The Dirac equation as a model of topological insulators

References

• R. Jackiw and C. Rebbi, Solitons with Fermion number 1/2, Phys. Rev. D 13 (1976), pp. 3398–3409. doi: 10.1103/PhysRevD.13.3398
   Web of Science ®Google Scholar
• L. Fu, C.L. Kane and E.J. Mele, Topological insulators in three dimensions, Phys. Rev. Lett. 98 (2007), pp. 106803. doi: 10.1103/PhysRevLett.98.106803
   PubMed Web of Science ®Google Scholar
• J.L. Moore and L. Balents, Topological invariants of time-reversal invariant band structures, Phys. Rev. B 75 (2007), pp. 121306(R). doi: 10.1103/PhysRevB.75.121306
   Web of Science ®Google Scholar
• R. Roy, Topological phases and the quantum spin-hall effect in three-dimensions, Phys. Rev. B 79 (2009), pp. 95322.
   Web of Science ®Google Scholar
• E.P. Wigner, The time reversal operation in quantum mechanics, Nachr. Akad. Ges. Wiss. Göttingen 31 (1932), pp. 546–559.
   Google Scholar
• H.A. Kramers, General theory of the paramagnetic rotation in crystals, Proc. Amsterdam Acad. 33 (1930), pp. 959.
   Google Scholar
• L. Fu and C.L. Kane, Topological insulators with inversion symmetry, Phys. Rev. B 76 (2007), pp. 045302. doi: 10.1103/PhysRevB.76.045302
   Web of Science ®Google Scholar
• F.D.M. Haldane, Model for a quantum hall effect without Landau levels, condensed matter-realization of the parity anomaly, Phys. Rev. Lett. 61 (1988), pp. 2015–2018. doi: 10.1103/PhysRevLett.61.2015
   PubMed Web of Science ®Google Scholar
• C.L. Kane and E.J. Mele, Z(2) topological order and the quantum spin hall effect, Phys. Rev. Lett. 95 (2005), pp. 146802. doi: 10.1103/PhysRevLett.95.146802
   PubMed Web of Science ®Google Scholar
• A.J. Niemi and G.W. Semenoff, Fermion number fractionalization in quantum-field theory, Phys. Rep. 135 (1986), pp. 99–193. doi: 10.1016/0370-1573(86)90167-5
   Google Scholar
• W.P. Su, J.R. Schrieffer and A.J. Heeger, Solitons in polyacetylene, Phys. Rev. Lett. 42 (1979), pp. 1698–1701. doi: 10.1103/PhysRevLett.42.1698
   Web of Science ®Google Scholar
• W.P. Su, J.R. Schrieffer and A.J. Heeger, Soliton excitations in polyactelylene, Phys. Rev. B 22 (1980), pp. 2099–2111. doi: 10.1103/PhysRevB.22.2099
   Web of Science ®Google Scholar
• D.J. Thouless, Quantization of particle transport, Phys. Rev. B 27 (1983), pp. 6083–6087. doi: 10.1103/PhysRevB.27.6083
   Web of Science ®Google Scholar
• R.B. Laughlin, Quantized hall conductivity in two-dimensions, Phys. Rev. B 23 (1981), pp. 5632–5633. doi: 10.1103/PhysRevB.23.5632
   Web of Science ®Google Scholar
• E.I. Rashba and V.I. Sheka, Symmetry of bands with spin-orbit interaction included, Fiz. Tverd. Tela. 2 (1959), pp. 162–176.
   Google Scholar
• M.V. Berry, Quantal phase-factors acompanying adiabatic changes, Proc. R. Soc. Lond. 392 (1984), pp. 45–57.
   Web of Science ®Google Scholar
• D. Hsieh, Y. Xia, L. Wray, D. Qian, A. Pal, J.H. Dil, J. Osterwalder, F. Meier, G. Bihlmayer, C.L. Kane, Y.S. Hor, R.J. Cava and M.Z. Hasan, Observation of unconventional quantum spin textures in topological insulators, Science 323 (2009), pp. 919–922. doi: 10.1126/science.1167733
   PubMed Web of Science ®Google Scholar
• A. Nishide, A.A. Taskin, Y. Takeichi, T. Okuda, A. Kakizaki, T. Hirahara, K. Nakatsuji, F. Komori, Y Ando and I. Matsuda, Direct mapping of the spin-filtered surface bands of a three-dimensional quantum spin hall insulator, Phys. Rev. B 81 (2010), pp. 041309. doi: 10.1103/PhysRevB.81.041309
   Web of Science ®Google Scholar
• Y.H. Wang, D. Hsieh, D. Pilon, L. Fu, D.R. Gardner, Y.S. Lee and N. Gedik, Observation of a warped helical spin texture in Bi<Se< from circular dichroism angle-resolved photoemission spectroscopy, Phys. Rev. Lett. 107 (2011), pp. 207602. doi: 10.1103/PhysRevLett.107.207602
   Google Scholar
• Z.H. Pan, E. Vescovo, A.V. Fedorov, D. Gardner, Y.S. Lee, S. Chu, G.D. Gu and T. Valla, Electronic structure of the topological insulator Bi<Se< using angle-resolved photoemission spectroscopy: Evidence for a nearly full surface spin polarization, Phys. Rev. Lett. 106 (2011), pp. 257004. doi: 10.1103/PhysRevLett.106.257004
   Google Scholar
• H.B. Nielsen and M. Ninomiya, The Adler-Bell-Jackiw anomaly and Weyl fermions in a crystal, Phys. Lett. B 130 (1983), pp. 389–396. doi: 10.1016/0370-2693(83)91529-0
   Web of Science ®Google Scholar
• Y. Aharonov and D. Bohm, Significance of electromagnetic potentials in the quantum theory, Phys. Rev. 115 (1959), pp. 485–491. doi: 10.1103/PhysRev.115.485
   Web of Science ®Google Scholar
• R.G. Chambers, Shift of interference pattern by enclosed magnetic flux, Phys. Rev. Lett. 5 (1960), pp. 3–5. doi: 10.1103/PhysRevLett.5.3
   Web of Science ®Google Scholar
• H. Boersch, H. Hamisch, K. Grohmann and D. Wohlleben, Experimenteller Nachweis Der Phasenscheibung Von Elektronwellen Durch Das Magnetische Vektorpotential, Z. Phys. 165 (1961), pp. 79–93. doi: 10.1007/BF01378089
   Google Scholar
• Y. Aharonov and L. Susskind, Observable sign change of spinors under <! rotations, Phys. Rev. 158 (1967), pp. 1237–1238. doi: 10.1103/PhysRev.158.1237
   Google Scholar
• S.S. Chern, A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds, Ann. Math. 45 (1944), pp. 747–752. doi: 10.2307/1969302
   Google Scholar
• S.S. Chern, On the curvatura integra in a Riemannian manifold, Ann. Math. 46 (1945), pp. 674–684. doi: 10.2307/1969203
   Web of Science ®Google Scholar
• M.F. Atiyah and I.M. Singer, Index of elliptic operators, Ann. Math. 87 (1968), pp. 484–530. doi: 10.2307/1970715
   Web of Science ®Google Scholar
• L. Onsager, Interpretation of the van Alphen–de Haas effect, Phil. Mag. 43 (1952), pp. 1006–1008. doi: 10.1080/14786440908521019
   Google Scholar
• M. Wilkinson, An example of phase holonomy in WKB theory, J. Phys. A 17 (1984), pp. 3459–3476. doi: 10.1088/0305-4470/17/18/016
   Google Scholar
• A.A. Taskin and Y. Ando, Quantum oscillations in a topological insulator Bi<Sb<, Phys. Rev. B 80 (2009), pp. 085303. doi: 10.1103/PhysRevB.80.085303
   Web of Science ®Google Scholar
• P. Cheng, C. Song, T. Zhang, Y. Zhang, Y. Wang, J.-F. Jia, J. Wang, Y. Wang, B.-F. Zhu, X. Chen, X. Ma, K. He, L. Wang, X. Dai, Z. Fang, X. Xie, X.-L. Qi, C.-X. Liu, S.-C. Zhang and Q.-K. Xue, Landau quantization of topological surface states in Bi<Se<, Phys. Rev. Lett. 105 (2010), pp. 076801. doi: 10.1103/PhysRevLett.105.076801
   PubMed Web of Science ®Google Scholar
• D.X. Qu, Y.S. Hor, J. Xiong, R.J. Cava and N.P. Ong, Quantum oscillations and hall anomaly of surface states in the topological insulator Bi<Te<, Science 329 (2010), pp. 821–824. doi: 10.1126/science.1189792
   PubMed Web of Science ®Google Scholar
• L.H. Bao, L. He, N. Meyer, X.F. Kou, P. Zhang, Z.G. Chen, A.V. Fedorov, J. Zou, T.M. Riedemann, T.A. Lograsso, K.L. Wang, G. Tuttle and F.X. Xiu, Weak anti-localization and quantum oscillations of surface states in topological insulator Bi<Se<Te, Sci. Rep. 2 (2012), pp. 726. doi: 10.1038/srep00726
   Google Scholar
• D.J. Thouless, M. Kohmoto, M.P. Nightingale and M. den Nijs, Quantized hall conductance in a two-dimensional periodic potential, Phys. Rev. Lett. 49 (1982), pp. 405–408. doi: 10.1103/PhysRevLett.49.405
   Web of Science ®Google Scholar
• K.S. Novoselov, A.E. Geim, S.V. Morozov, D. Jiang, M.I. Katsnelson, I.V. Grigorieva, S.V. Dubonos and A.A. Firsov, Two-dimensional gas of massless Dirac fermions in graphene, Nature 438 (2005), pp. 197–200. doi: 10.1038/nature04233
   PubMed Web of Science ®Google Scholar
• Y.B. Zhang, Y.W. Tan, H.L. Stormer and P. Kim, Experimental observation of the quantum hall effect and Berry's phase in graphene, Nature 438 (2005), pp. 201–204. doi: 10.1038/nature04235
   PubMed Web of Science ®Google Scholar
• V.P. Gusynin and S.G. Sharapov, Unconventional integer quantum hall effect in graphene, Phys. Rev. Lett. 95 (2005), pp. 146801. doi: 10.1103/PhysRevLett.95.146801
   PubMed Web of Science ®Google Scholar
• B. Simon, Holonomy, the quantum adiabatic theorem and Berry's phase, Phys. Rev. Lett. 51 (1983), pp. 2167–2170. doi: 10.1103/PhysRevLett.51.2167
   Web of Science ®Google Scholar
• P.G. Harper, Single band motion of conduction electrons in a uniform magnetic field, Proc. Phys. Soc. London, A 68 (1955), pp. 874–878. doi: 10.1088/0370-1298/68/10/304
   Google Scholar
• D.R. Hofstadter, Energy-levels and wave-functions of Bloch electrons in rational and irrational magnetic fields, Phys. Rev. B 14 (1976), pp. 2239–2249. doi: 10.1103/PhysRevB.14.2239
   Web of Science ®Google Scholar
• X. Zhang, J. Wang and S.C. Zhang, Topological insulators for high-performance terahertz to infra-red applications, Phys. Rev. B 82 (2010), pp. 245107. doi: 10.1103/PhysRevB.82.245107
   Web of Science ®Google Scholar
• P. Di Pietro, F.M. Vitucci, D. Nicoletti, L. Baldassarre, P. Calvani, R. Cava, Y.S. Hor, U. Schade and S. Lupi, Optical conductivity of bismuth-based topological insulators, Phys. Rev. B 86 (2012), pp. 045439. doi: 10.1103/PhysRevB.86.045439
   Web of Science ®Google Scholar
• W.K. Tse and A.H. MacDonald, Giant magneto-optical Kerr effect and universal Faraday effect in thin-film topological insulators, Phys. Rev. Lett. 105 (2010), pp. 057401. doi: 10.1103/PhysRevLett.105.057401
   PubMed Web of Science ®Google Scholar
• J. Maciejko, X.L. Qi, H.D. Drew and S.C. Zhang, Topological quantization in units of the fine structure constant, Phys. Rev. Lett. 105 (2010), pp. 166803. doi: 10.1103/PhysRevLett.105.166803
   PubMed Web of Science ®Google Scholar
• L. Wu, M. Salehi, N. Koirala, J. Moon, S. Oh and N.P. Armitage, Quantized Faraday and Kerr rotation and axion electrodynamics of a 3D topological insulator, Science 354 (2016), pp. 1124–1127. doi: 10.1126/science.aaf5541
   PubMed Web of Science ®Google Scholar
• T. Morimoto and N. Nagaosa, Topological nature of nonlinear optical effects in solids, Sci. Adv. 2 (2016), pp. e1501524. doi: 10.1126/sciadv.1501524
   PubMed Web of Science ®Google Scholar