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Bulletin of the American Physical Society

Many-Body Invaraints for Electric Multipoles in Higher-order Topological Insulators

Byungmin Kang, Hyunwoong Kwon, Kwon Park, Gil Young Cho

Published Mar 4, 2019• 0 citations
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Electric multipole moments and higher-order topological phases in crystalline insulators and superconductors

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This dissertation discusses extensions of the theory of polarization in crystalline insulators. The notion that electronic wave functions of energy bands in crystalline insulators have Berry phases associated with the positions of electronic charge is at the basis of our understanding of topological band theory. We show, by using the Berry phase formulation and generalizations of it, as well as by exact diagonalization of tight-binding models, that crystalline insulators can generate charge and polarization patterns compatible with the notion of quadrupole and octupole electric moments. In the presence of certain symmetries, these moments are quantized, and their boundary signatures fractionalized. These higher multipole moments then correspond to new symmetry-protected topological phases (SPTs). A salient signature of these SPTs is the existence of zero-dimensional, corner-localized mid-gap energy states. We then study topological crystalline superconductors, in which these zero-dimensional mid-gap states amount to the existence of Majorana bound states (MBS). We thoroughly classify these systems when rotational group symmetries are enforced, and investigate the existence of zero-dimensional MBS trapped at topological defects in these crystals. The finding of these higher multipole moments or generalizations of them to other platforms constitute a new paradigm for the realization of symmetry protected topological phases.

•Jul 6, 2018•0 citations
The concepts of multipolar insulators and multipole symmetries have hinted at tantalizing connections between the disparate fields of higher-order topological insulators and fracton phases. Here, the authors show that these connections are not superficial, but instead arise from a unified field theory description of the topological response of these systems to external fields, similar to the quantized Hall conductance of Chern insulators, and the quantized axion magnetoelectric polarizability in 3D topological insulators. These response theories are built from the recently introduced higher-rank tensor gauge potentials, and can be interpreted, in some scenarios, as the topological response to ordinary electric and magnetic field gradients.
This dissertation discusses extensions of the theory of polarization in crystalline insulators. The notion that electronic wave functions of energy bands in crystalline insulators have Berry phases associated with the positions of electronic charge is at the basis of our understanding of topological band theory. We show, by using the Berry phase formulation and generalizations of it, as well as by exact diagonalization of tight-binding models, that crystalline insulators can generate charge and polarization patterns compatible with the notion of quadrupole and octupole electric moments. In the presence of certain symmetries, these moments are quantized, and their boundary signatures fractionalized. These higher multipole moments then correspond to new symmetry-protected topological phases (SPTs). A salient signature of these SPTs is the existence of zero-dimensional, corner-localized mid-gap energy states. We then study topological crystalline superconductors, in which these zero-dimensional mid-gap states amount to the existence of Majorana bound states (MBS). We thoroughly classify these systems when rotational group symmetries are enforced, and investigate the existence of zero-dimensional MBS trapped at topological defects in these crystals. The finding of these higher multipole moments or generalizations of them to other platforms constitute a new paradigm for the realization of symmetry protected topological phases.