Extended SSH Model: Non-Local Couplings and Non-Monotonous Edge States
Abstract
:1. Introduction
2. SSH Model With Long-Range Hopping
3. Properties of the Edge States
3.1. Method 1: Edge-State Wave Functions Incorporating Zak Phase
3.2. Method 2: Exact Solutions of Zero-Energy Edge States
4. Possible Physical Realization Using Array of Nanoparticles
5. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Appendix A. Second-Order Difference Equations With Constant Coefficients
References
- Kane, C.L.; Mele, E.J. Z2 Topological Order and the Quantum Spin Hall Effect. Phys. Rev. Lett. 2005, 95, 146802. [Google Scholar] [CrossRef] [PubMed]
- Bernevig, B.A.; Hughes, T.L.; Zhang, S.C. Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells. Science 2006, 314, 1757–1761. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Pesin, D.; MacDonald, A.H. Spintronics and pseudospintronics in graphene and topological insulators. Nat. Mater. 2012, 11, 409–416. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Vobornik, I.; Manju, U.; Fujii, J.; Borgatti, F.; Torelli, P.; Krizmancic, D.; Hor, Y.S.; Cava, R.J.; Panaccione, G. Magnetic Proximity Effect as a Pathway to Spintronic Applications of Topological Insulators. Nano Lett. 2011, 11, 4079–4082. [Google Scholar] [CrossRef] [PubMed]
- Sinova, J.; Zutic, I. New moves of the spintronics tango. Nat. Mater. 2012, 11, 368–371. [Google Scholar] [CrossRef] [PubMed]
- Liu, M.; Cai, Z.R.; Hu, S.; Luo, A.P.; Zhao, C.J.; Zhang, H.; Xu, W.C.; Luo, Z.C. Dissipative rogue waves induced by long-range chaotic multi-pulse interactions in a fiber laser with a topological insulator-deposited microfiber photonic device. Opt. Lett. 2015, 40, 4767–4770. [Google Scholar] [CrossRef] [PubMed]
- Alicea, J.; Oreg, Y.; Refael, G.; von Oppen, F.; Fisher, M.P.A. Non-Abelian statistics and topological quantum information processing in 1D wire networks. Nat. Phys. 2011, 7, 412–417. [Google Scholar] [CrossRef] [Green Version]
- Hassler, F.; Akhmerov, A.R.; Hou, C.Y.; Beenakker, C.W.J. Anyonic interferometry without anyons: How a flux qubit can read out a topological qubit. New J. Phys. 2010, 12, 125002. [Google Scholar] [CrossRef]
- Ferreira, G.J.; Loss, D. Magnetically Defined Qubits on 3D Topological Insulators. Phys. Rev. Lett. 2013, 111, 106802. [Google Scholar] [CrossRef] [PubMed]
- Leijnse, M.; Flensberg, K. Quantum Information Transfer between Topological and Spin Qubit Systems. Phys. Rev. Lett. 2011, 107, 210502. [Google Scholar] [CrossRef] [PubMed]
- Su, W.P.; Schrieffer, J.R.; Heeger, A.J. Solitons in Polyacetylene. Phys. Rev. Lett. 1979, 42, 1698–1701. [Google Scholar] [CrossRef]
- Pérez-González, B.; Bello, M.; Gómez-León, Á.; Platero, G. SSH model with long-range hoppings: Topology, driving and disorder. arXiv, 2018; arXiv:1802.03973. [Google Scholar]
- Lu, L.; Joannopoulos, J.D.; Soljacic, M. Topological states in photonic systems. Nat. Phys. 2016, 12, 626–629. [Google Scholar] [CrossRef] [Green Version]
- Lu, L.; Joannopoulos, J.D.; Soljacic, M. Topological photonics. Nat. Photonics 2014, 8, 821–829. [Google Scholar] [CrossRef] [Green Version]
- Slobozhanyuk, A.P.; Poddubny, A.N.; Miroshnichenko, A.E.; Belov, P.A.; Kivshar, Y.S. Subwavelength Topological Edge States in Optically Resonant Dielectric Structures. Phys. Rev. Lett. 2015, 114, 123901. [Google Scholar] [CrossRef] [PubMed]
- Noh, C.; Rodriguez-Lara, B.M.; Angelakis, D.G. Proposal for realization of the Majorana equation in a tabletop experiment. Phys. Rev. A 2013, 87, 040102. [Google Scholar] [CrossRef]
- Duca, L.; Li, T.; Reitter, M.; Bloch, I.; Schleier-Smith, M.; Schneider, U. An Aharonov-Bohm interferometer for determining Bloch band topology. Science 2015, 347, 288–292. [Google Scholar] [CrossRef] [PubMed]
- Jotzu, G.; Messer, M.; Desbuquois, R.; Lebrat, M.; Uehlinger, T.; Greif, D.; Esslinger, T. Experimental realization of the topological Haldane model with ultracold fermions. Nature 2014, 515, 237–240. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Goldman, N.; Juzeliunas, G.; Ohberg, P.; Spielman, I.B. Light-induced gauge fields for ultracold atoms. Rep. Prog. Phys. 2014, 77, 126401. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Poddubny, A.; Miroshnichenko, A.; Slobozhanyuk, A.; Kivshar, Y. Topological Majorana States in Zigzag Chains of Plasmonic Nanoparticles. ACS Photonics 2014, 1, 101–105. [Google Scholar] [CrossRef]
- Ling, C.W.; Xiao, M.; Chan, C.T.; Yu, S.F.; Fung, K.H. Topological edge plasmon modes between diatomic chains of plasmonic nanoparticles. Opt. Express 2015, 23, 2021–2031. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Liu, C.X.; Dutt, M.V.G.; Pekker, D. Robust manipulation of light using topologically protected plasmonic modes. Opt. Express 2018, 26, 2857–2872. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Downing, C.A.; Weick, G. Topological collective plasmons in bipartite chains of metallic nanoparticles. Phys. Rev. B 2017, 95, 125426. [Google Scholar] [CrossRef]
- Koch, J.; Houck, A.A.; Le Hur, K.; Girvin, S.M. Time-reversal-symmetry breaking in circuit-QED-based photon lattices. Phys. Rev. A 2010, 82, 043811. [Google Scholar] [CrossRef] [Green Version]
- Nunnenkamp, A.; Koch, J.; Girvin, S.M. Synthetic gauge fields and homodyne transmission in Jaynes-Cummings lattices. New J. Phys. 2011, 13, 095008. [Google Scholar] [CrossRef]
- Mei, F.; You, J.B.; Nie, W.; Fazio, R.; Zhu, S.L.; Kwek, L.C. Simulation and detection of photonic Chern insulators in a one-dimensional circuit-QED lattice. Phys. Rev. A 2015, 92, 041805. [Google Scholar] [CrossRef]
- Mei, F.; Xue, Z.Y.; Zhang, D.W.; Tian, L.; Lee, C.H.; Zhu, S.L. Witnessing topological Weyl semimetal phase in a minimal circuit-QED lattice. Quantum Sci. Technol. 2016, 1, 015006. [Google Scholar] [CrossRef] [Green Version]
- Yang, Z.H.; Wang, Y.P.; Xue, Z.Y.; Yang, W.L.; Hu, Y.; Gao, J.H.; Wu, Y. Circuit quantum electrodynamics simulator of flat band physics in a Lieb lattice. Phys. Rev. A 2016, 93, 062319. [Google Scholar] [CrossRef]
- Tangpanitanon, J.; Bastidas, V.M.; Al-Assam, S.; Roushan, P.; Jaksch, D.; Angelakis, D.G. Topological Pumping of Photons in Nonlinear Resonator Arrays. Phys. Rev. Lett. 2016, 117, 213603. [Google Scholar] [CrossRef] [PubMed]
- Delplace, P.; Ullmo, D.; Montambaux, G. Zak phase and the existence of edge states in graphene. Phys. Rev. B 2011, 84, 195452. [Google Scholar] [CrossRef]
- Asbóth, J.K.; Oroszlány, L.; Pályi, A. A Short Course on Topological Insulators: Band-structure topology and edge states in one and two dimensions. arXiv, 2015; arXiv:1509.02295. [Google Scholar]
- Sinev, I.S.; Mukhin, I.S.; Slobozhanyuk, A.P.; Poddubny, A.N.; Miroshnichenko, A.E.; Samusev, A.K.; Kivshar, Y.S. Mapping plasmonic topological states at the nanoscale. Nanoscale 2015, 7, 11904–11908. [Google Scholar] [CrossRef] [PubMed]
- Cognet, L.; Tardin, C.; Boyer, D.; Choquet, D.; Tamarat, P.; Lounis, B. Single metallic nanoparticle imaging for protein detection in cells. Proc. Natl. Acad. Sci. USA 2003, 100, 11350–11355. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Assanov, G.S.; Zhanabaev, Z.Z.; Govorov, A.O.; Neiman, A.B. Modelling of photo-thermal control of biological cellular oscillators. Eur. Phys. J. Spec. Top. 2013, 222, 2697–2704. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Richardson, H.H.; Carlson, M.T.; Tandler, P.J.; Hernandez, P.; Govorov, A.O. Experimental and Theoretical Studies of Light-to-Heat Conversion and Collective Heating Effects in Metal Nanoparticle Solutions. Nano Lett. 2009, 9, 1139–1146. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Capobianco, J.A.; Vetrone, F.; D’Alesio, T.; Tessari, G.; Speghini, A.; Bettinelli, M. Optical spectroscopy of nanocrystalline cubic Y2O3: Er3+ obtained by combustion synthesis. Phys. Chem. Chem. Phys. 2000, 2, 3203–3207. [Google Scholar] [CrossRef]
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Li, C.; Miroshnichenko, A.E. Extended SSH Model: Non-Local Couplings and Non-Monotonous Edge States. Physics 2019, 1, 2-16. https://doi.org/10.3390/physics1010002
Li C, Miroshnichenko AE. Extended SSH Model: Non-Local Couplings and Non-Monotonous Edge States. Physics. 2019; 1(1):2-16. https://doi.org/10.3390/physics1010002
Chicago/Turabian StyleLi, Chao, and Andrey E. Miroshnichenko. 2019. "Extended SSH Model: Non-Local Couplings and Non-Monotonous Edge States" Physics 1, no. 1: 2-16. https://doi.org/10.3390/physics1010002