pith. sign in

arxiv: 2604.07459 · v1 · submitted 2026-04-08 · ❄️ cond-mat.mes-hall

Exploring topological phases with extended Su-Schrieffer-Heeger models

Raditya Weda Bomantara This is my paper

Pith reviewed 2026-05-10 17:22 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall
keywords Su-Schrieffer-Heeger modeltopological phasestight-binding latticezero energy modesedge statestopological insulatorsextensions of SSH
0
0 comments X

The pith

The Su-Schrieffer-Heeger model supports topological phases that become more sophisticated when the model is extended in dimensionality, unit cell size, or with additional physical terms.

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

The paper reviews ways to extend the basic one-dimensional SSH model, which has alternating hopping strengths and supports zero-energy states at the ends. Extensions are considered by making the lattice higher-dimensional, using larger repeating units, or including extra interactions. These modifications lead to topological features such as protected states in more complicated geometries. A reader would care because understanding these extensions helps in designing and predicting behavior in real topological materials like certain polymers or engineered lattices.

Core claim

The SSH model describes a 1D tight-binding lattice with alternating nearest-neighbor amplitudes and supports a topological phase with zero energy eigenstates localized at each end. Extensions by increasing dimensionality, enlarging the unit cell, or adding extra terms give rise to more sophisticated topological phenomena, as illustrated through case studies from existing literature where properties of topological origin are elaborated.

What carries the argument

Extended Su-Schrieffer-Heeger models obtained by modifying the original 1D alternating hopping lattice through added dimensions, larger unit cells, or physical effect terms, which carry the argument by exhibiting various boundary-localized modes and topological invariants.

If this is right

  • Models with increased dimensionality support topological phases in two or more dimensions with protected edge or surface states.
  • Enlarged unit cells allow for models with multiple bands and more complex topological classifications.
  • Adding extra terms representing physical effects introduces new types of topological protection or phase transitions.
  • These extended models provide concrete examples for studying phenomena absent in the minimal 1D SSH case.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The SSH framework acts as a simple base from which many known topological systems can be derived by systematic modifications.
  • Experimental realizations could involve cold atoms or photonic lattices tuned to match the extended Hamiltonians.
  • Further work might classify all possible extensions using symmetry indicators or topological invariants.

Load-bearing premise

That the chosen case studies from the literature cover a representative range of extensions and capture their key topological properties adequately.

What would settle it

A calculation or simulation of an extended SSH Hamiltonian showing no zero-energy boundary modes where the review predicts them would challenge the corresponding case study.

Figures

Figures reproduced from arXiv: 2604.07459 by Raditya Weda Bomantara.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic description of the SSH Hamiltonian with [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Schematic description of the SSH Hamiltonian in the [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a-c) The energy bands of [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The energy band structure of the 3D topological insulating model of Eq. (26) within the single particle subspace at a [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Schematic depiction of Weyl points with opposite [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Schematic description of a stack of SSH chains under [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The single particle energy spectrum of Eq. (35) un [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Schematic description of the extended SSH Hamilto [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. The spatial profile of the four edge modes in the [PITH_FULL_IMAGE:figures/full_fig_p010_10.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The single particle energy spectrum of the SSH3 [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. (a) The single particle energy spectrum of the SSH3 [PITH_FULL_IMAGE:figures/full_fig_p011_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. (a) The energy band structure of the square-root SSH model [PITH_FULL_IMAGE:figures/full_fig_p013_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. The single particle energy spectrum of the SSH3m [PITH_FULL_IMAGE:figures/full_fig_p013_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. The “phase diagram” of Eq. (68) based on the exis [PITH_FULL_IMAGE:figures/full_fig_p016_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15 [PITH_FULL_IMAGE:figures/full_fig_p017_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16 [PITH_FULL_IMAGE:figures/full_fig_p018_16.png] view at source ↗
read the original abstract

The Su-Schrieffer-Heeger (SSH) model describes a tight-binding one-dimensional (1D) lattice with alternating nearest-neighbor amplitudes. Despite its mathematically simple and physically intuitive structure, the SSH model is capable of supporting a 1D topological phase that is characterized by the presence of zero energy eigenstates (zero modes) localized at each end of the lattice. For this reason, many studies in the area of topological phases of matter often consider the SSH model as a subject for various extensions that give rise to more sophisticated topological phenomena. The purpose of this article is to review, in sufficient detail, existing approaches to extending the SSH model. This includes extensions by increasing the dimensionality of the lattice, enlarging the size of its unit cell, or adding extra terms that represent various physical effects. For each approach, some extended SSH models studied in relevant existing literature are discussed as case studies. Noteworthy properties of such models, which are of topological origin, are further comprehensively elaborated.

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

0 major / 3 minor

Summary. The manuscript is a review of the Su-Schrieffer-Heeger (SSH) model and its extensions for exploring topological phases of matter. It begins by recalling that the basic 1D SSH chain supports a topological phase with zero-energy edge-localized modes, then systematically examines three classes of extensions—increasing lattice dimensionality, enlarging the unit cell, and adding extra terms that encode additional physical effects—using selected case studies drawn from the existing literature to illustrate the resulting topological phenomena and invariants.

Significance. If the coverage is balanced and the case studies are accurately summarized, the review provides a useful entry point and reference for researchers working on 1D and higher-dimensional topological insulators and superconductors. By organizing extensions around a single, well-understood parent model, it can help readers map how simple modifications generate richer phenomenology such as higher-order topology, non-Hermitian effects, or fractionalized excitations.

minor comments (3)
  1. The abstract states that the review covers extensions 'in sufficient detail,' yet the manuscript does not include an explicit statement of selection criteria for the case studies or a table summarizing the topological invariants and edge-state properties across all discussed models; adding such a table would improve usability.
  2. Several figures (e.g., those depicting band structures or edge-mode wavefunctions for the extended models) lack labels for the topological phase boundaries or the value of the relevant winding number; this reduces clarity when the text refers to 'the topological regime.'
  3. The reference list appears to omit a few recent works on non-Hermitian extensions of the SSH model (post-2022) that are directly relevant to the 'adding extra terms' section; a brief check against current literature would strengthen completeness.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript as a useful entry point and reference for researchers studying topological phases via extensions of the SSH model. We appreciate the recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity: review of external literature

full rationale

The manuscript is a review that summarizes established results on the SSH model and its extensions from the existing literature. It presents no new derivations, predictions, or fitted parameters of its own; all case studies and noteworthy topological properties are explicitly drawn from cited external works. No load-bearing step reduces by construction to self-definition, self-citation chains, or renaming of inputs. The central claims restate well-known facts (zero-energy edge modes in the topological phase) without internal circular logic.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The review rests on the standard description of the SSH model and the assumption that selected literature examples illustrate the main extension strategies; no new free parameters or invented entities are introduced by the paper itself.

axioms (1)
  • domain assumption The SSH model is a tight-binding 1D lattice with alternating nearest-neighbor amplitudes that supports zero-energy end states in its topological phase.
    Directly stated in the abstract as the foundational model.

pith-pipeline@v0.9.0 · 5465 in / 1209 out tokens · 36076 ms · 2026-05-10T17:22:55.286894+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

180 extracted references · 180 canonical work pages

  1. [1]

    re- duced Brillouin zone

    Extension by changing the periodicity of the hopping amplitudes As was schematically shown in Fig. 1, the real space Hamiltonian of the SSH model consists of two alter- nating hopping amplitudesvandw. One of the most 10 FIG. 9. The single particle energy spectrum of the SSH3 model under (a,b,c) PBC and (d) OBC at varying values of w. The system parameters...

    work page

  2. [2]

    phase diagram

    Extension by applying a nontrivial square-root Another method for extending the SSH model is by appropriately defining a new HamiltonianH sqSSH, so that its square yields the real space Hamiltonian of the SSH model. This idea was first envisioned in Ref. [91], which considers a general tight-binding Hamiltonian for the square-root model as (in the first-q...

    work page

  3. [3]

    Jungwirth, Q

    T. Jungwirth, Q. Niu, and A. H. MacDonald, Anoma- lous Hall Effect in Ferromagnetic Semiconductors, Phys. Rev. Lett.88, 207208 (2002)

    work page 2002

  4. [4]

    Onoda and N

    M. Onoda and N. Nagaosa, Quantized Anomalous Hall Effect in Two-Dimensional Ferromagnets: Quantum Hall Effect in Metals, Phys. Rev. Lett.90, 206601 (2003)

    work page 2003

  5. [5]

    C. X. Liu, X. L. Qi, X. Dai, Z. Fang, and S. C. Zhang, Quantum Anomalous Hall Effect in Hg1−yMnyTe Quan- tum Wells, Phys. Rev. Lett.101, 146802 (2008)

    work page 2008

  6. [6]

    R. Yu, W. Zhang, H. Zhang, S. Zhang, X. Dai, and Z. Fang, Quantized Anomalous Hall Effect in Magnetic Topological Insulators, Science329, 61 (2010)

    work page 2010

  7. [7]

    C. L. Kane and E. J. Mele, Quantum Spin Hall Effect in Graphene, Phys. Rev. Lett.95, 226801 (2005)

    work page 2005

  8. [8]

    B. A. Bernevig, T. L. Hughes, and S. C. Zhang, Quan- tum spin Hall effect and topological phase transition in HgTe quantum wells, Science314, 1757 (2006)

    work page 2006

  9. [9]

    C. L. Kane and E. J. Mele,Z 2 Topological Order and the Quantum Spin Hall Effect, Phys. Rev. Lett.95, 146802 (2005)

    work page 2005

  10. [10]

    K¨ onig, S

    M. K¨ onig, S. Wiedmann, C. Br¨ une, A. Roth, H. Buh- mann, L. W. Molenkamp, X.-L. Qi, and S.-C. Zhang, Quantum Spin Hall Insulator State in HgTe Quantum Wells, Science318, 766 (2007)

    work page 2007

  11. [11]

    Hsieh, D

    D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava, and M. Z. Hasan, A topological Dirac insula- tor in a quantum spin Hall phase, Nature (London)452, 970 (2008)

    work page 2008

  12. [12]

    Zhang, C.-X

    H. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang, and S.-C. Zhang, Topological insulators in Bi 2Se3, Bi 2Te3 and Sb 2Te3 with a single Dirac cone on the surface, Nat. Phys.5, 438 (2009)

    work page 2009

  13. [13]

    Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, Observation of a large-gap topological- insulator class with a single Dirac cone on the surface, Nat. Phys.5, 398 (2009)

    work page 2009

  14. [14]

    Y. L. Chen, J. G. Analytis, J. H. Chu, Z. K. Liu, S. K. Mo, X. L. Qi, H. J. Zhang, D. H. Lu, X. Dai, Z. Fang, S. C. Zhang, I. R. Fisher, Z. Hus- sain, Z. X. Shen, Experimental Realization of a Three-Dimensional Topological Insulator, Bi 2Te3, Sci- ence325, 178 (2009)

    work page 2009

  15. [15]

    A. Y. Kitaev, Fault-tolerant quantum computation by anyons, Ann. Phys.303, 2-30 (2003)

    work page 2003

  16. [16]

    Nayak, S

    C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. D. Sarma, Non-Abelian anyons and topological quan- tum computation, Rev. Mod. Phys.80, 1083 (2008)

    work page 2008

  17. [17]

    A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Lud- wig, Classification of topological insulators and super- conductors in three spatial dimensions, Phys. Rev. B78, 195125 (2008)

    work page 2008

  18. [18]

    S. Ryu, A. Schnyder, A. Furusaki, and A. Ludwig, Topo- logical insulators and superconductors: ten-fold way and dimensional hierarchy, New J. Phys.12, 065010 (2010)

    work page 2010

  19. [19]

    Roy and F

    R. Roy and F. Harper, Periodic table for Floquet topo- logical insulators, Phys. Rev. B96, 155118 (2017)

    work page 2017

  20. [20]

    B. Wang, T. Chen, and X. Zhang, Experimental ob- servation of topologically protected bound states with vanishing chern numbers in a two-dimensional quantum walk, Phys. Rev. Lett.121, 100501 (2018)

    work page 2018

  21. [21]

    C. Chen, X. Ding, J. Qin, Y. He, Y.-H. Luo, M.- C. Chen, C. Liu, X.-L. Wang, W.-J. Zhang, H. Li, et al., Observation of topologically protected edge states in a photonic two-dimensional quantum walk. Phys. Rev. Lett.121, 100502 (2018)

    work page 2018

  22. [22]

    M. Xiao, G. Ma, Z. Yang, P. Sheng, Z. Q. Zhang, and C. T. Chan, Geometric phase and band inversion in pe- riodic acoustic systems. Nat. Phys.11, 240–244 (2015)

    work page 2015

  23. [23]

    B. K. Stuhl, H.-I. Lu, L. M. Aycock, D. Genkina, and I. B. Spielman. Visualizing edge states with an atomic bose gas in the quantum hall regime, Science349, 1514–1518 (2015)

    work page 2015

  24. [24]

    E. J. Meier, F. A. An, and B. Gadway, Ob- servation of the topological soliton state in the Su–Schrieffer–Heeger, Nat. Commun.7, 13986 (2016)

    work page 2016

  25. [25]

    Goldman, J

    N. Goldman, J. C. Budich, and P. Zoller, Topological quantum matter with ultracold gases in optical lattices, Nat. Phys.12, 639–645 (2016)

    work page 2016

  26. [26]

    Jotzu, M

    G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, and T. Esslinger. Experimen- tal realization of the topological haldane model with ultracold fermions. Nature515, 237–240 (2014)

    work page 2014

  27. [27]

    E. J. Meier, F. A. An, A. Dauphin, M. Maffei, P. Massig- nan, T. L. Hughes, and B. Gadway, Observation of the topological anderson insulator in disordered atomic wires, Science362, 929–933 (2018)

    work page 2018

  28. [28]

    X. Li, Y. Meng, X. Wu, S. Yan, Y. Huang, S. Wang, and W. Wen, Su-Schrieffer-Heeger model inspired acoustic interface states and edge states, Appl. Phys. Lett.113, 203501 (2018)

    work page 2018

  29. [29]

    Kiczynski, S

    M. Kiczynski, S. K. Gorman, H. Geng, M. B. Donnelly, Y. Chung, Y. He, J. G. Keizer, and M. Y. Simmons, Engineering topological states in atom-based semicon- ductor quantum dots, Nature606, 694 (2022)

    work page 2022

  30. [30]

    B.-Ocampo, B

    Y. B.-Ocampo, B. M.-Montanez, A. M. M.-Arg¨ uello, and R. A. M.- S´ anchez. Twofold topological phase tran- sitions induced by third-nearest-neighbor interactions in 1d chains, Phys. Rev. B109, 104111 (2024)

    work page 2024

  31. [31]

    Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zil- berberg, Topological States and Adiabatic Pumping in Quasicrystals, Phys. Rev. Lett.109, 106402 (2012). 22

    work page 2012

  32. [32]

    Verbin, O

    M. Verbin, O. Zilberberg, Y. Lahini, Y. E. Kraus, and Y. Silberberg, Topological pumping over a photonic Fi- bonacci quasicrystal, Phys. Rev. B91, 064201 (2015)

    work page 2015

  33. [33]

    Longhi, Topological pumping of edge states via adi- abatic passage, Phys

    S. Longhi, Topological pumping of edge states via adi- abatic passage, Phys. Rev. B99, 155150 (2019)

    work page 2019

  34. [34]

    Lohse, C

    M. Lohse, C. Schweizer, O. Zilberberg, M. Aidelsburger, and I. Bloch, A Thouless quantum pump with ultracold bosonic atoms in an optical superlattice, Nat. Phys.12, 350 (2016)

    work page 2016

  35. [35]

    Nakajima, T

    S. Nakajima, T. Tomita, S. Taie, T. Ichinose, H. Ozawa, L. Wang, M. Troyer, and Y. Takahashi, Topological Thouless pumping of ultracold fermions, Nat. Phys.12, 296 (2016)

    work page 2016

  36. [36]

    L. Wang, M. Troyer, and X. Dai, Topological Charge Pumping in a One-Dimensional Optical Lattice, Phys. Rev. Lett.111, 026802 (2013)

    work page 2013

  37. [37]

    Mei, J.-B

    F. Mei, J.-B. You, D.-W. Zhang, X. C. Yang, R. Fazio, S.-L. Zhu, and L. C. Kwek, Topological insulator and particle pumping in a one-dimensional shaken optical lattice, Phys. Rev. A90, 063638 (2014)

    work page 2014

  38. [38]

    H. Wang, L. Zhou, and J. Gong, Interband coherence induced correction to adiabatic pumping in periodically driven systems, Phys. Rev. B91, 085420 (2015)

    work page 2015

  39. [39]

    Y. Ke, X. Qin, F. Mei, H. Zhong, Y. S. Kivshar, and C. Lee, Topological phase transitions and Thou- less pumping of light in photonic waveguide arrays, Laser Photon. Rev.10, 995 (2016)

    work page 2016

  40. [40]

    H. H. Yap, L. Zhou, J.-S. Wang, and J. Gong, Com- putational study of the two-terminal transport of Flo- quet quantum Hall insulators, Phys. Rev. B96, 165443 (2017)

    work page 2017

  41. [41]

    Y. Ke, X. Qin, Y. S. Kivshar, and C. Lee, Multiparti- cle Wannier states and Thouless pumping of interacting bosons, Phys. Rev. A95, 063630 (2017)

    work page 2017

  42. [42]

    S. Hu, Y. Ke, Y. Deng, and C. Lee, Dispersion- suppressed topological Thouless pumping, Phys. Rev. B100, 064302 (2019)

    work page 2019

  43. [43]

    L. Lin, Y. Ke, and C. Lee, Interaction-induced topo- logical bound states and Thouless pumping in a one- dimensional optical lattice, Phys. Rev. A 101, 023620 (2020)

    work page 2020

  44. [44]

    S. Hu, Y. Ke, and C. Lee, Topological quantum trans- port and spatial entanglement distribution via a disor- dered bulk channel, Phys. Rev. A101, 052323 (2020)

    work page 2020

  45. [45]

    Lang and H

    N. Lang and H. P. Buchle, Topological networks for quantum communication between distant qubits, npj Quantum Information3, 47 (2017)

    work page 2017

  46. [46]

    Karzig, C

    T. Karzig, C. Knapp, R. M. Lutchyn, P. Bonderson, M. B. Hastings, C. Nayak, J. Alicea, K. Flensberg, S. Plugge,et al., Scalable designs for quasiparticle- poisoning-protected topological quantum computation with Majorana zero modes, Phys. Rev. B95, 235305 (2017)

    work page 2017

  47. [47]

    F. Mei, G. Chen, L. Tian, S.-L. Zhu, and S. Jia, Robust quantum state transfer via topological edge states in superconducting qubit chains, Phys. Rev. A98, 012331 (2018)

    work page 2018

  48. [48]

    R. W. Bomantara and J Gong, Quantum computation via Floquet topological edge modes, Phys. Rev. B98, 165421 (2018)

    work page 2018

  49. [49]

    Boross, J

    P. Boross, J. K. Asb´ oth, G. Sz´ echenyi, L. Oroszl´ any, and A. P´ alyi, Poor mans topological quantum gate based on the Su-Schrieffer-Heeger model, Phys. Rev. B100, 045414 (2019)

    work page 2019

  50. [50]

    S. Tan, R. W. Bomantara, and J. Gong, High-fidelity and long-distance entangled-state transfer with Floquet topological edge modes, Phys. Rev. A102, 022608 (2020)

    work page 2020

  51. [51]

    R. W. Bomantara and J. Gong, Combating quasi- particle poisoning with multiple Majorana fermions in a periodically-driven quantum wire, J. Phys.: Con- dens. Matter32, 435301 (2020)

    work page 2020

  52. [52]

    R. W. Bomantara and J. Gong, Measurement-only quantum computation with Floquet Majorana corner modes, Phys. Rev. B101, 085401 (2020)

    work page 2020

  53. [53]

    Matthies, J

    A. Matthies, J. Park, E. Berg, and A. Rosch, Stability of floquet majorana box qubits, Phys. Rev. Lett.128, 127702 (2022)

    work page 2022

  54. [54]

    A. H. Safwan and R. W. Bomantara, J. Phys. A: Math. Theor.58, 335302 (2025)

    work page 2025

  55. [55]

    Sun, Z.-C

    K. Sun, Z.-C. Gu, H. Katsura, and S. D. Sarma, Nearly-flat bands with nontrivial topology, Phys. Rev. Lett.106, 236803 (2011)

    work page 2011

  56. [56]

    Weeks and M

    C. Weeks and M. Franz, Topological Insulators on the Lieb and Perovskite Lattices, Phys. Rev. B82, 085310 (2010)

    work page 2010

  57. [57]

    Guo and M

    H.-M. Guo and M. Franz, Topological insulator on the kagome lattice, Phys. Rev. B80, 113102 (2009)

    work page 2009

  58. [58]

    N. H. Lindner, G. Refael, and V. Galitski, Nat. Phys.7, 490 (2011)

    work page 2011

  59. [59]

    D. Y. H. Ho and J. Gong, Topological effects in chi- ral symmetric driven systems, Phys. Rev. B90, 195419 (2014)

    work page 2014

  60. [60]

    L. Fu, C. L. Kane, and E. J. Mele, Topological Insula- tors in Three Dimensions, Phys. Rev. Lett.98, 106803 (2007)

    work page 2007

  61. [61]

    J. E. Moore and L. Balents, Topological invariants of time-reversal-invariant band structures, Phys. Rev. B 75, 121306(R) (2007)

    work page 2007

  62. [62]

    Roy, Topological phases and the quantum spin Hall effect in three dimensions, Phys

    R. Roy, Topological phases and the quantum spin Hall effect in three dimensions, Phys. Rev. B79, 195322 (2009)

    work page 2009

  63. [63]

    Roushan, J

    P. Roushan, J. Seo, C. V. Parker, Y. S. Hor, D. Hsieh, D. Qian, A. Richardella, M. Z. Hasan, R. J. Cava, and A. Yazdani, Topological surface states protected from backscattering by chiral spin texture, Nature460, 1106–1109 (2009)

    work page 2009

  64. [64]

    M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators, Rev. Mod. Phys.82, 3045 (2010)

    work page 2010

  65. [65]

    Ringel, Y

    Z. Ringel, Y. E. Kraus, and A. Stern, Phys. Rev. B86, 045102 (2012)

    work page 2012

  66. [66]

    A. Y. Kitaev, Unpaired majorana fermions in quantum wires, Physics-uspekhi44, 131 (2001)

    work page 2001

  67. [67]

    Fendley, Parafermionic edge zero modes in Zn- invariant spin chains, J

    P. Fendley, Parafermionic edge zero modes in Zn- invariant spin chains, J. Stat. Mech. P11020 (2012)

    work page 2012

  68. [68]

    Ganeshan, K

    S. Ganeshan, K. Sun and S. D. Sarma, Topological Zero- Energy Modes in Gapless Commensurate Aubry-Andr´ e- Harper Models, Phys. Rev. Lett.110, 180403 (2013)

    work page 2013

  69. [69]

    R. W. Bomantara, G. N. Raghava, L. Zhou, and J. Gong, Floquet topological semimetal phases of an ex- tended kicked Harper model, Phys. Rev. E93, 022209 (2016)

    work page 2016

  70. [70]

    R. W. Bomantara and J. Gong, Generating con- trollable type-II Weyl points via periodic driving, Phys. Rev. B94, 235447 (2016)

    work page 2016

  71. [71]

    H. Q. Wang, M. N. Chen, R. W. Bomantara, J. Gong, and D. Y. Xing, Line nodes and surface Majorana flat bands in static and kickedp-wave superconducting 23 Harper model, Phys. Rev. B95, 075136 (2017)

    work page 2017

  72. [72]

    Prodan and H

    E. Prodan and H. S.-Baldes, Bulk and Boundary In- variants for Complex Topological Insulators: From K- Theory to Physics, Mathematical Physics Studies (Vol. 185), Springer, Cham (2016)

    work page 2016

  73. [73]

    W. P. Su, J. R. Schrieffer, and A. J. Heeger, Solitons in Polyacetylene, Phys. Rev. Lett.42, 1698 (1979)

    work page 1979

  74. [74]

    Zak, Berry’s phase for energy bands in solids, Phys

    J. Zak, Berry’s phase for energy bands in solids, Phys. Rev. Lett.62, 2747 (1988)

    work page 1988

  75. [75]

    Delplace, D

    P. Delplace, D. Ullmo, and G. Montambaux, The Zak phase and the existence of edge states in graphene, Phys. Rev. B84, 195452 (2011)

    work page 2011

  76. [76]

    Jackiw and C

    R. Jackiw and C. Rebbi, Solitons with fermion number 1 2, Phys. Rev. D13, 3398 (1975)

    work page 1975

  77. [77]

    Wen, Y.-S

    X.-G. Wen, Y.-S. Wu, and Y. Hatsugai, Chiral opera- tor product algebra and edge excitations of a fractional quantum Hall droplet, Nucl. Phys. B422, 476 (1994)

    work page 1994

  78. [78]

    Qi, Y.-S

    X.-L. Qi, Y.-S. Wu, S.-C. Zhang, Topological quantiza- tion of the spin hall effect in two dimensional paramag- netic semiconductors, Phys. Rev. B74, 085308 (2006)

    work page 2006

  79. [79]

    Shen, Topological Insulators: Dirac Equation in Condensed Matter, 2nd ed., (Springer Singapore, 2017)

    S.-Q. Shen, Topological Insulators: Dirac Equation in Condensed Matter, 2nd ed., (Springer Singapore, 2017)

    work page 2017

  80. [80]

    Yang, Y.-M

    K.-Y. Yang, Y.-M. Lu, and Y. Ran, Quantum Hall ef- fects in a Weyl semimetal: Possible application in py- rochlore iridates, Phys. Rev. B84, 075129 (2011)

    work page 2011

Showing first 80 references.