Quantization of fractional corner charge in ${C}_{n}$-symmetric higher-order topological crystalline insulators (2024)

Physical Review B

covering condensed matter and materials physics

Highlights
Recent
Accepted
Collections
Authors
Referees
Search
Press
About
Editorial Team

Editors' Suggestion

Quantization of fractional corner charge in $C_{n}$ -symmetric higher-order topological crystalline insulators

Wladimir A. Benalcazar, Tianhe Li, and Taylor L. Hughes

Phys. Rev. B 99, 245151 – Published 26 June 2019

Article
References
Citing Articles (477)
Supplemental Material

PDFHTMLExport Citation

$Quantization of fractional corner charge in ${C}_{n}$-symmetric higher-order topological crystalline insulators (1)$

Abstract

Authors

Article Text

CLASSIFICATION

PRIMITIVE GENERATORS

FILLING ANOMALY AND CHARGE…

CONSTRUCTION OF THE TOPOLOGICAL INDICES…

FRACTIONAL CORNER CHARGE IN TCIS WITHOUT…

FRACTIONAL CHARGE AT TOPOLOGICAL DEFECTS

DISCUSSION AND CONCLUSION

ACKNOWLEDGMENTS

Supplemental Material

References

$Quantization of fractional corner charge in ${C}_{n}$-symmetric higher-order topological crystalline insulators (2)$

Abstract

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.

$Quantization of fractional corner charge in ${C}_{n}$-symmetric higher-order topological crystalline insulators (3)$
$Quantization of fractional corner charge in ${C}_{n}$-symmetric higher-order topological crystalline insulators (4)$
$Quantization of fractional corner charge in ${C}_{n}$-symmetric higher-order topological crystalline insulators (5)$
$Quantization of fractional corner charge in ${C}_{n}$-symmetric higher-order topological crystalline insulators (6)$
$Quantization of fractional corner charge in ${C}_{n}$-symmetric higher-order topological crystalline insulators (7)$
$Quantization of fractional corner charge in ${C}_{n}$-symmetric higher-order topological crystalline insulators (8)$
$Quantization of fractional corner charge in ${C}_{n}$-symmetric higher-order topological crystalline insulators (9)$

Received 30 October 2018
Revised 6 June 2019

DOI:https://doi.org/10.1103/PhysRevB.99.245151

Physics Subject Headings (PhySH)

Research Areas

ChargeDefectsEdge statesElectric polarizationElectronic structureFractionalizationSymmetry protected topological statesTopological materials

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Wladimir A. Benalcazar^1,2,*, Tianhe Li², and Taylor L. Hughes²

¹Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA
²Department of Physics and Institute for Condensed Matter Theory, University of Illinois at Urbana-Champaign, Illinois 61801, USA

^*wladimir.benalcazar@psu.edu

Article Text (Subscription Required)

Click to Expand

Supplemental Material (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 99, Iss. 24 — 15 June 2019

$Quantization of fractional corner charge in ${C}_{n}$-symmetric higher-order topological crystalline insulators (10)$

Reuse & Permissions

Access Options

Buy Article »
Log in with individual APS Journal Account »
Log in with a username/password provided by your institution »
Get access through a U.S. public or high school library »

$Quantization of fractional corner charge in ${C}_{n}$-symmetric higher-order topological crystalline insulators (11)$

Article Available via CHORUS

Download Accepted Manuscript

$Quantization of fractional corner charge in ${C}_{n}$-symmetric higher-order topological crystalline insulators (14)$

Authorization Required

Other Options

Buy Article »
Find an Institution with the Article »

Download & Share

PDFExportReuse & PermissionsCiting Articles (477)

Images

$Quantization of fractional corner charge in ${C}_{n}$-symmetric higher-order topological crystalline insulators (15)$
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 $\frac{3 | e |}{4}$ , (b)a $C_{4}$ -symmetric TCI with corner charge $\frac{| e |}{2}$ , (c)a $C_{6}$ -symmetric TCI with corner charge $\frac{| e |}{6}$ , and (d)a $C_{3}$ -symmetric TCI with corner charge $\frac{| e |}{3}$ . 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.
Reuse & Permissions
$Quantization of fractional corner charge in ${C}_{n}$-symmetric higher-order topological crystalline insulators (16)$
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).
Reuse & Permissions
$Quantization of fractional corner charge in ${C}_{n}$-symmetric higher-order topological crystalline insulators (17)$
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.
Reuse & Permissions
$Quantization of fractional corner charge in ${C}_{n}$-symmetric higher-order topological crystalline insulators (18)$
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.
Reuse & Permissions
$Quantization of fractional corner charge in ${C}_{n}$-symmetric higher-order topological crystalline insulators (19)$
Figure 5
Filling anomaly in the $C_{4}$ -symmetric insulator of Eq.(9). (a)A unit cell with charge $3 | e |$ at position $1 a$ and three electrons with Wannier centers at positions $b$ (red circle) and $c, c^{'}$ (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.
Reuse & Permissions
$Quantization of fractional corner charge in ${C}_{n}$-symmetric higher-order topological crystalline insulators (20)$
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 $c^{'}$ . (c)Two electrons at positions $b$ and $b^{'}$ . (d)Three electrons at positions $c, c^{'}$ , 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 $e)$ .
Reuse & Permissions
$Quantization of fractional corner charge in ${C}_{n}$-symmetric higher-order topological crystalline insulators (21)$
Figure 7
Quantized fractionalization of charge at the core of disclinations. (a)Disclination in the lattice of primitive generator $h_{3 c}^{(6)}$ . (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 $\frac{| e |}{2}$ . 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.
Reuse & Permissions