Abstract
We consider Toeplitz operators defined on a concave cornershaped subset of the square lattice. We obtain a necessary and sufficient condition for these operators to be Fredholm. We further construct a Fredholm concave corner Toeplitz operator of index one. By using this, a relation between Fredholm indices of quarterplane and concave corner Toeplitz operators is clarified. As an application, topological invariants and corner states for some bulkedges gapped Hamiltonians on twodimensional (2D) class AIII and 3D class A systems with concave corners are studied. Explicit examples clarify that these topological invariants depend on the shape of the system. We discuss the Benalcazar–Bernevig–Hughes’ 2D Hamiltonian and see that there still exist topologically protected corner states even if we break some symmetries as long as the chiral symmetry is preserved.
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Notes
 1.
 2.
The square lattice \(\mathbb {Z}^2\) is naturally embedded in the Euclidean space \(\mathbb {R}^2\). As a subset of \(\mathbb {R}^2\), what we called convex corners are not convex sets. We here use the words convex and concave just to distinguish the two models of corners indicated in Fig. 1.
 3.
In order to distinguish these two cases, we use hat “\(\wedge \)” for objects associated with convex corners and check “\(\vee \)” for those with concave corners (e.g., \(\hat{\mathcal {H}}^{\alpha ,\beta }\) and \(\check{\mathcal {H}}^{\alpha ,\beta }\)).
 4.
In what follows, Kgroups of \(C^*\)algebras \(\check{\mathcal {T}}^{\alpha ,\beta }\) and \(\check{\mathcal {C}}^{\alpha ,\beta }\) are computed, and the results for \(\mathcal {S}^{\alpha , \beta }\), \(\mathcal {C}^\alpha \) and \(\mathcal {C}^\beta \) are presented corresponding to the values of \(\alpha \) and \(\beta \). The case of \(\alpha = \infty \) or \(\beta = + \infty \) is the same as that of rational \(\alpha \) or rational \(\beta \).
 5.
If A is an algebra, \(M_r(A)\) denotes the algebra of all \(r \times r\) matrices with entries in A.
 6.
\(P^{\alpha }h P^{\alpha }/P^{\alpha }h P^{\alpha }\) is defined by the continuous functional calculus by the continuous function \(\mathbb {C}\setminus \{ 0\} \rightarrow \mathbb {C}\) given by \(z \mapsto z/z\).
 7.
This element does not depend on the choice of the identification \(V \cong \mathbb {C}^N\).
 8.
We here give the definition of edge topological invariants for 1D class AIII topological insulators. By the bulkedge correspondence, this coincides with the bulk topological invariant which is defined as the winding number of the determinant of its symbol, that is \(\mathrm {Wind}(\{\det h_j(t)\}_{t \in \mathbb {T}})\) where \(H_j = \begin{pmatrix} 0 &{} h_j^*\\ h_j &{} 0 \end{pmatrix}\). (see [23], for example).
 9.
We here regard \(\mathbb {T}\) as the unit circle in the complex plane and fix the counterclockwise orientation.
 10.
\( \sigma _1 = \begin{pmatrix} 0 &{}\quad 1\\ 1 &{}\quad 0 \end{pmatrix}, \ \sigma _2 = \begin{pmatrix} 0 &{}\quad i\\ i &{}\quad 0 \end{pmatrix}, \ \sigma _3 = \begin{pmatrix} 1 &{}\quad 0\\ 0 &{}\quad 1 \end{pmatrix}. \)
 11.
We here employ the following identification: \( \begin{pmatrix} a &{} b\\ c &{} d \end{pmatrix} \otimes A = \begin{pmatrix} aA &{} bA\\ cA &{} dA \end{pmatrix}. \)
 12.
Note that \(r_4 (\sigma _2 \otimes 1) r_4^* =  \sigma _3 \otimes \sigma _2\), \(r_4(\sigma _1 \otimes 1)r_4^* = \sigma _3 \otimes \sigma _1\), \( r_4(\sigma _3 \otimes \sigma _1)r_4^* = \sigma _1 \otimes 1\) and \(r_4 (\sigma _3 \otimes \sigma _2) r_4^* = \sigma _2 \otimes 1\) holds.
 13.
Note that we have \(U(\sigma _1 \otimes 1) U^* = \sigma _1 \otimes 1\), \(U(\sigma _2 \otimes 1) U^* =  \sigma _2 \otimes \sigma _3\), \(U(\sigma _3 \otimes \sigma _1) U^* =  \sigma _2 \otimes \sigma _2\) and \(U(\sigma _3 \otimes \sigma _2) U^* =  \sigma _2 \otimes \sigma _1\).
 14.
Specifically, they are \(U \varTheta U^* = \varTheta \), \(U \varXi U^* = (\sigma _3 \otimes \sigma _1) \circ \varTheta \), \(U \varPi U^* = \sigma _3 \otimes \sigma _1\), \(U m_x U^* = \sigma _1 \otimes \sigma _3\), \(U m_y U^* = \sigma _1 \otimes \sigma _1\) and \(U r_4 U^* = \begin{pmatrix} 0 &{}\quad 1 \\ i\sigma _2 &{}\quad 0 \end{pmatrix}\), respectively.
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Acknowledgements
The author would like to thank Takeshi Nakanishi and Yukinori Yoshimura for showing him the result of a numerical calculation, which convinced him about the content of this paper. He also would like to thank KenIchiro Imura and Ryo Okugawa for many discussions concerning [1] and Max Lein for sharing the information regarding [25]. The author acknowledges the support of the Erwin Schrödinger Institute where part of this work was conducted. He would like to thank organizers of the workshop “Bivariant Ktheory in Geometry and Physics” for their hospitability. This work was supported by JSPS KAKENHI Grant Nos. JP17H06461 and JP19K14545.
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A some variants
A some variants
As in Remark 1, most results in this paper also hold in the cases in which the corners (or edges) do not necessarily include lattice points on lines \(y=\alpha x\) and \(y=\beta x\). In this appendix, we make this statement precise by fixing the setups and clarifying the corresponding results. Although the proofs of the corresponding results are parallel with those contained in the main body of this paper, some parts of the discussions are based on the explicit construction of an example, especially the constructions of rankone projections (Lemma 2), and that of the Fredholm concave corner Toeplitz operator of index one (Theorem 3). For these reasons, we collect the corresponding results in this appendix. The corresponding results for quarterplane Toeplitz operators, briefly mentioned in [17], are also included for completeness.
Since we consider two edges, corresponding to whether the edge includes lattice points on boundaries, we can consider four cases. Each case corresponds to the case in which closed subspaces \(\mathcal {H}^{\alpha }\) and \(\mathcal {H}^{\beta }\) of \(\mathcal {H}\) are spanned by the following sets:

Case 1 : \(\{ { e_{m,n}} \mid \alpha m + n \ge 0 \}\) and \(\{ { e_{m,n}} \mid \beta m + n \le 0 \}\), respectively.

Case 2 : \(\{ { e_{m,n}} \mid \alpha m + n > 0 \}\) and \(\{ { e_{m,n}} \mid \beta m + n \le 0 \}\), respectively.

Case 3 : \(\{ { e_{m,n}} \mid \alpha m + n \ge 0 \}\) and \(\{ { e_{m,n}} \mid \beta m + n < 0 \}\), respectively.

Case 4 : \(\{ { e_{m,n}} \mid \alpha m + n > 0 \}\) and \(\{ { e_{m,n}} \mid \beta m + n < 0 \}\), respectively.
For these cases, we associate concave corners and define concave corner \(C^*\)algebras \(\check{\mathcal {T}}^{\alpha ,\beta }\) in the same way as in Sect. 2. Note that Case 1 is already treated in the main body of this paper. In the following, we assume the condition (\(\dagger \)) for \(\alpha \) and \(\beta \).
We first collect constructions of rankone projections in Cases 2–4. They correspond to Lemma 2 in Case 1. As in Lemma 2, we take \(N \in \{ 2,3,\ldots \}\) such that \(\frac{1}{N+1} < \alpha \le \frac{1}{N}\).
Lemma 4
In Case 2–4, some \(\tilde{\mathcal {P}}_k\) is a rankone projection. Explicitly, we have the following results.
In Case 2, \({\left\{ \begin{array}{ll} \text {when} \ \frac{1}{N+1}< \alpha \le \frac{1}{N} \ \text {and} \ \beta = 1, \text {we have} \ \tilde{\mathcal {P}}_{N1} = p_{N1,1}.\\ \text {when} \ \frac{1}{N+1}< \alpha \le \frac{1}{N} \ \text {and} \ 1< \beta < \infty , \text {we have} \ \tilde{\mathcal {P}}_N = p_{N1,1}. \end{array}\right. }\)
In Case 3, \({\left\{ \begin{array}{ll} \text {when} \ \alpha = \frac{1}{N} \ \text {and} \ 1< \beta \le \infty , \ \text {we have} \ \tilde{\mathcal {P}}_{N1} = p_{N,1}.\\ \text {when} \ \frac{1}{N+1}< \alpha< \frac{1}{N} \ \text {and} \ 1 < \beta \le \infty , \text {we have} \ \tilde{\mathcal {P}}_N = p_{N1,1}. \end{array}\right. }\)
In Case 4, \({\left\{ \begin{array}{ll} \text {when} \ \alpha = \frac{1}{N} \ \text {and} \ \beta = 1, \ \text {we have} \ \tilde{\mathcal {P}}_{1} = p_{1,0}.\\ \text {in the other cases (under } (\dagger )), \text { we have} \ \tilde{\mathcal {P}}_{N} = p_{N1,1}. \end{array}\right. }\)
We here write down the result of computing the Fredholm index of the following operator in Cases 2–4 which corresponds to Theorem 3 in Case 1.
Proposition 6
In Cases 2–4, \(\check{A}\) is a surjective Fredholm operator whose Fredholm index is 1. We also have \(\check{A}  1 \in \check{\mathcal {C}}^{\alpha ,\beta }\). Its kernel is given as follows:
In Case 2, \({\left\{ \begin{array}{ll} \text {when} \ 0< \alpha \le \frac{1}{2} \ \text {and} \ \beta = 1, {\mathrm {Ker}}\check{A} = \mathbb {C}({\varvec{e}}_{2, 0}  {\varvec{e}}_{1,0}).\\ \text {when} \ 0< \alpha \le \frac{1}{2} \ \text {and} \ 1< \beta < \infty , {\mathrm {Ker}}\check{A} = \mathbb {C}({\varvec{e}}_{1,0}  {\varvec{e}}_{0,0}). \end{array}\right. }\)
In Case 3, under the assumption \((\dagger )\), we have \({\mathrm {Ker}}\check{A} = \mathbb {C}({\varvec{e}}_{1,0}  {\varvec{e}}_{0,0})\).
In Case 4, under the assumption \((\dagger )\), we have \({\mathrm {Ker}}\check{A} = \mathbb {C}({\varvec{e}}_{0,1}  {\varvec{e}}_{1,1})\).
We next consider the following quarterplane Toeplitz operator in Cases 1–4.
Note that \(\hat{A} \in \hat{\mathcal {T}}^{\alpha ,\beta }\). Jiang shows in [17] that, under the assumption (\(\dagger \)), this is an isometric Fredholm operator and compute its Fredholm index mainly in the Case 1. The other cases are briefly mentioned (Remark (1) in p2828 of [17]), though their Fredholm indices are stated as \(\pm 1\). We here need to fix its sign in order to obtain the corresponding result for Corollary 1 especially in Cases 2–4. For this reason, we (re)state necessary results in the following. Its proof is totally parallel with that of Jiang [17].
Proposition 7
(Jiang [17]) In Cases 1–4, \(\hat{A}\) is an isometric Fredholm operator whose Fredholm index is \(1\). Its cokernel is given as follows:
In Case 1, under the assumption \((\dagger )\), we have \({\mathrm {Coker}}\hat{A} = \mathbb {C}{\varvec{e}}_{0,0}\).
In Case 2, under the assumption \((\dagger )\), we have \({\mathrm {Coker}}\hat{A} = \mathbb {C}{\varvec{e}}_{1,1}\).
In Case 3, \({\left\{ \begin{array}{ll} \text {when} \ 0< \alpha \le \frac{1}{2} \ \text {and} \ \beta = 1, {\mathrm {Coker}}\hat{A} = \mathbb {C}{\varvec{e}}_{2,1}.\\ \text {when} \ 0< \alpha \le \frac{1}{2} \ \text {and} \ 1< \beta < \infty , {\mathrm {Coker}}\hat{A} = \mathbb {C}{\varvec{e}}_{1,1}. \end{array}\right. }\)
In Case 4, \({\left\{ \begin{array}{ll} \text {when} \ \alpha = \frac{1}{2} \ \text {and} \ \beta = 1, {\mathrm {Coker}}\hat{A} = \mathbb {C}{\varvec{e}}_{3,2}.\\ \text {when} \ 0< \alpha< \frac{1}{2} \ \text {and} \ \beta = 1, {\mathrm {Coker}}\hat{A} = \mathbb {C}{\varvec{e}}_{2,1}.\\ \text {when} \ 0< \alpha \le \frac{1}{2} \ \text {and} \ 1< \beta < \infty , {\mathrm {Coker}}\hat{A} = \mathbb {C}{\varvec{e}}_{1,1}. \end{array}\right. }\)
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Hayashi, S. Toeplitz operators on concave corners and topologically protected corner states. Lett Math Phys 109, 2223–2254 (2019). https://doi.org/10.1007/s1100501901184w
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Keywords
 Toeplitz operators on concave corners
 Topologically protected corner states
 Bulkedge and corner correspondence
 Ktheory and index theory
Mathematics Subject Classification
 19K56
 47B35
 81V99