Quantization of fractional corner charge in $C_n$-symmetric higher-order topological crystalline insulators

In the presence of crystalline symmetries, certain topological insulators present a filling anomaly: a mismatch between the number of electrons in an energy band and the number of electrons required for charge neutrality. In this paper, we show that a filling anomaly can arise when corners are introduced in $C_n$-symmetric crystalline insulators with vanishing polarization, having as a consequence the existence of corner-localized charges quantized in multiples of $\frac{e}{n}$. We characterize the existence of this charge systematically and build topological indices that relate the symmetry representations of the occupied energy bands of a crystal to the quanta of fractional charge robustly localized at its corners. When an additional chiral symmetry is present, $\frac{e}{2}$ corner charges are accompanied by zero-energy corner-localized states. We show the application of our indices in a number of atomic and fragile topological insulators and discuss the role of fractional charges bound to disclinations as bulk probes for these crystalline phases.

Figure 1

Quantized fractional corner charge in $C_n$-symmetric TCIs. The plots show the total (electronic and ionic) charge density of two-dimensional TCIs. (a) a $C_4$-symmetric TCI with corner charge $\frac{3\left|e\right|}{4}$, (b) a $C_4$-symmetric TCI with corner charge $\frac{\left|e\right|}{2}$, (c) a $C_6$-symmetric TCI with corner charge $\frac{\left|e\right|}{6}$, and (d) a $C_3$-symmetric TCI with corner charge $\frac{\left|e\right|}{3}$. In all cases, the bulk and edges are neutral. These charge patterns are obtained by stacking the primitive generator models as described in Sec. 5.

Figure 2

(a), (b) Maximal Wyckoff positions for (a) $C_4$- and (b) $C_2$-symmetric unit cells. (c)–(e) Lattices for the three primitive generators that span the classification of $C_4$-symmetric TCIs. The lattices for the primitive generators for the classification of $C_2$-symmetric TCIs are those in (c), (e), and (f).

Figure 3

(a), (b) Maximal Wyckoff positions for (a) $C_6$- and (b) $C_3$-symmetric unit cells. (c), (d) Primitive generators that span the classification of $C_6$-symmetric TCIs. (e), (f) Primitive generators for the classification of $C_3$-symmetric TCIs.

Figure 4

Filling anomaly in the reflection symmetric Su-Schrieffer-Hegger model with the Bloch Hamiltonian of Eq. (5) and open boundaries. (a) Trivial atomic limit. Charges are balanced. (b) Obstructed atomic limit. Positive and negative charges are unbalanced. For $N$ positive ions, there are $N-1$ electrons (left) or $N+1$ electrons (right). Solid (dimmer) circles represent bulk (boundary) Wannier centers.

Figure 5

Filling anomaly in the $C_4$-symmetric insulator of Eq. (9). (a) A unit cell with charge $3\left|e\right|$ at position $1a$ and three electrons with Wannier centers at positions $b$ (red circle) and $c,c^{\prime }$ (blue circles). (b) A $4\times 4$ lattice formed by tiling the unit cell shown in (a) along $x$ and $y$. The configuration is neutral but breaks $C_4$ symmetry. (c) A deformation of (b) as an attempt to restore $C_4$ symmetry along the edges; symmetry is still broken at corners. (d),(e) Two choices that restore full $C_4$ symmetry in the lattice by either removing three corner electrons (d) or adding one (e); in either case, charge neutrality is lost.

Figure 6

Edge and corner fractional charges for TCIs with Wannier centers at maximal Wyckoff positions for (a), (b) $C_4$-symmetric, (c), (d) $C_6$-symmetric, and (e), (f) $C_3$-symmetric lattices. (a) One electron at position $b$ of each unit cell. (b) Two electrons at positions $c$ and $c^{\prime }$. (c) Two electrons at positions $b$ and $b^{\prime }$. (d) Three electrons at positions $c,c^{\prime }$, and $c^{″}$. (e) One electron at position $b$. (f) One electron at position $c$. Solid colored circles represent bulk electrons; dimmed colored circles represent boundary electrons for a particular choice of $C_n$-symmetry breaking; white circles represent atomic ions. Bulk unit cells are always neutral. Electronic charges at edge and corner unit cells after the removal of the symmetry breaking electrons are indicated mod 1 (in units of the electron charge $e)$.

Figure 7

Quantized fractionalization of charge at the core of disclinations. (a) Disclination in the lattice of primitive generator $h_{3c}^{\left(6\right)}$. (b) Wannier centers for the lattice in (a). There is an overall fractional electronic charge (each hollow circle contributes $\frac{e}{2}$ charge) within the region of darker unit cells which enclose the core of the disclination. (c) Charge density for the disclination in (a). All corners and the core of the disclination have charges of $\frac{\left|e\right|}{2}$. The simulation is done over 276 unit cells with added intra-unit cell hoppings between nearest neighbors of $\frac{1}{4}$ the amplitude of the interunit cell hoppings.