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Electric multipole moments, topological multipole moment pumping, and chiral hinge states in crystalline insulators

Wladimir A. Benalcazar, B. Andrei Bernevig, and Taylor L. Hughes
Phys. Rev. B 96, 245115 – Published 11 December 2017
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Abstract

We extend the theory of dipole moments in crystalline insulators to higher multipole moments. As first formulated in Benalcazar et al. [Science 357, 61 (2017)], we show that bulk quadrupole and octupole moments can be realized in crystalline insulators. In this paper, we expand in great detail the theory presented previously [Benalcazar et al., Science 357, 61 (2017)] and extend it to cover associated topological pumping phenomena, and a class of three-dimensional (3D) insulator with chiral hinge states. We start by deriving the boundary properties of continuous classical dielectrics hosting only bulk dipole, quadrupole, or octupole moments. In quantum mechanical crystalline insulators, these higher multipole bulk moments manifest themselves by the presence of boundary-localized moments of lower dimension, in exact correspondence with the electromagnetic theory of classical continuous dielectrics. In the presence of certain symmetries, these moments are quantized, and their boundary signatures are fractionalized. These multipole moments then correspond to new symmetry-protected topological phases. The topological structure of these phases is described by “nested” Wilson loops, which we define. These Wilson loops reflect the bulk-boundary correspondence in a way that makes evident a hierarchical classification of the multipole moments. Just as a varying dipole generates charge pumping, a varying quadrupole generates dipole pumping, and a varying octupole generates quadrupole pumping. For nontrivial adiabatic cycles, the transport of these moments is quantized. An analysis of these interconnected phenomena leads to the conclusion that a new kind of Chern-type insulator exists, which has chiral, hinge-localized modes in 3D. We provide the minimal models for the quantized multipole moments, the nontrivial pumping processes, and the hinge Chern insulator, and describe the topological invariants that protect them.

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  • Received 28 August 2017

DOI:https://doi.org/10.1103/PhysRevB.96.245115

©2017 American Physical Society

Physics Subject Headings (PhySH)

Condensed Matter, Materials & Applied Physics

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This article appears in the following collection:

Physical Review B 50th Anniversary Milestones

These Milestone studies represent lasting contributions to physics by way of reporting significant discoveries, initiating new areas of research, or substantially enhancing the conceptual tools for making progress in the burgeoning field of condensed matter physics.

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Topological Insulators Turn a Corner

Published 11 December 2017

Theorists have discovered topological insulators that are insulating in their interior and on their surfaces but have conducting channels at corners or along edges.

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Authors & Affiliations

Wladimir A. Benalcazar1, B. Andrei Bernevig2,*, and Taylor L. Hughes1,†

  • 1Department of Physics and Institute for Condensed Matter Theory, University of Illinois at Urbana-Champaign, Illinois 61801, USA
  • 2Department of Physics, Princeton University, Princeton, New Jersey 08544, USA

  • *On leave from Donostia International Physics Center, P. Manuel de Lardizabal 4, 20018 Donostia-San Sebastián, Spain; Laboratoire Pierre Aigrain, École Normale Supérieure-PSL Research University, CNRS, Université Pierre et Marie Curie-Sorbonne Universités, Université Paris Diderot-Sorbonne Paris Cité, 24 rue Lhomond, 75231 Paris Cedex 05, France; and Sorbonne Universités, UPMC Univ Paris 06, UMR 7589, LPTHE, F-75005, Paris, France.
  • To whom correspondence should be addressed: hughest@illinois.edu

See Also

Reflection-Symmetric Second-Order Topological Insulators and Superconductors

Josias Langbehn, Yang Peng, Luka Trifunovic, Felix von Oppen, and Piet W. Brouwer
Phys. Rev. Lett. 119, 246401 (2017)

(d2)-Dimensional Edge States of Rotation Symmetry Protected Topological States

Zhida Song, Zhong Fang, and Chen Fang
Phys. Rev. Lett. 119, 246402 (2017)

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Vol. 96, Iss. 24 — 15 December 2017

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