Toeplitz operators on concave corners and topologically protected corner states

Abstract

We consider Toeplitz operators defined on a concave corner-shaped 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 quarter-plane and concave corner Toeplitz operators is clarified. As an application, topological invariants and corner states for some bulk-edges gapped Hamiltonians on two-dimensional (2-D) class AIII and 3-D 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’ 2-D 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.

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 rank-one 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 quarter-plane 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 rank-one 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 rank-one projection. Explicitly, we have the following results.

In Case 2, ${\left\{ \begin{array}{ll} \text {when} \ \frac{1}{N+1}< \alpha \le \frac{1}{N} \ \text {and} \ \beta = 1, \text {we have} \ \tilde{\mathcal {P}}_{N-1} = p_{-N-1,-1}.\\ \text {when} \ \frac{1}{N+1}< \alpha \le \frac{1}{N} \ \text {and} \ 1< \beta < \infty , \text {we have} \ \tilde{\mathcal {P}}_N = p_{-N-1,-1}. \end{array}\right. }$

In Case 3, ${\left\{ \begin{array}{ll} \text {when} \ \alpha = \frac{1}{N} \ \text {and} \ 1< \beta \le \infty , \ \text {we have} \ \tilde{\mathcal {P}}_{N-1} = p_{-N,-1}.\\ \text {when} \ \frac{1}{N+1}< \alpha< \frac{1}{N} \ \text {and} \ 1 < \beta \le \infty , \text {we have} \ \tilde{\mathcal {P}}_N = p_{-N-1,-1}. \end{array}\right. }$

In Case 4, ${\left\{ \begin{array}{ll} \text {when} \ \alpha = \frac{1}{N} \ \text {and} \ \beta = 1, \ \text {we have} \ \tilde{\mathcal {P}}_{1} = p_{-1,0}.\\ \text {in the other cases (under } (\dagger )), \text { we have} \ \tilde{\mathcal {P}}_{N} = p_{-N-1,-1}. \end{array}\right. }$

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.

$$\begin{aligned} \check{A} := \check{\mathcal {P}}_{0,1} + M_{1,1}(1 - \check{\mathcal {P}}_{-1,0}) + M_{1,0}(\check{\mathcal {P}}_{-1,0} - \check{\mathcal {P}}_{0,1}). \end{aligned}$$

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 quarter-plane Toeplitz operator in Cases 1–4.

$$\begin{aligned} \hat{A} := \hat{\mathcal {P}}_{0,1} + M_{1,1}(1 - \hat{\mathcal {P}}_{-1,0}) + M_{1,0}(\hat{\mathcal {P}}_{-1,0} - \hat{\mathcal {P}}_{0,1}). \end{aligned}$$

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, ${\left\{ \begin{array}{ll} \text {when} \ \alpha = \frac{1}{2} \ \text {and} \ \beta = 1, {\mathrm {Coker}}\hat{A} = \mathbb {C}{\varvec{e}}_{3,2}.\\ \text {when} \ 0< \alpha< \frac{1}{2} \ \text {and} \ \beta = 1, {\mathrm {Coker}}\hat{A} = \mathbb {C}{\varvec{e}}_{2,1}.\\ \text {when} \ 0< \alpha \le \frac{1}{2} \ \text {and} \ 1< \beta < \infty , {\mathrm {Coker}}\hat{A} = \mathbb {C}{\varvec{e}}_{1,1}. \end{array}\right. }$