Crystallographic bulk-edge correspondence: glide reflections and twisted mod 2 indices

Abstract

A 2-torsion topological phase exists for Hamiltonians symmetric under the wallpaper group with glide reflection symmetry, corresponding to the unorientable cycle of the Klein bottle fundamental domain. We prove a mod 2 twisted Toeplitz index theorem, which implies a bulk-edge correspondence between this bulk phase and the exotic topological zero modes that it acquires along a boundary glide axis.

Appendices

A Appendix: Computation of $K^{\bullet +\widehat{\nu }}_{{\mathbb {Z}}_2}({\mathcal {B}})$ and $K^{\bullet + c + \widehat{\nu }}_{\mathbb {Z}_2}(\mathcal {B}_x)$

1.1 Computation of $K^{\bullet +\widehat{\nu }}_{{\mathbb {Z}}_2}({\mathcal {B}})$

Let $\mathcal {B}_y = \mathbb {R}/2\pi \mathbb {Z}$ be the circle with the involution $k_y \mapsto -k_y$, and $R({\mathbb {Z}}_2)={\mathbb {Z}}[t]/(1-t^2)$ be the representation ring of ${\mathbb {Z}}_2$ with t the sign representation. The following result is known, for example, in [34]:

Lemma A.1

We have the following identifications of $R(\mathbb {Z}_2)$-modules

$$\begin{aligned} K^0_{\mathbb {Z}_2}(\mathcal {B}_y)&\cong \overbrace{R(\mathbb {Z}_2)}^{\mathbb {Z}^2} \oplus \overbrace{(1 - t)}^{\mathbb {Z}},&K^1_{\mathbb {Z}_2}(\mathcal {B}_y)&= 0. \end{aligned}$$

Proof

To apply the Mayer–Vietoris exact sequence, let us consider the $\mathbb {Z}_2$-invariant subspaces U and V given by

$$\begin{aligned} U&= \{ k_y \in \mathcal {B}_y |\ -\pi /2 \le k_y \le \pi /2 \},&V&= \{ k_y \in \mathcal {B}_y |\ \pi /2 \le k_y \le 3\pi /2 \}. \end{aligned}$$

These spaces are equivariantly contractible, so that

$$\begin{aligned} K^n_{\mathbb {Z}_2}(U) \cong K^n_{\mathbb {Z}_2}(V) \cong K^n_{\mathbb {Z}_2}({\mathrm {pt}}) \cong \left\{ \begin{array}{ll} R(\mathbb {Z}_2) &{}\quad (n = 0), \\ 0 &{}\quad (n = 1). \end{array}\right. \end{aligned}$$

The intersection $U \cap V$ is the space consisting of two points with free action of $\mathbb {Z}_2$, so that

$$\begin{aligned} K^n_{\mathbb {Z}_2}(U \cap V) \cong K^n_{\mathbb {Z}_2}({\mathrm {pt}}\sqcup {\mathrm {pt}}) \cong K^n({\mathrm {pt}}) \cong \left\{ \begin{array}{ll} \mathbb {Z}&{}\quad (n = 0), \\ 0 &{}\quad (n = 1). \end{array}\right. \end{aligned}$$

Now, the Mayer–Vietoris exact sequence is

The homomorphism $\Delta$ is realised as $\Delta (u, v) = i_U^*u - i_V^*v$, where $i_U^*$ and $i_V^*$ are induced from the inclusions $i_U : U \cap V \rightarrow U$ and $i_V : U \cap V \rightarrow V$. In the present case, we can identify $i^*_U$ as well as $i^*_V$ with the “dimension” $R(\mathbb {Z}_2) \rightarrow \mathbb {Z}$ given by $f(t) \mapsto f(1)$. This is surjective, and so is $\Delta$. As a result, we get $K^1_{\mathbb {Z}_2}(\mathcal {B}_y) \cong {\mathrm {Coker}}(\Delta ) = 0$. We also get $K^0_{\mathbb {Z}_2}(\mathcal {B}_x) \cong {\mathrm {Ker}}(\Delta ) \cong \mathbb {Z}^3$. As a basis of this abelian group, we can choose

$$\begin{aligned} \{ (1, 1), (t, t), (0, 1 - t) \} \subset R(\mathbb {Z}_2) \oplus R(\mathbb {Z}_2). \end{aligned}$$

The former two base elements generate the $R(\mathbb {Z}_2)$-module $R(\mathbb {Z}_2)$, whereas the latter element $(0, 1 - t)$ generates the $R(\mathbb {Z}_2)$-module $(1 - t)$. $\square$

Let $\mathcal {B} = \mathcal {B}_x \times \mathcal {B}_y$ be the 2-dimensional torus $(\mathbb {R}/2\pi \mathbb {Z}) \times (\mathbb {R}/2\pi \mathbb {Z})$ with the $\mathbb {Z}_2$-action $(k_x, k_y) \mapsto (k_x, -k_y)$, and $\widehat{\nu } \in Z^2(\mathbb {Z}_2, C(\mathcal {B}, U(1)))$ the group 2-cocycle (7) induced from the wallpaper group pg.

Proposition A.2

We have the following identifications of $R(\mathbb {Z}_2)$-modules

$$\begin{aligned} K^{0 + \widehat{\nu }}_{\mathbb {Z}_2}(\mathcal {B})&\cong \overbrace{(1 + t)}^{\mathbb {Z}},&K^{1 + \widehat{\nu }}_{\mathbb {Z}_2}(\mathcal {B})&\cong \overbrace{(1 + t)}^{\mathbb {Z}} \oplus \,\mathbb {Z}/2. \end{aligned}$$

where $\mathbb {Z}/2$ is the unique $R(\mathbb {Z}_2)$-module whose underlying abelian group is $\mathbb {Z}/2$.

Proof

We cover $\mathcal {B}$ by the $\mathbb {Z}_2$-invariant subspaces

$$\begin{aligned} U&= \{ k_x \in \mathcal {B}_x |\ -\pi /2 \le k_x \le \pi /2 \} \times \mathcal {B}_y, \\ V&= \{ k_x \in \mathcal {B}_x |\ \pi /2 \le k_x \le 3\pi /2 \} \times \mathcal {B}_y, \end{aligned}$$

each of which is equivariantly homotopic to $\mathcal {B}_y$. Their intersection is identified with the disjoint union $U \cap V = \mathcal {B}_y ^+ \sqcup \mathcal {B}_y^-$ of two copies $\mathcal {B}_y^{\pm }$ of $\mathcal {B}_y$. The Mayer–Vietoris exact sequence associated to this cover is

where, for example, $\widehat{\nu }|_U \in Z^2(\mathbb {Z}_2, C(U, U(1)))$ is the restriction of $\widehat{\nu }$ to $U \subset \mathcal {B}$, and $\Delta$ is realised as $\Delta (u, v) = i_U^*u - i_V^*v$ by using the inclusions $i_U : U \cap V \rightarrow U$ and $i_V : U \cap V \rightarrow V$. The restricted cocycles can be trivialised, and a choice of these trivialisations induces the isomorphisms

$$\begin{aligned} K^{n + \widehat{\nu }|_{U}}_{\mathbb {Z}_2}(U)&\cong K^{n}_{\mathbb {Z}_2}(U) \cong K^n_{\mathbb {Z}_2}(\mathcal {B}_y), \\ K^{n + \widehat{\nu }|_{V}}_{\mathbb {Z}_2}(V)&\cong K^{n}_{\mathbb {Z}_2}(V) \cong K^n_{\mathbb {Z}_2}(\mathcal {B}_y), \\ K^{n + \widehat{\nu }|_{U \cap V}}_{\mathbb {Z}_2}(U \cap V)&\cong K^{n}_{\mathbb {Z}_2}(U \cap V) \cong K^n_{\mathbb {Z}_2}(\mathcal {B}_y^+ \sqcup \mathcal {B}_y^-). \end{aligned}$$

Notice, however, that there is no (global) trivialisation of $\widehat{\nu }$. At best, we can choose the (local) trivialisations on U, V and $U \cap V$ so that the trivialisations on U and V agree with that on $\mathcal {B}_y^+$ and their discrepancy on $\mathcal {B}_y^-$ is the sign representation that is a group 1-cocycle in $Z^1(\mathbb {Z}_2, U(1))$. This sign representation acts on $K^n_{\mathbb {Z}_2}(\mathcal {B}_y^-)$ as an automorphism and is realised by multiplication by $t \in R(\mathbb {Z}_2)$. Taking the effects of trivialisations into account, we can identify $\Delta$ as the homomorphism

$$\begin{aligned} \Delta&: K^0_{\mathbb {Z}_2}(\mathcal {B}_x) \oplus K^0_{\mathbb {Z}_2}(\mathcal {B}_x) \rightarrow K^0_{\mathbb {Z}_2}(\mathcal {B}_x^+) \oplus K^0_{\mathbb {Z}_2}(\mathcal {B}_x^-),&\Delta (u, v)&= (u - v, u - tv). \end{aligned}$$

With this expression, the Mayer–Vietoris sequence is folded to

$$\begin{aligned} 0 \rightarrow K^{0 + \widehat{\nu }}_{\mathbb {Z}_2}(\mathcal {B}) \rightarrow K^0_{\mathbb {Z}_2}(\mathcal {B}_y) \overset{\delta }{\rightarrow } K^0_{\mathbb {Z}_2}(\mathcal {B}_y) \rightarrow K^{1 + \widehat{\nu }}_{\mathbb {Z}_2}(\mathcal {B}) \rightarrow 0, \end{aligned}$$

in which $\delta (w) = (1 - t)w$. Now, we can readily see that

$$\begin{aligned} K^{0 + \widehat{\nu }}_{\mathbb {Z}_2}(\mathcal {B})&\cong (1 + t), \\ K^{1 + \widehat{\nu }}_{\mathbb {Z}_2}(\mathcal {B})&\cong R(\mathbb {Z}_2)/(1 - t) \oplus (1 - t)/(2 - 2t) \cong (1 + t) \oplus \mathbb {Z}/2, \end{aligned}$$

as claimed. $\square$

1.2 Computation of $K^{\bullet + c + \widehat{\nu }}_{\mathbb {Z}_2}(\mathcal {B}_x)$

As in Sect. 6.2, let $\mathcal {B}_x$ be the circle with trivial involution, $c:{\mathbb {Z}}_2\rightarrow {\mathbb {Z}}_2$ the identity map, and $\widehat{\nu }$ the cocycle (13).

Proposition A.3

We have the following identifications of $R(\mathbb {Z}_2)$-modules

$$\begin{aligned} K^{0 + c + \widehat{\nu }}_{\mathbb {Z}_2}(\mathcal {B}_x)&\cong \mathbb {Z}/2,&K^{1 + c + \widehat{\nu }}_{\mathbb {Z}_2}(\mathcal {B}_x)&=0 \end{aligned}$$

where $\mathbb {Z}/2$ is the unique $R(\mathbb {Z}_2)$-module whose underlying abelian group is $\mathbb {Z}/2$.

Proof

The computation is similar to that in the proof of Proposition A.2: we cover $\mathcal {B}_x$ by two closed intervals U and V so that $U \cap V \cong {\mathrm {pt}} \sqcup {\mathrm {pt}}$, which are invariant with respect to the trivial $\mathbb {Z}_2$-actions. Trivialising the twists $\widehat{\nu }|_U$, $\widehat{\nu }|_V$ and $\widehat{\nu }|_{U \cap V}$, we have

$$\begin{aligned} K^{n + c + \widehat{\nu }|_{U}}_{\mathbb {Z}_2}(U)&\cong K^{n + c}_{\mathbb {Z}_2}(U) \cong K^{n + c}_{\mathbb {Z}_2}({\mathrm {pt}}), \\ K^{n + c + \widehat{\nu }|_{V}}_{\mathbb {Z}_2}(V)&\cong K^{c + n}_{\mathbb {Z}_2}(V) \cong K^{c + n}_{\mathbb {Z}_2}({\mathrm {pt}}), \\ K^{n + c + \widehat{\nu }|_{U \cap V}}_{\mathbb {Z}_2}(U \cap V)&\cong K^{c + n}_{\mathbb {Z}_2}(U \cap V) \cong K^{c + n}_{\mathbb {Z}_2}({\mathrm {pt}}) \oplus K^{c + n}_{\mathbb {Z}_2}({\mathrm {pt}}). \end{aligned}$$

The K-theory $K^{c + n}_{\mathbb {Z}_2}({\mathrm {pt}})$ twisted by the identity homomorphism $c : \mathbb {Z}_2 \rightarrow \mathbb {Z}_2$ is isomorphic to $K^n_{\pm }({\mathrm {pt}})$ in [54], and is known to be [17]

$$\begin{aligned} K^{c + 0}_{\mathbb {Z}_2}({\mathrm {pt}})&= 0,&K^{c + 1}_{\mathbb {Z}_2}({\mathrm {pt}}) \cong \overbrace{(1 - t)}^{\mathbb {Z}}. \end{aligned}$$

Taking care of the choice of the local trivialisations, we can reduce the Mayer–Vietoris exact sequence to

$$\begin{aligned} 0 \rightarrow K^{1 + c + \widehat{\nu }}_{\mathbb {Z}_2}(\mathcal {B}_x) \rightarrow \overbrace{(1 - t)}^{{\mathbb {Z}}} \overset{1 - t}{\rightarrow } \overbrace{(1 - t)}^{{\mathbb {Z}}} \rightarrow K^{0 + c + \widehat{\nu }}_{\mathbb {Z}_2}(\mathcal {B}_x) \rightarrow 0, \end{aligned}$$

from which $K^{n + c + \widehat{\nu }}_{\mathbb {Z}_2}(\mathcal {B}_x)$ is determined immediately. $\square$

B Appendix: Bulk-edge correspondence of integer indices for ${\textsf {pg}}$-symmetric Hamiltonians

There is another natural surjective index map $K^{1 + \widehat{\nu }}_{\mathbb {Z}_2}(\mathcal {B})={\mathbb {Z}}\oplus {\mathbb {Z}}/2\rightarrow K^0(\mathcal {B}_y)\cong {\mathbb {Z}}$ which is most conveniently formulated using $C^*$-algebraic language, as briefly mentioned in Sect. 7.3. Namely, $K^{1 + \widehat{\nu }}_{\mathbb {Z}_2}(\mathcal {B})$ is also the operator algebraic $K_1(C^*({\textsf {pg}}))$. Now ${\textsf {pg}}\cong {\mathbb {Z}}_y\rtimes {\mathbb {Z}}_g$ where we have written ${\mathbb {Z}}_y\cong {\mathbb {Z}}$ and ${\mathbb {Z}}_g\cong {\mathbb {Z}}$ for the subgroups generated by vertical lattice translation and by glide reflection, respectively, and the semidirect product is given by the nontrivial reflection action of ${\mathbb {Z}}_g$ on ${\mathbb {Z}}_y$. We can rewrite

$$\begin{aligned} C^*({\textsf {pg}})\cong C^*({\mathbb {Z}}_y)\rtimes _\alpha {\mathbb {Z}}_g\cong C(\mathcal {B}_y)\rtimes _\alpha {\mathbb {Z}}_g \end{aligned}$$

as a crossed product in which ${\mathbb {Z}}_g$ acts on $C(\mathcal {B}_y)$ by the reflection automorphism $\alpha$ taking $(\alpha \cdot f)(k_y)=f(-k_y)$. The Pimsner–Voiculescu (PV) exact sequence [40] for this crossed product is

If we write $K_1(C(\mathcal {B}_y))\cong {\mathbb {Z}}[u_y]$ with $u_y$ the generating unitary corresponding to vertical translation (i.e. winds around the Fourier transformed space $\mathcal {B}_y$ once), and $K_0(C(\mathcal {B}_y))={\mathbb {Z}}[\mathbf{1}]$ with $\mathbf{1}$ the generating trivial projection given by the identity, then the PV sequence simplifies to

in which $\partial ^{(0)}_{\mathrm{PV}}$ is the zero map while the other ${\mathbb {Z}}$-valued connecting “index” map

$$\begin{aligned} \partial ^{(1)}_{\mathrm{PV}}:K^{1 + \widehat{\nu }}_{\mathbb {Z}_2}(\mathcal {B})=K_1(C^*({\textsf {pg}}))\rightarrow K_0(C(\mathcal {B}_y))=K^0(\mathcal {B}_y) \end{aligned}$$

is surjective. The free generator of $K_1(C^*({\textsf {pg}}))\cong {\mathbb {Z}}\oplus {\mathbb {Z}}/2$ can be taken to be the unitary corresponding to the generating glide reflection, i.e. the $[U_\mathrm{r}]$ of Sect. 3.2.1 specifying the Hamiltonian $H_\mathrm{r}$ (this has winding number $-1$ around $\mathcal {B}_x$).

Thus the index map $\partial ^{(1)}_{\mathrm{PV}}$ counts the winding around $\mathcal {B}_x$, and in our tight-binding model, we see from Fig. 6 that it has the analytic interpretation of counting the number of black zero modes (per unit cell) left behind after truncating to the half-space on the right side of a vertical edge. The PV connecting homomorphism, where available, is an important ingredient in the formulation of bulk-edge correspondences of weak phases via generalised Connes–Chern character formulae, as studied in [42]. From that point of view, $[U_\mathrm{r}]$ is a “weak” topological phase, but as the Klein phase in our ${\textsf {pg}}$ example shows, the notions of “weak” and “strong” topological insulators are somewhat blurred in the presence of crystallographic symmetries.