Bulk-and-edge to corner correspondence

We show that two-dimensional band insulators, with vanishing bulk polarization, obey bulk-and-edge to corner charge correspondence, stating that the knowledge of the bulk and the two corresponding ribbon band structures uniquely determines a fractional part of the corner charge irrespective of the corner termination. Moreover, physical observables related to macroscopic charge density of a terminated crystal can be obtained by representing the crystal as collection of polarized edge regions with polarizations ${\vec{P}}_{\alpha }^{\text{edge}}$, where the integer $\alpha$ enumerates the edges. We introduce a particular manner of cutting a crystal, dubbed “Wannier cut,” which allows us to compute ${\vec{P}}_{\alpha }^{\text{edge}}$. We find that ${\vec{P}}_{\alpha }^{\text{edge}}$ consists of two pieces: the bulk piece expressed via quadrupole tensor of the bulk Wannier functions' charge density and the edge piece corresponding to the Wannier edge polarization—the polarization of the edge subsystem obtained by Wannier cut. For a crystal with $n$ edges, out of $2n$ independent components of ${\vec{P}}_{\alpha }^{\text{edge}}$, only $2n-1$ are independent of the choice of Wannier cut and correspond to physical observables: corner charges and edge dipoles.

Figure 1

Illustration of a crystal with boundary. To obtain a fractional part of each of the five corners (hatched regions), the bulk and the five edge terminations (dotted regions) need to be specified, while the termination around the corners need not be specified. The edges run along the lattice vectors ${\vec{e}}_{\alpha }$, with the unit normal vectors ${\vec{n}}_{\alpha }$ pointing outward, $\alpha =1,\cdots ,5$. The corner charge, where the edges along ${\vec{e}}_{\alpha }$ and ${\vec{e}}_{\alpha +1}$ meet, is denoted by $Q_{\alpha }^{\text{c}}$. The macroscopic charge density of a crystal can be seen to be generated by a collection of polarized edge regions (dotted) with polarizations ${\vec{P}}_{\alpha }^{\text{edge}}$. The charge neutrality and vanishing bulk polarization are assumed; hence, ${\sum }_{\alpha }{Q}_{\alpha }^{\text{c}}=0$, whereas ${\sum }_{\alpha }{\vec{P}}_{\alpha }^{\text{edge}}$ need not vanish.

Figure 2

The macroscopic charge density ${\rho }^{\text{macro}}\left(\vec{r}\right)$ for $\vec{r}$ in the bulk, the two edges, and the corner regions. For clarity, the crystal is divided into cells (containing many unit cells) and the total charge in each cell is shown. The corner is defined by the two edges with the unit normal vectors ${\vec{n}}_{\alpha }$, $\alpha =1,2$. When either $D_1^{\text{edge}}$ or $D_2^{\text{edge}}$ are nonvanishing, integrating ${\rho }^{\text{macro}}$ over different corner regions ${\mathcal{R}}^{\text{c}}$, denoted by dashed lines and the hatched region, results in different corner charge.

Figure 3

A ribbon infinite in ${\vec{e}}_1$ direction with $N_2$ unit cells in ${\vec{e}}_2$ direction (a). Nonzero polarization $\vec{P}$ implies end charge $Q^{\text{e}}$ once the ribbon is terminated (b). The end charge $Q^{\text{e}}$, defined as the charge contained in the region bounded by the dashed lines, can be computed by multiplying the charge density $\rho$ in Eq. (9) by continuous ramp function $f_1\left(r_1\right)$, which takes values 0 (dotted region), $r_1-r_{01}$ (hatched region), and 1 (filled region) (b), followed by integration over the whole space. The region where the ramp function takes nonzero values is denoted by dashed line $(\vec{r}-{\vec{r}}_0)·\vec{n}=0$, which crosses the ribbon far from ends.

Figure 4

A charge neutral flake with total bulk polarization zero. The flake is semi-infinite in both directions. Two cuts (wavy lines) along a charge neutral lines $(\vec{r}-{\vec{r}}_0)·{\vec{n}}_{\alpha }=0$, $\alpha =1,2$, split the flake into four subsystems: the two edges (dotted regions), the corner (hatched region), and the remaining bulk subsystem. A fractional part of the upper-right corner charge is given by Eq. (20). If the cuts are chosen to be Wannier cuts, the resulting polarization of the edge subsystem is called Wanner edge polarization ${\vec{\mathfrak{P}}}_{\alpha }^{\text{edge}}$. The corner region ${\mathcal{R}}^{\text{c}}$ is enclosed by dashed line. The end charge of the two edge subsystems is denoted by $Q^{\text{e1}}$ and $Q^{\text{e2}}$.

Figure 5

A flake with the regions $\square$ (dotted region with a thick solid line) and ${\mathcal{R}}^{\text{c}}$ (hatched region with dashed line). The bulk subsystem, located away from the edges (thin solid lines), is obtained by tiling $\square$ with the occupied bulk WFs (wavy star-shaped objects). The integer $R_{\text{WF}}$ is the “radius” of the bulk WFs; outside of it the charge density of the bulk WFs can be neglected.

Figure 6

A corner defined by the edges along lattice vectors ${\vec{e}}_{1,2}$ and unit-normal vectors ${\vec{n}}_{1,2}$. The corner region ${\mathcal{R}}^{\text{c}}$ is defined by the (dashed) lines $(\vec{r}-{\vec{r}}_0)·{\vec{n}}_{\alpha }^{\prime }=0$, where the point ${\vec{r}}_0$ lies in the bulk.

Figure 7

The ribbon corresponding to the BBH model (41) with theorist boundary conditions in the $y$ direction. The bulk Wannier functions can be chosen to be localized on four sites and are represented by the squares (a). The projector ${\mathcal{P}}_{k_x}^{L=1}$ is obtain by removing the bulk Wannier functions belonging to the unit cell at $R_2=0$ (b); see Eq. (34). Dashed circles group the sites that belong to the same unit cell and integers correspond to $\gamma$.

Figure 8

Three different corners for the system described by the Hamiltonian (50) with the corresponding corner regions (dashed lines). The corner between the edges along the primitive vectors ${\vec{a}}_1$ and ${\vec{a}}_2$, with the corner region parallel to the edges (a). The corner defined by the lattice vectors ${\vec{e}}_1=2{\vec{a}}_1+{\vec{a}}_2$ and ${\vec{a}}_2$ [(b), (c)]. Choosing the corner region above the line along ${\vec{a}}_1$ (b) results in a different corner charge compared to the corner region parallel to the edges (c), i.e., $Q_b^{\text{c}}\ne Q_c^{\text{c}}$. The same as in panels (b) and (c) for the corner between the edges along ${\vec{e}}_1^{\prime }=2{\vec{a}}_1-{\vec{a}}_2$ and ${\vec{a}}_2$ is shown in panels (d) and (e). The unit normal vectors ${\vec{n}}_1$, ${\vec{n}}_2$ [$-{\vec{n}}_2$ for panel (e)], $\vec{n}$ and ${\vec{n}}^{\prime }$ are oriented to point toward the corner.