Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1812.06999 · Physical review. B./Physical review. B

Many-body order parameters for multipoles in solids

Byungmin Kang, Ken Shiozaki, Gil Young Cho

Published Dec 20, 2019• 142 citations
arXivHTMLPDF

Abstract

We propose many-body order parameters for bulk multipoles in crystalline systems, whenever the bulk multipole moments can be defined in terms of the nontrivial topology of the nested Wilson loop. The many-body order parameters are designed to measure multipolar charge distribution in a crystalline unit cell, and they match the localized corner charge originating from the multipoles when the symmetries quantizing the multipoles are protected. Our many-body order parameters provide a complementary view to the nested Wilson loop approaches and even go beyond, as our many-body order parameters are readily applicable to interacting quantum many-body systems. Furthermore, even when the symmetries quantizing multipoles are lost so that the nested Wilson loop spectrum does not exactly reproduce the physical multipole moments, our many-body order parameters faithfully measure the physical multipole moments, which is confirmed numerically during Thouless pumping processes. We provide analytic arguments and numerical demonstration of the order parameters for various higher-order insulators having bulk multipole moments. Finally, we discuss the applicability of our many-body order parameters in the cases where the Wannier gap is absent so that the bulk multiple moments cannot be defined in terms of the nested Wilson loop.

DOI
10.1103/physrevb.100.245134

Related papers

Many-body multipole index and bulk-boundary correspondence

Yasuhiro Tada, Masaki Oshikawa

We propose dipole and quadrupole indices for interacting insulators with point group symmetries. The proposed indices are defined in terms of many-body quantum multipole operators combined with the generator of the point group symmetry. Unlike the original multipole operators, these combined operators commute with the Hamiltonian under symmetry and therefore their eigenvalues are quantized. This enables a clear identification of nontrivial multipolar states. We calculate the multipole indices in representative models and show their effectiveness as order parameters. Furthermore, we demonstrate a bulk-boundary correspondence: a nonzero index implies the existence of edge/corner states under the point group symmetry.

arXiv:2302.00800•Physical review. B./Physical review. B•Dec 15, 2023•7 citations

Electric multipole moments and higher-order topological phases in crystalline insulators and superconductors

Wladimir A. Benalcazar

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

Many-body higher-order topological invariant for Cn -symmetric insulators

Ammar Jahin, Yuan-Ming Lu, Yuxuan Wang

Higher-order topological insulators in two spatial dimensions display fractional corner charges. While fractional charges in one dimension are known to be captured by a many-body bulk invariant, computed by the Resta formula, a many-body bulk invariant for higher-order topology and the corresponding fractional corner charges remains elusive despite several attempts. Inspired by recent work by Tada and Oshikawa [arXiv:2302.00800], we propose a well-defined, many-body bulk invariant for ${C}_{n}$-symmetric higher-order topological insulators that is valid for both noninteracting and interacting systems. Instead of relating them to the bulk quadrupole moment as was previously done, we show that in the presence of ${C}_{n}$ rotational symmetry, this bulk invariant can be directly identified with quantized fractional corner charges. In particular, we prove that the corner charge is quantized as $e/n$ with ${C}_{n}$ symmetry, leading to a ${\mathbb{Z}}_{n}$ classification for higher-order topological insulators in two dimensions.

Physical review. B./Physical review. B•May 8, 2024•6 citations

Measuring higher-order bulk topology in a programmable quantum simulator

Pratik Sathe, Niclas Heinsdorf, Majid Kheirkhah +13 more

Topological insulators lie outside the standard Landau paradigm: Instead of being characterized by local order parameters and spontaneous symmetry breaking, their defining properties are encoded in the global structure of their quantum states and manifested as robust boundary states. Crystalline higher-order topological insulators can extend this principle by hosting localized corner modes in two and higher dimensions. Although these boundary modes can be detected spectroscopically through the local density of states, determining the underlying bulk topology, for example through nested Wilson loops, remains an outstanding experimental challenge. Here we use up to 728 qubits in a superconducting analogue-digital quantum simulator to realize and probe a hierarchy of one-, two-, and three-dimensional topological insulators governed by effective tight-binding Hamiltonians. By reconstructing complex propagators from real-space measurements, we obtain complementary boundary and bulk characterizations: local densities of states reveal localized boundary modes, whereas experimentally reconstructed band projectors yield bulk topological invariants, culminating in a three-dimensional octupole moment extracted through third nested Wilson loops. These results establish a route to realizing, characterizing, and manipulating topologically protected states in programmable three-dimensional quantum matter.

quant-pharXiv:2610.16046•Oct 13, 2026•0 citations

Many-body higher-order topological invariant for $C_n$-symmetric insulators

Ammar Jahin, Yuan-Ming Lu, Yuxuan Wang

Higher-order topological insulators in two spatial dimensions display fractional corner charges. While fractional charges in one dimension are known to be captured by a many-body bulk invariant, computed by the Resta formula, a many-body bulk invariant for higher-order topology and the corresponding fractional corner charges remains elusive despite several attempts. Inspired by recent work by Tada and Oshikawa, we propose a well-defined many-body bulk invariant for $C_n$ symmetric higher-order topological insulators, which is valid for both non-interacting and interacting systems. Instead of relating them to the bulk quadrupole moment as was previously done, we show that in the presence of $C_n$ rotational symmetry, this bulk invariant can be directly identified with quantized fractional corner charges. In particular, we prove that the corner charge is quantized as $e/n$ with $C_n$ symmetry, leading to a $\mathbb{Z}_n$ classification for higher-order topological insulators in two dimensions.

arXiv:2401.00050•Dec 29, 2023•0 citations

Quantized electric multipole insulators

Wladimir A. Benalcazar, Bogdan Andrei Bernevig, Taylor L. Hughes

The Berry phase provides a modern formulation of electric polarization in crystals. We extend this concept to higher electric multipole moments and determine the necessary conditions and minimal models for which the quadrupole and octupole moments are topologically quantized electromagnetic observables. Such systems exhibit gapped boundaries that are themselves lower-dimensional topological phases. Furthermore, they host topologically protected corner states carrying fractional charge, exhibiting fractionalization at the boundary of the boundary. To characterize these insulating phases of matter, we introduce a paradigm in which "nested" Wilson loops give rise to topological invariants that have been overlooked. We propose three realistic experimental implementations of this topological behavior that can be immediately tested. Our work opens a venue for the expansion of the classification of topological phases of matter.

arXiv:1611.07987•Science•Jul 7, 2017•2,031 citations
We propose many-body order parameters for bulk multipoles in crystalline systems, whenever the bulk multipole moments can be defined in terms of the nontrivial topology of the nested Wilson loop. The many-body order parameters are designed to measure multipolar charge distribution in a crystalline unit cell, and they match the localized corner charge originating from the multipoles when the symmetries quantizing the multipoles are protected. Our many-body order parameters provide a complementary view to the nested Wilson loop approaches and even go beyond, as our many-body order parameters are readily applicable to interacting quantum many-body systems. Furthermore, even when the symmetries quantizing multipoles are lost so that the nested Wilson loop spectrum does not exactly reproduce the physical multipole moments, our many-body order parameters faithfully measure the physical multipole moments, which is confirmed numerically during Thouless pumping processes. We provide analytic arguments and numerical demonstration of the order parameters for various higher-order insulators having bulk multipole moments. Finally, we discuss the applicability of our many-body order parameters in the cases where the Wannier gap is absent so that the bulk multiple moments cannot be defined in terms of the nested Wilson loop.
We propose dipole and quadrupole indices for interacting insulators with point group symmetries. The proposed indices are defined in terms of many-body quantum multipole operators combined with the generator of the point group symmetry. Unlike the original multipole operators, these combined operators commute with the Hamiltonian under symmetry and therefore their eigenvalues are quantized. This enables a clear identification of nontrivial multipolar states. We calculate the multipole indices in representative models and show their effectiveness as order parameters. Furthermore, we demonstrate a bulk-boundary correspondence: a nonzero index implies the existence of edge/corner states under the point group symmetry.
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.
Higher-order topological insulators in two spatial dimensions display fractional corner charges. While fractional charges in one dimension are known to be captured by a many-body bulk invariant, computed by the Resta formula, a many-body bulk invariant for higher-order topology and the corresponding fractional corner charges remains elusive despite several attempts. Inspired by recent work by Tada and Oshikawa [arXiv:2302.00800], we propose a well-defined, many-body bulk invariant for Cn{C}_{n}Cn​-symmetric higher-order topological insulators that is valid for both noninteracting and interacting systems. Instead of relating them to the bulk quadrupole moment as was previously done, we show that in the presence of Cn{C}_{n}Cn​ rotational symmetry, this bulk invariant can be directly identified with quantized fractional corner charges. In particular, we prove that the corner charge is quantized as e/ne/ne/n with Cn{C}_{n}Cn​ symmetry, leading to a Zn{\mathbb{Z}}_{n}Zn​ classification for higher-order topological insulators in two dimensions.
Topological insulators lie outside the standard Landau paradigm: Instead of being characterized by local order parameters and spontaneous symmetry breaking, their defining properties are encoded in the global structure of their quantum states and manifested as robust boundary states. Crystalline higher-order topological insulators can extend this principle by hosting localized corner modes in two and higher dimensions. Although these boundary modes can be detected spectroscopically through the local density of states, determining the underlying bulk topology, for example through nested Wilson loops, remains an outstanding experimental challenge. Here we use up to 728 qubits in a superconducting analogue-digital quantum simulator to realize and probe a hierarchy of one-, two-, and three-dimensional topological insulators governed by effective tight-binding Hamiltonians. By reconstructing complex propagators from real-space measurements, we obtain complementary boundary and bulk characterizations: local densities of states reveal localized boundary modes, whereas experimentally reconstructed band projectors yield bulk topological invariants, culminating in a three-dimensional octupole moment extracted through third nested Wilson loops. These results establish a route to realizing, characterizing, and manipulating topologically protected states in programmable three-dimensional quantum matter.
Many-body higher-order topological invariant for CnC_nCn​-symmetric insulators
Higher-order topological insulators in two spatial dimensions display fractional corner charges. While fractional charges in one dimension are known to be captured by a many-body bulk invariant, computed by the Resta formula, a many-body bulk invariant for higher-order topology and the corresponding fractional corner charges remains elusive despite several attempts. Inspired by recent work by Tada and Oshikawa, we propose a well-defined many-body bulk invariant for CnC_nCn​ symmetric higher-order topological insulators, which is valid for both non-interacting and interacting systems. Instead of relating them to the bulk quadrupole moment as was previously done, we show that in the presence of CnC_nCn​ rotational symmetry, this bulk invariant can be directly identified with quantized fractional corner charges. In particular, we prove that the corner charge is quantized as e/ne/ne/n with CnC_nCn​ symmetry, leading to a Zn\mathbb{Z}_nZn​ classification for higher-order topological insulators in two dimensions.
The Berry phase provides a modern formulation of electric polarization in crystals. We extend this concept to higher electric multipole moments and determine the necessary conditions and minimal models for which the quadrupole and octupole moments are topologically quantized electromagnetic observables. Such systems exhibit gapped boundaries that are themselves lower-dimensional topological phases. Furthermore, they host topologically protected corner states carrying fractional charge, exhibiting fractionalization at the boundary of the boundary. To characterize these insulating phases of matter, we introduce a paradigm in which "nested" Wilson loops give rise to topological invariants that have been overlooked. We propose three realistic experimental implementations of this topological behavior that can be immediately tested. Our work opens a venue for the expansion of the classification of topological phases of matter.