Electric Multipole Moments, Topological Multipole Moment Pumping, and Chiral Hinge States in Crystalline Insulators

TL;DR

This paper generalizes the theory of electric multipole moments in crystalline insulators to include topological pumping phenomena and introduces a new class of 3D insulators with chiral hinge states, expanding the understanding of topological phases.

Contribution

It develops a comprehensive theory of higher multipole moments, topological pumping, and chiral hinge states in crystalline insulators, with minimal models and topological invariants.

Findings

Higher multipole moments manifest as boundary-localized lower-dimensional moments.

Quantized multipole moments lead to hierarchical topological phases.

A new 3D Chern-type insulator with chiral hinge states is identified.

Abstract

We extend the theory of dipole moments in crystalline insulators to higher multipole moments. In this paper, we expand in great detail the theory presented in Ref. 1, and extend it to cover associated topological pumping phenomena, and a novel class of 3D insulator with chiral hinge states. In quantum-mechanical crystalline insulators, 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 SPT phases. The topological structure of these phases is described by "nested" Wilson loops, which 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 non-trivial 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 non-trivial pumping processes and the hinge Chern insulator, and describe the topological invariants that protect them.