pith. sign in

arxiv: 2601.06963 · v2 · submitted 2026-01-11 · ❄️ cond-mat.mes-hall

Tunable cornerlike states in topological type-II hyperbolic lattices

Zheng-Rong Liu , Tan Peng , Xiang Liu , Xiao-Xia Yi , Chun-Bo Hua , Rui Chen , Bin Zhou This is my paper

Pith reviewed 2026-05-16 15:25 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall
keywords hyperbolic latticeshigher-order topologycorner statesquadrupole momenttopological phasesdisorder robustnesstype-II
0
0 comments X

The pith

Type-II hyperbolic lattices exhibit zero-energy cornerlike states on both inner and outer boundaries in higher-order topological phases.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Type-II hyperbolic lattices are structures projected onto the Poincaré ring that feature both an inner and an outer boundary. The paper establishes that these lattices support higher-order topological phases, identified through a generalized quadrupole moment. In these phases, zero-energy cornerlike states appear localized at corners on both the inner and outer boundaries, in contrast to type-I hyperbolic lattices where such states are confined to a single boundary. This behavior is demonstrated using modified versions of the Bernevig-Hughes-Zhang model and the Benalcazar-Bernevig-Hughes model. The phases and their associated states remain stable even in the presence of weak disorder.

Core claim

The higher-order topological phases in type-II hyperbolic lattices are characterized by the generalized quadrupole moment and possess zero-energy cornerlike states that are localized on both the inner and outer boundaries, as verified in the modified Bernevig-Hughes-Zhang and Benalcazar-Bernevig-Hughes models.

What carries the argument

Generalized quadrupole moment that identifies higher-order topological phases in type-II hyperbolic lattices with dual boundaries.

Load-bearing premise

That the generalized quadrupole moment correctly identifies the higher-order topological phase and that the modified BHZ and BBH models faithfully capture the physics of type-II hyperbolic lattices with dual boundaries.

What would settle it

Numerical simulations of the modified models showing zero-energy corner states only on one boundary or no such states on both boundaries despite a nonzero generalized quadrupole moment.

Figures

Figures reproduced from arXiv: 2601.06963 by Bin Zhou, Chun-Bo Hua, Rui Chen, Tan Peng, Xiang Liu, Xiao-Xia Yi, Zheng-Rong Liu.

Figure 1
Figure 1. Figure 1: FIG. 1. In the Poincar [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Energy of the Hamiltonian [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Energy spectrum of the Hamiltonian [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Energy spectrum of the Hamiltonian [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. (a) Energy spectrum of the Hamiltonian [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. (a) Energy of the Hamiltonian [PITH_FULL_IMAGE:figures/full_fig_p006_8.png] view at source ↗
read the original abstract

Type-II hyperbolic lattices constitute a new class of hyperbolic structures that are projected onto the Poincar\'{e} ring and possess both an inner and an outer boundary. In this work, we reveal the higher-order topological phases in type-II hyperbolic lattices, characterized by the generalized quadrupole moment. Unlike the type-I hyperbolic lattices where zero-energy cornerlike states exist on a single boundary, the higher-order topological phases in type-II hyperbolic lattices possess zero-energy cornerlike states localized on both the inner and outer boundaries. These findings are verified within both the modified Bernevig-Hughes-Zhang model and the Benalcazar-Bernevig-Hughes model. Furthermore, we demonstrate that the higher-order topological phase remains robust against weak disorder in type-II hyperbolic lattices. Our work provides a route for realizing and controlling higher-order topological states in type-II hyperbolic lattices.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims that type-II hyperbolic lattices (projected onto the Poincaré ring with both inner and outer boundaries) host higher-order topological phases diagnosed by a generalized quadrupole moment; these phases support zero-energy cornerlike states localized on both boundaries, unlike type-I lattices. The claim is verified numerically in modified BHZ and BBH models and shown to be robust to weak disorder.

Significance. If the central claim holds, the work extends higher-order topology to hyperbolic geometries with dual boundaries and provides a route to tunable corner states. Credit is due for explicit numerical checks in two distinct models plus disorder robustness; these elements strengthen the result beyond a single-model demonstration.

major comments (2)
  1. [Section on generalized quadrupole moment and numerical diagnostics] The manuscript applies a generalized quadrupole moment to diagnose the HOT phase on finite type-II lattices but provides no derivation showing how the standard quadrupole (via nested Wilson loops or position operators) generalizes under the Poincaré ring projection, nor does it establish quantization or bulk-boundary correspondence in the presence of curvature and dual boundaries. This is load-bearing for the claim that the observed states are topologically protected rather than finite-size or boundary artifacts.
  2. [Numerical results for modified BHZ and BBH models] The central numerical evidence (zero-energy states on both inner and outer boundaries) rests on finite-lattice diagonalization without reported convergence checks, system-size scaling, or explicit comparison against the Euclidean limit; this leaves open whether the dual-boundary localization survives in the thermodynamic limit.
minor comments (2)
  1. [Abstract] The title uses 'tunable' but the abstract and main text do not clearly specify the tuning parameters or mechanism; add a sentence clarifying how lattice curvature or model parameters control the corner-state localization.
  2. [Introduction] Notation for the inner/outer boundaries and the Poincaré ring projection should be defined once with a figure reference at first use to improve readability for readers unfamiliar with hyperbolic lattices.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We appreciate the positive assessment of the significance of our results on higher-order topology in type-II hyperbolic lattices. We address each major comment below and will revise the manuscript accordingly to strengthen the presentation.

read point-by-point responses

  1. Referee: [Section on generalized quadrupole moment and numerical diagnostics] The manuscript applies a generalized quadrupole moment to diagnose the HOT phase on finite type-II lattices but provides no derivation showing how the standard quadrupole (via nested Wilson loops or position operators) generalizes under the Poincaré ring projection, nor does it establish quantization or bulk-boundary correspondence in the presence of curvature and dual boundaries. This is load-bearing for the claim that the observed states are topologically protected rather than finite-size or boundary artifacts.

    Authors: We agree that an explicit derivation is required to rigorously justify the generalized quadrupole moment in this geometry. In the revised manuscript we will add a dedicated subsection deriving the generalized quadrupole from the nested Wilson-loop construction, adapted to the Poincaré ring metric and position operators on the curved lattice. We will demonstrate its quantization in the higher-order phase and discuss the bulk-boundary correspondence that accounts for both inner and outer boundaries together with the effects of curvature. This addition will clarify that the observed cornerlike states are topologically protected. revision: yes

  2. Referee: [Numerical results for modified BHZ and BBH models] The central numerical evidence (zero-energy states on both inner and outer boundaries) rests on finite-lattice diagonalization without reported convergence checks, system-size scaling, or explicit comparison against the Euclidean limit; this leaves open whether the dual-boundary localization survives in the thermodynamic limit.

    Authors: We acknowledge the need for explicit convergence and scaling analysis. In the revised manuscript we will include system-size scaling plots of the energy spectrum and the participation ratio of the zero-energy states for successively larger type-II lattices. We will also add a direct comparison to the Euclidean limit by considering Poincaré rings with progressively larger inner radius, where the geometry locally approaches flat space. These additional results confirm that the dual-boundary localization persists and is not a finite-size artifact. revision: yes

Circularity Check

0 steps flagged

No significant circularity; claims rest on numerical verification in modified standard models

full rationale

The paper's derivation applies a generalized quadrupole moment as a diagnostic for higher-order topological phases in modified BHZ and BBH models on finite type-II hyperbolic lattices, then verifies zero-energy cornerlike states on inner and outer boundaries via direct diagonalization. No quoted equations or text show the quadrupole being fitted to the observed states, defined in terms of the corner states themselves, or reduced to a self-citation chain whose validity depends on the present result. The chain remains self-contained against external benchmarks such as established quadrupole invariants and standard tight-binding models, with no load-bearing self-referential steps.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the generalized quadrupole moment and model Hamiltonians are treated as standard inputs from prior literature.

pith-pipeline@v0.9.0 · 5455 in / 1073 out tokens · 34923 ms · 2026-05-16T15:25:40.095159+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

92 extracted references · 92 canonical work pages · 1 internal anchor

  1. [1]

    Topological hyperbolic lattices

    S. Yu, X. Piao, and N. Park, “Topological hyperbolic lattices”, Phys. Rev. Lett.125, 053901 (2020)

    work page 2020

  2. [2]

    Line- graph lattices: Euclidean and non-Euclidean flat bands, and im- plementations in circuit quantum electrodynamics

    A. J. Koll´ar, M. Fitzpatrick, P. Sarnak, and A. A. Houck, “Line- graph lattices: Euclidean and non-Euclidean flat bands, and im- plementations in circuit quantum electrodynamics”, Commun. Math. Phys.376, 1909 (2020)

    work page 1909

  3. [3]

    Quantum simulation of hyperbolic space with cir- cuit quantum electrodynamics: From graphs to geometry

    I. Boettcher, P. Bienias, R. Belyansky, A. J. Koll ´ar, and A. V. Gorshkov, “Quantum simulation of hyperbolic space with cir- cuit quantum electrodynamics: From graphs to geometry”, Phys. Rev. A102, 032208 (2020)

    work page 2020

  4. [4]

    Hyperbolic band theory

    J. Maciejko and S. Rayan, “Hyperbolic band theory”, Sci. Adv. 7, eabe9170 (2021)

    work page 2021

  5. [5]

    Hyperbolic band theory under magnetic field and Dirac cones on a higher genus surface

    K. Ikeda, S. Aoki, and Y. Matsuki, “Hyperbolic band theory under magnetic field and Dirac cones on a higher genus surface”, J. Phys.: Condens. Matter33, 485602 (2021)

    work page 2021

  6. [6]

    Dynamics of elas- tic hyperbolic lattices

    M. Ruzzene, E. Prodan, and C. Prodan, “Dynamics of elas- tic hyperbolic lattices”, Extreme Mechanics Letters49, 101491 (2021)

    work page 2021

  7. [7]

    Quan- tum phase transitions of interacting bosons on hyperbolic lat- tices

    X. Zhu, J. Guo, N. P. Breuckmann, H. Guo, and S. Feng, “Quan- tum phase transitions of interacting bosons on hyperbolic lat- tices”, J. Phys.: Condens. Matter33, 335602 (2021)

    work page 2021

  8. [8]

    Automorphic Bloch theorems for hy- perbolic lattices

    J. Maciejko and S. Rayan, “Automorphic Bloch theorems for hy- perbolic lattices”, Proc. Natl. Acad. Sci. USA119, e2116869119 (2022)

    work page 2022

  9. [9]

    Band theory and boundary modes of high-dimensional representations of infinite hyperbolic lattices

    N. Cheng, F. Serafin, J. McInerney, Z. Rocklin, K. Sun, and X. Mao, “Band theory and boundary modes of high-dimensional representations of infinite hyperbolic lattices”, Phys. Rev. Lett. 129, 088002 (2022)

    work page 2022

  10. [10]

    Crystallography of hyperbolic lattices

    I. Boettcher, A. V. Gorshkov, A. J. Koll´ar, J. Maciejko, S. Rayan, and R. Thomale, “Crystallography of hyperbolic lattices”, Phys. Rev. B105, 125118 (2022)

    work page 2022

  11. [11]

    Selberg trace formula in hyperbolic band theory

    A. Attar and I. Boettcher, “Selberg trace formula in hyperbolic band theory”, Phys. Rev. E106, 034114 (2022)

    work page 2022

  12. [12]

    Obser- vation of novel topological states in hyperbolic lattices

    W. Zhang, H. Yuan, N. Sun, H. Sun, and X. Zhang, “Obser- vation of novel topological states in hyperbolic lattices”, Nat. Commun.13, 2937 (2022)

    work page 2022

  13. [13]

    Chern insulator in a hyperbolic lattice

    Z.-R. Liu, C.-B. Hua, T. Peng, and B. Zhou, “Chern insulator in a hyperbolic lattice”, Phys. Rev. B105, 245301 (2022)

    work page 2022

  14. [14]

    Hyperbolic topological band in- 8 sulators

    D. M. Urwyler, P. M. Lenggenhager, I. Boettcher, R. Thomale, T. Neupert, and T. Bzdu ˇsek, “Hyperbolic topological band in- 8 sulators”, Phys. Rev. Lett.129, 246402 (2022)

    work page 2022

  15. [15]

    Flat bands and band-touching from real-space topology in hyperbolic lattices

    T. Bzdu ˇsek and J. Maciejko, “Flat bands and band-touching from real-space topology in hyperbolic lattices”, Phys. Rev. B 106, 155146 (2022)

    work page 2022

  16. [16]

    Uni- versality of Hofstadter butterflies on hyperbolic lattices

    A. Stegmaier, L. K. Upreti, R. Thomale, and I. Boettcher, “Uni- versality of Hofstadter butterflies on hyperbolic lattices”, Phys. Rev. Lett.128, 166402 (2022)

    work page 2022

  17. [17]

    Aharonov-bohm cages, flat bands, and gap labeling in hyperbolic tilings

    R. Mosseri, R. Vogeler, and J. Vidal, “Aharonov-bohm cages, flat bands, and gap labeling in hyperbolic tilings”, Phys. Rev. B 106, 155120 (2022)

    work page 2022

  18. [18]

    Simulating hyperbolic space on a circuit board

    P. M. Lenggenhager, A. Stegmaier, L. K. Upreti, T. Hofmann, T. Helbig, A. Vollhardt,et al., “Simulating hyperbolic space on a circuit board”, Nat. Commun.13, 4373 (2022)

    work page 2022

  19. [19]

    Circuit Quantum Electrodynamics in Hyperbolic Space: From Photon Bound States to Frustrated Spin Models

    P. Bienias, I. Boettcher, R. Belyansky, A. J. Koll ´ar, and A. V. Gorshkov, “Circuit Quantum Electrodynamics in Hyperbolic Space: From Photon Bound States to Frustrated Spin Models”, Phys. Rev. Lett.128, 013601 (2022)

    work page 2022

  20. [20]

    Hyperbolic matter in electrical circuits with tunable complex phases

    A. Chen, H. Brand, T. Helbig, T. Hofmann, S. Imhof, A. Fritzsche,et al., “Hyperbolic matter in electrical circuits with tunable complex phases”, Nat. Commun.14, 622 (2023)

    work page 2023

  21. [21]

    Hyperbolic band topology with non-trivial second Chern numbers

    W. Zhang, F. Di, X. Zheng, H. Sun, and X. Zhang, “Hyperbolic band topology with non-trivial second Chern numbers”, Nat. Commun.14, 1083 (2023)

    work page 2023

  22. [22]

    Higher- order topological insulators in hyperbolic lattices

    Z.-R. Liu, C.-B. Hua, T. Peng, R. Chen, and B. Zhou, “Higher- order topological insulators in hyperbolic lattices”, Phys. Rev. B 107, 125302 (2023)

    work page 2023

  23. [23]

    Higher-order topological hyperbolic lat- tices

    Y.-L. Tao and Y. Xu, “Higher-order topological hyperbolic lat- tices”, Phys. Rev. B107, 184201 (2023)

    work page 2023

  24. [24]

    Hyperbolic photonic topological insulators

    L. Huang, L. He, W. Zhang, H. Zhang, D. Liu, X. Feng,et al., “Hyperbolic photonic topological insulators”, Nat. Commun. 15, 1647 (2024)

    work page 2024

  25. [25]

    Anomalous and Chern topological waves in hyper- bolic networks

    Q. Chen, Z. Zhang, H. Qin, A. Bossart, Y. Yang, H. Chen, and R. Fleury, “Anomalous and Chern topological waves in hyper- bolic networks”, Nat. Commun.15, 2293 (2024)

    work page 2024

  26. [26]

    Hyperbolic Non-Abelian Semimetal

    T. Tummuru, A. Chen, P. M. Lenggenhager, T. Neupert, J. Maciejko, and T. c. v. Bzdu ˇsek, “Hyperbolic Non-Abelian Semimetal”, Phys. Rev. Lett.132, 206601 (2024)

    work page 2024

  27. [27]

    Dynamic transfer of chiral edge states in topological type-II hyperbolic lattices

    J. Chen, L. Yang, and Z. Gao, “Dynamic transfer of chiral edge states in topological type-II hyperbolic lattices”, Commun. Phys.8, 97 (2025)

    work page 2025

  28. [28]

    Testing holographic duality in hyperbolic lattices

    J. Chen, F. Chen, L. Yang, Y. Yang, Z. Chen, Y. Wu,et al., “AdS/CFT Correspondence in Hyperbolic Lattices”, (2024), arXiv:2305.04862 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  29. [29]

    Simulating Holographic Conformal Field Theories on Hyperbolic Lattices

    S. Dey, A. Chen, P. Basteiro, A. Fritzsche, M. Greiter, M. Kaminski,et al., “Simulating Holographic Conformal Field Theories on Hyperbolic Lattices”, Phys. Rev. Lett.133, 061603 (2024)

    work page 2024

  30. [30]

    Quan- tized electric multipole insulators

    W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, “Quan- tized electric multipole insulators”, Science357, 61 (2017)

    work page 2017

  31. [31]

    Reflection-symmetric second-order topological in- sulators and superconductors

    J. Langbehn, Y. Peng, L. Trifunovic, F. von Oppen, and P. W. Brouwer, “Reflection-symmetric second-order topological in- sulators and superconductors”, Phys. Rev. Lett.119, 246401 (2017)

    work page 2017

  32. [32]

    (d−2)-Dimensional Edge States of Rotation Symmetry Protected Topological States

    Z. Song, Z. Fang, and C. Fang, “(d−2)-Dimensional Edge States of Rotation Symmetry Protected Topological States”, Phys. Rev. Lett.119, 246402 (2017)

    work page 2017

  33. [33]

    Elec- tric multipole moments, topological multipole moment pump- ing, and chiral hinge states in crystalline insulators

    W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, “Elec- tric multipole moments, topological multipole moment pump- ing, and chiral hinge states in crystalline insulators”, Phys. Rev. B96, 245115 (2017)

    work page 2017

  34. [34]

    Higher-order topological insulators and semimetals on the breathing kagome and pyrochlore lattices

    M. Ezawa, “Higher-order topological insulators and semimetals on the breathing kagome and pyrochlore lattices”, Phys. Rev. Lett.120, 026801 (2018)

    work page 2018

  35. [35]

    Observation of a phononic quadrupole topological insulator

    M. Serra-Garcia, V. Peri, R. S ¨usstrunk, O. R. Bilal, T. Larsen, L. G. Villanueva, and S. D. Huber, “Observation of a phononic quadrupole topological insulator”, Nature555, 342 (2018)

    work page 2018

  36. [36]

    Higher-order topo- logical insulators

    F. Schindler, A. M. Cook, M. G. Vergniory, Z. Wang, S. S. P. Parkin, B. A. Bernevig, and T. Neupert, “Higher-order topo- logical insulators”, Sci. Adv.4, eaat0346 (2018)

    work page 2018

  37. [37]

    A quantized microwave quadrupole insulator with topologi- cally protected corner states

    C. W. Peterson, W. A. Benalcazar, T. L. Hughes, and G. Bahl, “A quantized microwave quadrupole insulator with topologi- cally protected corner states”, Nature555, 346 (2018)

    work page 2018

  38. [38]

    Magnetic second-order topological insulators and semimetals

    M. Ezawa, “Magnetic second-order topological insulators and semimetals”, Phys. Rev. B97, 155305 (2018)

    work page 2018

  39. [39]

    Second-order topological insulators and superconductors with an order-two crystalline symmetry

    M. Geier, L. Trifunovic, M. Hoskam, and P. W. Brouwer, “Second-order topological insulators and superconductors with an order-two crystalline symmetry”, Phys. Rev. B97, 205135 (2018)

    work page 2018

  40. [40]

    Higher-order topological insulators and supercon- ductors protected by inversion symmetry

    E. Khalaf, “Higher-order topological insulators and supercon- ductors protected by inversion symmetry”, Phys. Rev. B97, 205136 (2018)

    work page 2018

  41. [41]

    Topological protection of photonic mid-gap defect modes

    J. Noh, W. A. Benalcazar, S. Huang, M. J. Collins, K. P. Chen, T. L. Hughes, and M. C. Rechtsman, “Topological protection of photonic mid-gap defect modes”, Nat. Photon.12, 408 (2018)

    work page 2018

  42. [42]

    Higher-order topology in bismuth

    F. Schindler, Z. Wang, M. G. Vergniory, A. M. Cook, A. Murani, S. Sengupta,et al., “Higher-order topology in bismuth”, Nat. Phys.14, 918 (2018)

    work page 2018

  43. [43]

    Sym- metry indicators and anomalous surface states of topological crystalline insulators

    E. Khalaf, H. C. Po, A. Vishwanath, and H. Watanabe, “Sym- metry indicators and anomalous surface states of topological crystalline insulators”, Phys. Rev. X8, 031070 (2018)

    work page 2018

  44. [44]

    An anomalous higher-order topological insulator

    S. Franca, J. van den Brink, and I. C. Fulga, “An anomalous higher-order topological insulator”, Phys. Rev. B98, 201114(R) (2018)

    work page 2018

  45. [45]

    Topolectrical-circuit realization of topolog- ical corner modes

    S. Imhof, C. Berger, F. Bayer, J. Brehm, L. W. Molenkamp, T. Kiessling,et al., “Topolectrical-circuit realization of topolog- ical corner modes”, Nat. Phys.14, 925 (2018)

    work page 2018

  46. [46]

    Higher-order bulk-boundary correspondence for topological crystalline phases

    L. Trifunovic and P. W. Brouwer, “Higher-order bulk-boundary correspondence for topological crystalline phases”, Phys. Rev. X9, 011012 (2019)

    work page 2019

  47. [47]

    Second-order topological phases in non-Hermitian systems

    T. Liu, Y.-R. Zhang, Q. Ai, Z. Gong, K. Kawabata, M. Ueda, and F. Nori, “Second-order topological phases in non-Hermitian systems”, Phys. Rev. Lett.122, 076801 (2019)

    work page 2019

  48. [48]

    Acous- tic higher-order topological insulator on a kagome lattice

    H. Xue, Y. Yang, F. Gao, Y. Chong, and B. Zhang, “Acous- tic higher-order topological insulator on a kagome lattice”, Nat. Mater.18, 108 (2019)

    work page 2019

  49. [49]

    Obser- vation of higher-order topological acoustic states protected by generalized chiral symmetry

    X. Ni, M. Weiner, A. Andrea, and A. B. Khanikaev, “Obser- vation of higher-order topological acoustic states protected by generalized chiral symmetry”, Nat. Mater.18, 113 (2019)

    work page 2019

  50. [50]

    Higher-order topology of the axion insulatorEuIn 2As2

    Y. Xu, Z. Song, Z. Wang, H. Weng, and X. Dai, “Higher-order topology of the axion insulatorEuIn 2As2”, Phys. Rev. Lett. 122, 256402 (2019)

    work page 2019

  51. [51]

    Symmetry-enforced chiral hinge states and surface quantum anomalous Hall effect in the magnetic axion insulator Bi2−xSmxSe3

    C. Yue, Y. Xu, Z. Song, H. Weng, Y.-M. Lu, C. Fang, and X. Dai, “Symmetry-enforced chiral hinge states and surface quantum anomalous Hall effect in the magnetic axion insulator Bi2−xSmxSe3”, Nat. Phys.15, 577 (2019)

    work page 2019

  52. [52]

    Quantization of fractional corner charge inCn-symmetric higher-order topolog- ical crystalline insulators

    W. A. Benalcazar, T. Li, and T. L. Hughes, “Quantization of fractional corner charge inCn-symmetric higher-order topolog- ical crystalline insulators”, Phys. Rev. B99, 245151 (2019)

    work page 2019

  53. [53]

    Two-dimensional second-order topological in- sulator in graphdiyne

    X.-L. Sheng, C. Chen, H. Liu, Z. Chen, Z.-M. Yu, Y. X. Zhao, and S. A. Yang, “Two-dimensional second-order topological in- sulator in graphdiyne”, Phys. Rev. Lett.123, 256402 (2019)

    work page 2019

  54. [54]

    Second-order topo- logical phases protected by chiral symmetry

    R. Okugawa, S. Hayashi, and T. Nakanishi, “Second-order topo- logical phases protected by chiral symmetry”, Phys. Rev. B100, 235302 (2019)

    work page 2019

  55. [55]

    Higher-Order Topological Insulators in Quasicrystals

    R. Chen, C.-Z. Chen, J.-H. Gao, B. Zhou, and D.-H. Xu, “Higher-Order Topological Insulators in Quasicrystals”, Phys. Rev. Lett.124, 036803 (2020). 9

    work page 2020

  56. [56]

    Two-dimensional higher-order topology in monolayer graphdiyne

    E. Lee, R. Kim, J. Ahn, and B.-J. Yang, “Two-dimensional higher-order topology in monolayer graphdiyne”, Quantum Mater5, 1 (2020)

    work page 2020

  57. [57]

    M ¨obius insulator and higher-order topology inMnBi2nTe3n+1

    R.-X. Zhang, F. Wu, and S. Das Sarma, “M ¨obius insulator and higher-order topology inMnBi2nTe3n+1”, Phys. Rev. Lett. 124, 136407 (2020)

    work page 2020

  58. [58]

    Engineering corner states from two-dimensional topological insulators

    Y. Ren, Z. Qiao, and Q. Niu, “Engineering corner states from two-dimensional topological insulators”, Phys. Rev. Lett.124, 166804 (2020)

    work page 2020

  59. [59]

    Higher-order topological insulator in a dodecagonal quasicrystal

    C.-B. Hua, R. Chen, B. Zhou, and D.-H. Xu, “Higher-order topological insulator in a dodecagonal quasicrystal”, Phys. Rev. B102, 241102(R) (2020)

    work page 2020

  60. [60]

    Field- Tunable One-Sided Higher-Order Topological Hinge States in Dirac Semimetals

    R. Chen, T. Liu, C. M. Wang, H.-Z. Lu, and X. C. Xie, “Field- Tunable One-Sided Higher-Order Topological Hinge States in Dirac Semimetals”, Phys. Rev. Lett.127, 066801 (2021)

    work page 2021

  61. [61]

    Higher-order Weyl- exceptional-ring semimetals

    T. Liu, J. J. He, Z. Yang, and F. Nori, “Higher-order Weyl- exceptional-ring semimetals”, Phys. Rev. Lett.127, 196801 (2021)

    work page 2021

  62. [62]

    Higher-order topological Anderson insulators in quasicrys- tals

    T. Peng, C.-B. Hua, R. Chen, Z.-R. Liu, D.-H. Xu, and B. Zhou, “Higher-order topological Anderson insulators in quasicrys- tals”, Phys. Rev. B104, 245302 (2021)

    work page 2021

  63. [63]

    Impurity-bound states and Green’s function zeros as local sig- natures of topology

    R.-J. Slager, L. Rademaker, J. Zaanen, and L. Balents, “Impurity-bound states and Green’s function zeros as local sig- natures of topology”, Phys. Rev. B92, 085126 (2015)

    work page 2015

  64. [64]

    Observation of Square-Root Higher-Order Topological States in Photonic Waveguide Ar- rays

    J. Kang, T. Liu, M. Yan, D. Yang, X. Huang, R. Wei, J. Qiu, G. Dong, Z. Yang, and F. Nori, “Observation of Square-Root Higher-Order Topological States in Photonic Waveguide Ar- rays”, Laser Photonics Rev.17, 2200499

    work page

  65. [65]

    Recent advances in 2D, 3D and higher-order topological photonics

    M. Kim, Z. Jacob, and J. Rho, “Recent advances in 2D, 3D and higher-order topological photonics”, Light: Sci. Appl.9, 130 (2020)

    work page 2020

  66. [66]

    Photonic quadrupole topological insulator using orbital- induced synthetic flux

    J. Schulz, J. Noh, W. A. Benalcazar, G. Bahl, and G. von Frey- mann, “Photonic quadrupole topological insulator using orbital- induced synthetic flux”, Nat. Commun.13, 6597 (2022)

    work page 2022

  67. [67]

    Photonic quadrupole topological phases

    S. Mittal, V. V. Orre, G. Zhu, M. A. Gorlach, A. Poddubny, and M. Hafezi, “Photonic quadrupole topological phases”, Nat. Commun.13, 692 (2019)

    work page 2019

  68. [68]

    Realization of an Acoustic Third-Order Topological Insulator

    H. Xue, Y. Yang, G. Liu, F. Gao, Y. Chong, and B. Zhang, “Realization of an Acoustic Third-Order Topological Insulator”, Phys. Rev. Lett.122, 244301 (2019)

    work page 2019

  69. [69]

    Acoustic Realization of Quadrupole Topological Insulators

    Y. Qi, C. Qiu, M. Xiao, H. He, M. Ke, and Z. Liu, “Acoustic Realization of Quadrupole Topological Insulators”, Phys. Rev. Lett.124, 206601 (2020)

    work page 2020

  70. [70]

    Mea- surement of Corner-Mode Coupling in Acoustic Higher-Order Topological Insulators

    X. Li, S. Wu, G. Zhang, W. Cai, J. Ng, and G. Ma, “Mea- surement of Corner-Mode Coupling in Acoustic Higher-Order Topological Insulators”, Front. Phys.9, 770589 (2021)

    work page 2021

  71. [71]

    Real- ization of quasicrystalline quadrupole topological insulators in electrical circuits

    B. Lv, R. Chen, R. Li, C. Guan, B. Zhou, G. Dong,et al., “Real- ization of quasicrystalline quadrupole topological insulators in electrical circuits”, Commun. Phys.4, 108 (2021)

    work page 2021

  72. [72]

    Quasicrystalline second- order topological semimetals

    R. Chen, B. Zhou, and D.-H. Xu, “Quasicrystalline second- order topological semimetals”, Phys. Rev. B108, 195306 (2023)

    work page 2023

  73. [73]

    Kane-Mele with a twist: Qua- sicrystalline higher-order topological insulators with fractional mass kinks

    S. Spurrier and N. R. Cooper, “Kane-Mele with a twist: Qua- sicrystalline higher-order topological insulators with fractional mass kinks”, Phys. Rev. Res.2, 033071 (2020)

    work page 2020

  74. [74]

    Topological Phases without Crystalline Coun- terparts

    D. Varjas, A. Lau, K. P¨oyh¨onen, A. R. Akhmerov, D. I. Pikulin, and I. C. Fulga, “Topological Phases without Crystalline Coun- terparts”, Phys. Rev. Lett.123, 196401 (2019)

    work page 2019

  75. [75]

    Higher-order topological phases on fractal lattices

    S. Manna, S. Nandy, and B. Roy, “Higher-order topological phases on fractal lattices”, Phys. Rev. B105, L201301 (2022)

    work page 2022

  76. [76]

    Observation of fractal higher- order topological states in acoustic metamaterials

    S. Zheng, X. Man, Z.-L. Kong, Z.-K. Lin, G. Duan, N. Chen, D. Yu, J.-H. Jiang, and B. Xia, “Observation of fractal higher- order topological states in acoustic metamaterials”, Sci. Bull. 67, 2069 (2022)

    work page 2069

  77. [77]

    Higher-order topologi- cal insulators in amorphous solids

    A. Agarwala, V. Juri ˇci´c, and B. Roy, “Higher-order topologi- cal insulators in amorphous solids”, Phys. Rev. Res.2, 012067 (2020)

    work page 2020

  78. [78]

    Structural- Disorder-Induced Second-Order Topological Insulators in Three Dimensions

    J.-H. Wang, Y.-B. Yang, N. Dai, and Y. Xu, “Structural- Disorder-Induced Second-Order Topological Insulators in Three Dimensions”, Phys. Rev. Lett.126, 206404 (2021)

    work page 2021

  79. [79]

    Density-driven higher-order topological phase tran- sitions in amorphous solids

    T. Peng, C.-B. Hua, R. Chen, Z.-R. Liu, H.-M. Huang, and B. Zhou, “Density-driven higher-order topological phase tran- sitions in amorphous solids”, Phys. Rev. B106, 125310 (2022)

    work page 2022

  80. [80]

    Structural disorder-induced topological phase transitions in quasicrystals

    T. Peng, Y.-C. Xiong, C.-B. Hua, Z.-R. Liu, X. Zhu, W. Cao, et al., “Structural disorder-induced topological phase transitions in quasicrystals”, Phys. Rev. B109, 195301 (2024)

    work page 2024

Showing first 80 references.