inside the brackets is the tensor products of P ≥0 except for the i-th tensor product replaced by the identity. By using these projections, we define the following operators: for 1 ≤ i ≤ k and for t ∈ T d . These two are our model for a codimension k corner and codimension k − 1 corners, respectively. When we fix a basis on V , we have (H 1 (t), . . . , H k (t)) ∈ M N (S k ) by the construction. We assume the following condition in this subsection. Assumption 5.9 (Spectral Gap Condition). We assume that our models for codimension k − 1 corners H 1 , . . . , H k are invertible.
Under this assumption, the model for the bulk, surfaces and corners of codimension less than k, whose intersection makes our codimension-k corner, are invertible. As in Sect. 5.1, let h = (H 1 , . . . , H k ).
Definition 5.10. For a Hamiltonian in class ♠ = AI, BDI, D, DIII, AII, CII, C or CI satisfying Assumption 5.9, let u be a unitary defined by using this h in place of that in Sect. 5.2. As in Sect. 5.2, this unitary u satisfies the relation R i(♠) where i(♠) is as indicated in Table 10. We denote its class [u] in the KO-group  Tables 10 and 11. For a Hamiltonian H in class ♠ satisfying Assumption 5.9, we denote I n,k,♠ Gapless (H) for the class [c(♠)H c ] in KO i(♠)−1 (C(T d ), τ T ). We call I n,k,♠ Gapless (H) the gapless corner invariant. We next discuss a relation between these two topological invariants. As in Sect. 5.4, we consider a tensor product of the extension (4.4) and (C(T d ), τ T ) and , τ ) be the boundary map associated with it expressed through exponentials. Since H c is a self-adjoint lift of (H 1 , . . . , H k ), the following relation holds, as in Theorem 5.6. Theorem 5.12. ∂ i(♠) (I n,k,♠ Gapped (H)) = I n,k,♠ Gapless (H). Remark 5.13. As in Remark 5.7, under Assumption 5.9, some gapped invariants related to corner states for corners of codimension < k are also defined, though, by Remark 4.7, they are trivial except for a component corresponding to KO-groups of a point.
Remark 5.14. Gapless corner invariants for each systems are elements of the group id). As in the case of (first-order) topological insulators [42], we call the component KO i−d (C, id) strong and others weak.
Complex cases can also be discussed in a similar way 11 . For class A systems with a codimension ≥ 3 corner, under the Assumption 5.9, we define gapped and gapless invariants as elements of K 0 (S k ⊗ C(T d )) and K 1 (C(T d )), respectively, and the boundary map ∂ 0 : K 0 (S k ⊗ C(T d )) → K 1 (K k ⊗ C(T d )) associated with (4.4) relates these two, which is surjective by Proposition 4.6. In class AIII systems, we use ∂ 1 : , and we call the component K i−d (C) strong and others weak.
Strong invariants for each system are classified in Table 1. 5.6. Numerical Corner Invariants. Our gapless corner invariants are defined as elements of some KO-group. In this subsection, we introduce Z-or Z 2 -valued numerical corner invariants for our systems in cases where k = n and k = n − 1 to make the relation between our gapped invariants and corner states more explicit. From Table 11, we discuss Hamiltonians in classes BDI, D, DIII, and CII when k = n and D, DIII, AII and CII when k = n − 1 satisfying our spectral gap condition. 5.6.1. Case of k = n. In this case, our model of the corner H c is a self-adjoint Fredholm operator which has some symmetry corresponding to its Altland-Zirnbauer class 12 . An appropriate definition of numerical topological invariants is introduced in [9] and we put them in our framework.
In class BDI, the operator H c is an element of the fixed point set (Fred (0,2) * ) rΘ of the involution r Θ , where the Clifford action of Cl 0,1 on the Hilbert space is given by ǫ 1 = Π (see also Lemma A.10 and Remark A.11). We express Π and Θ as in (5.3) and express H c as follows.
The operator h c is a Fredholm operator that commutes with C and thus is a real Fredholm operator. Its Fredholm index is where the right-hand side is the trace of Π restricted to Ker(H c ). The Fredholm index induces an isomorphism ind BDI : [(pt, id), (Fred (0,2) * , r Θ )] Z2 → Z. 11 In [32,33], there is a mistake in the computations of the group K 0 (S α,β ) in the case where α and β are rational numbers (there is a torsion part in general, as in KO 0 (S α,β , τ S ) computed in Sect. 4), which is correctly stated in [55]. The author would like to thank Guo Chuan Thiang for pointing this mistake out. 12 In what follows, we also write H c forĤ α,β in k = 2 case.
In class D, iH c commutes with the real structure Ξ and is a real skew-adjoint Fredholm operator. Its mod 2 index [9] is ind 1 (iH c ) = rank C Ker(H c ) mod 2, which induces the isomorphism ind D : [(pt, id), (Fred (1,0) In class DIII, H c is an element of (Fred (1,1) * ) qΘ , where the action of Cl 1,0 is given by e 1 = iΠ. The operator iH c and e 1 commute with the real structure Ξ; thus, iH c is a real skew-adjoint Fredholm operator that anticommutes with e 1 . Its mod 2 index [9] is which induces the isomorphism ind DIII : [(pt, id), (Fred (1,1) * , q Θ )] Z2 → Z 2 . In class CII, the operator H c is an element of (Fred (0,2) * ) qΘ , where the Clifford action of Cl 0,1 is given by ǫ 1 = Π. We express Θ and Π as in (5.5) and express H c as in (5.6). The operator h c commutes with J and is a quaternionic Fredholm operator. Its Fredholm index ind(h c ) is an even integer that induces an isomorphism ind CII : [(pt, id), (Fred (0,2) * , q Θ )] Z2 → 2Z.
Definition 5.15. For n-D systems with codimension-n corners in classes BDI, D, DIII and CII, we define the numerical corner invariant as follows.
• In class BDI, let N n,n,BDI • In class CII, let N n,n,CII Gapless (H) = ind(h c ) ∈ 2Z. Note that by these definitions, they are images of gapless corner invariants I n,n,♠ Gapless (H) for each class ♠ = BDI, D, DIII and CII through the isomorphism ind ♠ . In each case, the numerical corner invariant is computed through Ker(H c ) and is related to the number of corner states. 5.6.2. Case of k = n − 1. In this case, {H c (t)} t∈T is a continuous family of selfadjoint Fredholm operators preserving some symmetry. The numerical corner invariants are given by using (Z-valued) spectral flow [8] and its Z 2 -valued variants [19,18]. We first review Z-and Z 2 -valued spectral flow.
Spectral flow is, roughly speaking, the net number of crossing points of eigenvalues of a continuous family of self-adjoint Fredholm operators with zero [8]. The following definition of spectral flow is due to Phillips [56].
Spectral flow is independent of the choice made and depends only on the homotopy class of the path A leaving the endpoints fixed. Thus the spectral flow induces a map sf : [T, Fred (0,1) * ] → Z which is a group isomorphism. We next discuss Z 2 -valued spectral flow. Let ζ 0 be an involution on the interval [−1, 1] given by ζ 0 (s) = −s. Let A be a Z 2 -map from ([−1, 1], ζ 0 ) to (Fred (0,1) * , q). Then, the spectrum sp(A s ) of A s is symmetric with respect to ζ 0 , and roughly speaking, Z 2 -valued spectral flow counts the mod 2 of the net number of pairs of crossing points of sp(A s ) with zero. Z 2 -valued spectral flow is studied in [19,18,21,15] and we give one definition following [56,19].
In class D, we have a Z 2 -map iH c : (T, ζ) → (Fred (1,0)   In class DIII, H c : (T, ζ) → (Fred (1,1) * , q Θ ) is a Z 2 -map, where the action of the Clifford algebra on the right-hand side is given by e 1 = iΠ. The Z 2 -homotopy classes [(T, ζ), (Fred (1,1 Combined with the Z 2 -valued spectral flow, we obtain the following map: For b = 1 or −1, let i b be the inclusion {b} ֒→ T, and let w b be the composite of the following maps: , q ′ ). The operator C 1 is contained in the space of self-adjoint unitaries on V ′ that anticommutes with e 1 ⊕ e ′ 1 . As in [9], this space of unitaries is contractible by Kuiper's theorem. We embed [−1, 1] into T by s → exp( πis 2 ) and, as in Example 5.18, extend C onto T through this contractible space of unitaries to obtain a Z 2 -map D : (T, ζ) → (Fred Note that image the image of sf C are even integers since each eigenspace corresponding to the crossing points of the spectrum of −iA t with zero has a quaternionic vector space structure given by Ξ.   In class DIII cases, we have three surjections sf DIII , w + and w − from Z 2 ⊕ Z 2 to Z 2 . There is the following relation between them.  Remark 5.26. In Definition 5.25, the numerical corner invariants for both class DIII and class AII are defined by using Z 2 -valued spectral flow, though these two Z 2 are different from the viewpoint of index theory in the sense that they sit in different Bott clock. A similar remark holds for, e.g., cases of n = k in classes BDI and CII, where both of these numerical corner invariants are defined as Fredholm indices. 5.7. Product Formula. In Sect. 4 of [32], a construction of the second-order topological insulators of 3-D class A systems is proposed, which is given by using the Hamiltonians of 2-D class A and 1-D class AIII topological insulators. In this subsection, we generalize this construction to other pairs in the Altland-Zirnbauer classification. This provides a way to construct nontrivial examples of each entry in Table 1 from the Hamiltonians of (first-order) topological insulators 13 . For this purpose, we use an exterior product of topological KR-groups [6].
For j = 1, 2, let H j be a bulk Hamiltonian of an n j -D k j -th order topological insulator 14 in real Altland-Zirnbauer class ♠ j (AI, BDI, D, DIII, AII, CII, C or CI). Let n = n 1 + n 2 , k = k 1 + k 2 and d j = n j − k j for j = 1, 2, and let d = Corresponding to the class in the Altland-Zirnbauer classification (for which we write ♠ j ) to which the Hamiltonian belongs, it preserves the symmetries as (even/odd) TRS, (even/odd) PHS or chiral symmetry. We write Θ j , Ξ j and Π j for the symmetry operator for H j . As in Sect. 5, the models of corners H c i lead to a continuous family of self-adjoint or skew-adjoint Fredholm operators and defines an element of the KO-group KO i ′ (♠j ) (C(T dj ), τ ) where i ′ (♠ j ) = i(♠ j ) − 1. As in Appendix A.1, we have an exterior product of KO-groups described through these Fredholm operators. By using this form of the product, we obtain an explicit form of the product of the gapless invariants of H 1 and H 2 . As a result, we can write down a bulk Hamiltonian H of an n-D k-th order topological insulator of class ♠. The lattice on which we consider H c as a model of the codimension-k corner is introduced as the product of that 15 of H c 1 and that of H c 2 . By this construction, we have the following relation between gapless invariants.
Note that Theorem 5.27 is the product formula at the level of KO-group elements and accounts for both strong and weak invariants. In order to show this theorem, we need to write down the explicit form of H. In the following, we discuss them for some classes.
Let us consider the case where ♠ 1 = BDI and ♠ 2 = BDI. In this case, each H j has even TRS, even PHS and chiral symmetry. We now consider the following n-dimensional Hamiltonian: which satisfies even TRS given by Θ = Θ 1 ⊗ Θ 2 , even PHS given by Ξ = Ξ 1 ⊗ Ξ 2 and the chiral symmetry given by Π = Π 1 ⊗ Π 2 . Thus, the Hamiltonian H belongs to the class ♠ = BDI. The model of the codimension-k corner H c of H is written by using the model H c j of the codimension-k j corner as follows: , where t j is an element of the d j -dimensional torus (momentum space) corresponding to a direction parallel to the corner of H c j for j = 1, 2. Note that (t 1 , t 2 ) constitute the parameter of the d-dimensional momentum space in a direction parallel to the corner of H c . By our assumption, H c j (t j ) is an element of the space Fred (0,2) * and gives a Z 2 -map (T dj , ζ) → (Fred (0,2) * , r Θj ). The operator H c (t 1 , t 2 ) is the image of the pair (H c 1 , H c 2 ) through the map, 15 When k j = 1, the lattice is Z ≥0 × Z d j , where H c j is the compression of the bulk Hamiltonian onto this half-space. Topological invariants for them are the one discussed in topological insulators. To clarify our sign choices, we mention that they are obtained by applying the discussion in Sect. 5 to the Toeplitz extension (4.1) in place of (2.2) or (4.2). in (A.2), where the action of Cl 0,1 to define the left-hand side is given by ǫ j = Π j and that for the right-hand side is given by ǫ = ǫ 1 ⊗ ǫ 2 = Π. Since this map induces the exterior product of KO-groups (Appendix A.1), we obtain Theorem 5.27 in this case.
The other cases are computed in a similar way, and the results are summarized in Table 12, where we write  [32,33].
Our product formula (Theorem 5.27) and the graded ring structure of KO * (C, id) (Theorem 6.9 of [7]) lead to the following product formula for numerical corner invariants. We collect the results here where the form of H is as indicated in Table 12.
Corollary 5.28 (Cases of n = k). The case of k 1 = n 1 and k 2 = n 2 .
Remark A.3. Banach Z 2 -spaces and its open Z 2 -subspaces are Z 2 -absolute neighborhood retracts [3], and have the homotopy type of Z 2 -CW complexes [46]. The path spaces and loop spaces we discuss in the following also have the homotopy type of Z 2 -CW complexes [67]. By the equivariant Whitehead theorem, weak Z 2homotopy equivalences between these spaces are Z 2 -homotopy equivalences [50,4].
Let V be a separable infinite-dimensional complex Hilbert space. Let B(V) be the space of bounded complex linear operators on V equipped with the norm topology. Let GL(V), U (V), Fred(V) and K(V) be subspaces of B(V) consisting of invertible, unitary, Fredholm and compact operators on V, respectively. We assume that our Hilbert space V has a real structure r or a quaternionic structure q, that is, an antiunitary operator on V satisfying r 2 = 1 or q 2 = −1, respectively. Correspondingly, the space B(V) has an (antilinear) involution r = Ad r or q = Ad q . These involutions induce involutions on GL(V), U (V), Fred(V) and K(V), for which we also write r or q. We write a for r or q and a for r or q. We also assume that there is a complex linear action of the Clifford algebra Cl k,l on the Hilbert space V that commutes with the real or the quaternionic structure. For an element v ∈ Cl k,l , we also write v for its action on V, for simplicity. When k − l ≡ 3 mod 4, we further assume that each of the two inequivalent irreducible real or quaternionic representations of Cl k,l has infinite multiplicity. In the following, we discuss the subspaces of B(V); we may abbreviate the Hilbert space V from its notation when it is clear from the context. When the Hilbert space V is such a Cl k,l -module, let B (k+1,l) sk (resp. B  ; e k , −e k ), a), where α 1 (A)(t) = e k cos(πt) + A sin(πt) for 0 ≤ t ≤ 1.