Emerging topological bound states in Haldane model zigzag nanoribbons
Source abstract
Abstract
Abstract Zigzag nanoribbons hosting the Haldane Chern insulator model are considered. In this context, a reentrant topological phase, characterized by the emergence of quasi zero dimensional in-gap states, is discussed. The bound states, which reside in the gap opened by the hybridization of the counter-propagating edge modes of the Haldane phase, are localized at the ends of the strip and are found to be robust against on-site disorder. These findings are supported by the behavior of the Zak phase over the parameter space, which exhibits jumps of π in correspondence to the phase transitions between the trivial and the non-trivial phases. The effective mass inversion leading to the jumps in the Zak phase is interpreted in a low energy framework. Setups with non-uniform parameters also show topological bound states via the Jackiw-Rebbi mechanism. All the properties reported are shown to be extremely sensitive to the strip width.
Source: crossref · Quality: baseline
Research context
OpenAlex topics: Topological Materials and Phenomena · Quantum many-body systems · Graphene research and applications
SciVX discovery tags are terminology-based. OpenAlex topics are source-provided classifications.
References (59)
Show source-provided references
1. Qi, X.-L. & Zhang, S.-C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011).10.1103/RevModPhys.83.1057 ↗
2. Bernevig, B. A., Hughes, T. L. & Zhang, S.-C. Quantum spin hall effect and topological phase transition in hgte quantum wells. Science 314, 1757–1761 (2006).10.1126/science.1133734 ↗
3. Ryu, S., Schnyder, A. P., Furusaki, A. & Ludwig, A. W. W. Topological insulators and superconductors: tenfold way and dimensional hierarchy. New J. Phys. 12, 065010 (2010).10.1088/1367-2630/12/6/065010 ↗
4. Schindler, F. et al. Higher-order topological insulators. Sci. Adv. 4, eaat0346 (2018).10.1126/sciadv.aat0346 ↗
5. Bergholtz, E. J., Budich, J. C. & Kunst, F. K. Exceptional topology of non-hermitian systems. Rev. Mod. Phys. 93, 015005 (2021).10.1103/RevModPhys.93.015005 ↗
6. Strunz, J. et al. Interacting topological edge channels. Nat. Phys. 16, 83–88 (2020).10.1038/s41567-019-0692-4 ↗
7. Stühler, R. et al. Effective lifting of the topological protection of quantum spin hall edge states by edge coupling. Nat. Commun. 13, 3480 (2022).10.1038/s41467-022-30996-z ↗
8. Zhou, B., Lu, H.-Z., Chu, R.-L., Shen, S.-Q. & Niu, Q. Finite size effects on helical edge states in a quantum spin-hall system. Phys. Rev. Lett. 101, 246807 (2008).10.1103/PhysRevLett.101.246807 ↗
9. Kobayashi, K., Yoshimura, Y., Imura, K.-I. & Ohtsuki, T. Dimensional crossover of transport characteristics in topological insulator nanofilms. Phys. Rev. B 92, 235407 (2015).10.1103/PhysRevB.92.235407 ↗
10. Matsugatani, A. & Watanabe, H. Connecting higher-order topological insulators to lower-dimensional topological insulators. Phys. Rev. B 98, 205129 (2018).10.1103/PhysRevB.98.205129 ↗
11. Väyrynen, J. I. & Ojanen, T. Chiral topological phases and fractional domain wall excitations in one-dimensional chains and wires. Phys. Rev. Lett. 107, 166804 (2011).10.1103/PhysRevLett.107.166804 ↗
12. Rodriguez, R. H. et al. Relaxation and revival of quasiparticles injected in an interacting quantum hall liquid. Nat. Commun. 11, 2426 (2020).10.1038/s41467-020-16331-4 ↗
13. Fleckenstein, C., Ziani, N. T., Calzona, A., Sassetti, M. & Trauzettel, B. Formation and detection of majorana modes in quantum spin hall trenches. Phys. Rev. B 103, 125303 (2021).10.1103/PhysRevB.103.125303 ↗
14. Traverso, S., Sassetti, M. & Ziani, N. T. Role of the edges in a quasicrystalline haldane model. Phys. Rev. B 106, 125428 (2022).10.1103/PhysRevB.106.125428 ↗
15. Vigliotti, L. et al. Effects of the spatial extension of the edge channels on the interference pattern of a helical josephson junction. Nanomaterials 13, 569 (2023).10.3390/nano13030569 ↗
16. Choi, S.-J. & Trauzettel, B. Stacking-induced symmetry-protected topological phase transitions. Phys. Rev. B 107, 245409 (2023).10.1103/PhysRevB.107.245409 ↗
17. Zhu, D., Kheirkhah, M. & Yan, Z. Sublattice-enriched tunability of bound states in second-order topological insulators and superconductors. Phys. Rev. B 107, 085407 (2023).10.1103/PhysRevB.107.085407 ↗
18. Saha, S., Nag, T. & Mandal, S. Multiple higher-order topological phases with even and odd pairs of zero-energy corner modes in a c3symmetry broken model. Europhys. Lett. 142, 56002 (2023).10.1209/0295-5075/acd71a ↗
19. Potter, A. C. & Lee, P. A. Multichannel generalization of kitaev’s majorana end states and a practical route to realize them in thin films. Phys. Rev. Lett. 105, 227003 (2010).10.1103/PhysRevLett.105.227003 ↗
20. Zhang, J. et al. Observation of dimension-crossover of a tunable 1d dirac fermion in topological semimetal nbsixte2. npj Quantum Mater. 7, 54 (2022).10.1038/s41535-022-00462-6 ↗
21. Cao, T., Zhao, F. & Louie, S. G. Topological phases in graphene nanoribbons: Junction states, spin centers, and quantum spin chains. Phys. Rev. Lett. 119, 076401 (2017).10.1103/PhysRevLett.119.076401 ↗
22. Li, J. et al. Topological phase transition in chiral graphene nanoribbons: from edge bands to end states. Nat. Commun. 12, 5538 (2021).10.1038/s41467-021-25688-z ↗
23. Boyd, E. E. & Westervelt, R. M. Extracting the density profile of an electronic wave function in a quantum dot. Phys. Rev. B 84, 205308 (2011).10.1103/PhysRevB.84.205308 ↗
24. Ziani, N. T., Cavaliere, F. & Sassetti, M. Theory of the stm detection of wigner molecules in spin-incoherent cnts. Europhys. Lett. 102, 47006 (2013).10.1209/0295-5075/102/47006 ↗
25. Gambetta, F. M., Ziani, N. T., Barbarino, S., Cavaliere, F. & Sassetti, M. Anomalous friedel oscillations in a quasihelical quantum dot. Phys. Rev. B 91, 235421 (2015).10.1103/PhysRevB.91.235421 ↗
26. Klitzing, K. V., Dorda, G. & Pepper, M. New method for high-accuracy determination of the fine-structure constant based on quantized hall resistance. Phys. Rev. Lett. 45, 494–497 (1980).10.1103/PhysRevLett.45.494 ↗
27. Haldane, F. D. M. Model for a quantum hall effect without landau levels: condensed-matter realization of the “parity anomaly”. Phys. Rev. Lett. 61, 2015–2018 (1988).10.1103/PhysRevLett.61.2015 ↗
28. Kane, C. L. & Mele, E. J. Quantum spin hall effect in graphene. Phys. Rev. Lett. 95, 226801 (2005).10.1103/PhysRevLett.95.226801 ↗
29. Kane, C. L. & Mele, E. J. Z2 topological order and the quantum spin hall effect. Phys. Rev. Lett. 95, 146802 (2005).10.1103/PhysRevLett.95.146802 ↗
30. Reis, F. et al. Bismuthene on a sic substrate: a candidate for a high-temperature quantum spin hall material. Science 357, 287–290 (2017).10.1126/science.aai8142 ↗
31. Bampoulis, P. et al. Quantum spin hall states and topological phase transition in germanene. Phys. Rev. Lett. 130, 196401 (2023).10.1103/PhysRevLett.130.196401 ↗
32. Zak, J. Berry’s phase for energy bands in solids. Phys. Rev. Lett. 62, 2747–2750 (1989).10.1103/PhysRevLett.62.2747 ↗
33. Rhim, J.-W., Behrends, J. & Bardarson, J. H. Bulk-boundary correspondence from the intercellular zak phase. Phys. Rev. B 95, 035421 (2017).10.1103/PhysRevB.95.035421 ↗
34. Jeong, S.-G. & Kim, T.-H. Topological and trivial domain wall states in engineered atomic chains. npj Quantum Mater. 7, 22 (2022).10.1038/s41535-022-00424-y ↗
35. Jackiw, R. & Rebbi, C. Solitons with fermion number $$\frac{1}{2}$$. Phys. Rev. D 13, 3398–3409 (1976).10.1103/PhysRevD.13.3398 ↗
36. Qi, X.-L., Hughes, T. L. & Zhang, S.-C. Fractional charge and quantized current in the quantum spin hall state. Nat. Phys. 4, 273–276 (2008).10.1038/nphys913 ↗
37. Traverso Ziani, N., Fleckenstein, C., Dolcetto, G. & Trauzettel, B. Fractional charge oscillations in quantum dots with quantum spin hall effect. Phys. Rev. B 95, 205418 (2017).10.1103/PhysRevB.95.205418 ↗
38. Fu, B., Zou, J.-Y., Hu, Z.-A., Wang, H.-W. & Shen, S.-Q. Quantum anomalous semimetals. npj Quantum Mater. 7, 94 (2022).10.1038/s41535-022-00503-0 ↗
39. Cheng, S.-g, Wu, Y., Jiang, H., Sun, Q.-F. & Xie, X. C. Transport measurement of fractional charges in topological models. npj Quantum Mater. 8, 30 (2023).10.1038/s41535-023-00558-7 ↗
40. Bernevig, B. & Hughes, T.Topological Insulators and Topological Superconductors (Princeton University Press, 2013).10.1515/9781400846733 ↗
41. Hao, N., Zhang, P., Wang, Z., Zhang, W. & Wang, Y. Topological edge states and quantum hall effect in the haldane model. Phys. Rev. B 78, 075438 (2008).10.1103/PhysRevB.78.075438 ↗
42. Doh, H. & Jeon, G. S. Bifurcation of the edge-state width in a two-dimensional topological insulator. Phys. Rev. B 88, 245115 (2013).10.1103/PhysRevB.88.245115 ↗
43. Cano-Cortés, L., Ortix, C. & van den Brink, J. Fundamental differences between quantum spin hall edge states at zigzag and armchair terminations of honeycomb and ruby nets. Phys. Rev. Lett. 111, 146801 (2013).10.1103/PhysRevLett.111.146801 ↗
44. Ohyama, Y., Tsuchiura, H. & Sakuma, A. Finite size effects on the quantum spin hall state. J. Phys. Conf. Ser. 266, 012103 (2011).10.1088/1742-6596/266/1/012103 ↗
45. Berry, M. V. Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. London. A. Math. Phys. Sci. 392, 45–57 (1984).
46. Resta, R. Macroscopic polarization in crystalline dielectrics: the geometric phase approach. Rev. Mod. Phys. 66, 899–915 (1994).10.1103/RevModPhys.66.899 ↗
47. Yusufaly, T., Vanderbilt, D. & Coh, S. Tight-binding formalism in the context of the pythtb package. https://www.physics.rutgers.edu/pythtb/formalism.html.
48. Vanderbilt, D. Berry Phases in Electronic Structure Theory: Electric Polarization, Orbital Magnetization and Topological Insulators (Cambridge University Press, 2018).10.1017/9781316662205 ↗
49. Atala, M. et al. Direct measurement of the zak phase in topological bloch bands. Nat. Phys. 9, 795–800 (2013).10.1038/nphys2790 ↗
50. Cayssol, J. & Fuchs, J. N. Topological and geometrical aspects of band theory. J. Phys. Mater. 4, 034007 (2021).10.1088/2515-7639/abf0b5 ↗
51. Ziani, N. T., Fleckenstein, C., Vigliotti, L., Trauzettel, B. & Sassetti, M. From fractional solitons to majorana fermions in a paradigmatic model of topological superconductivity. Phys. Rev. B 101, 195303 (2020).10.1103/PhysRevB.101.195303 ↗
52. Zhou, B. & Shen, S.-Q. Crossover from majorana edge- to end-states in quasi-one-dimensional p-wave superconductors. Phys. Rev. B 84, 054532 (2011).10.1103/PhysRevB.84.054532 ↗
53. Kitaev, A. Y. Unpaired majorana fermions in quantum wires. Phys-Usp. 44, 131 (2001).10.1070/1063-7869/44/10S/S29 ↗
54. Albrecht, S. M. et al. Exponential protection of zero modes in majorana islands. Nature 531, 206–209 (2016).10.1038/nature17162 ↗
55. Mourik, V. et al. Signatures of majorana fermions in hybrid superconductor-semiconductor nanowire devices. Science 336, 1003–1007 (2012).10.1126/science.1222360 ↗
56. Fendley, P. Parafermionic edge zero modes in zn-invariant spin chains. J. Stat. Mech. 2012, P11020 (2012).10.1088/1742-5468/2012/11/P11020 ↗
57. Alicea, J. & Fendley, P. Topological phases with parafermions: theory and blueprints. Annu. Rev. Condens. Matter Phys. 7, 119–139 (2016).10.1146/annurev-conmatphys-031115-011336 ↗
58. Calzona, A., Meng, T., Sassetti, M. & Schmidt, T. L. F4 parafermions in one-dimensional fermionic lattices. Phys. Rev. B 98, 201110 (2018).10.1103/PhysRevB.98.201110 ↗
59. Moldovan, D., Anđelković, M. & Peeters, F. pybinding v0.9.5: a Python package for tight- binding calculations https://doi.org/10.5281/zenodo.4010216 (2020). This work was supported by the Flemish Science Foundation (FWO-Vl) and the Methusalem Funding of the Flemish Government.10.5281/zenodo.4010216 ↗