Topologically Protected States in One-Dimensional Systems
About this Title
C. L. Fefferman, J. P. Lee-Thorp and M. I. Weinstein
Publication: Memoirs of the American Mathematical Society
Publication Year:
2017; Volume 247, Number 1173
ISBNs: 978-1-4704-2323-0 (print); 978-1-4704-3707-7 (online)
DOI: https://doi.org/10.1090/memo/1173
Published electronically: February 1, 2017
Keywords:Schrödinger equation, Dirac equation, Floquet-Bloch theory, topological
protection, edge states, Hill’s equation, domain wall
Table of Contents
Chapters
- Chapter 1. Introduction and Outline
- Chapter 2. Floquet-Bloch and Fourier Analysis
- Chapter 3. Dirac Points of 1D Periodic Structures
- Chapter 4. Domain Wall Modulated Periodic Hamiltonian and Formal Derivation of Topologically Protected Bound States
- Chapter 5. Main Theorem—Bifurcation of Topologically Protected States
- Chapter 6. Proof of the Main Theorem
- Appendix A. A Variant of Poisson Summation
- Appendix B. 1D Dirac points and Floquet-Bloch Eigenfunctions
- Appendix C. Dirac Points for Small Amplitude Potentials
- Appendix D. Genericity of Dirac Points - 1D and 2D cases
- Appendix E. Degeneracy Lifting at Quasi-momentum Zero
- Appendix F. Gap Opening Due to Breaking of Inversion Symmetry
- Appendix G. Bounds on Leading Order Terms in Multiple Scale Expansion
- Appendix H. Derivation of Key Bounds and Limiting Relations in the Lyapunov-Schmidt Reduction
Abstract
We study a class of periodic Schrödinger operators, which in distinguished cases can be proved to have linear band-crossings or âĂIJDirac pointsâĂİ. We then show that the introduction of an âĂIJedgeâĂİ, via adiabatic modulation of these periodic potentials by a domain wall, results in the bifurcation of spatially localized âĂIJedge statesâĂİ. These bound states are associated with the topologically protected zero-energy mode of an asymptotic one-dimensional Dirac operator. Our model captures many aspects of the phenomenon of topologically protected edge states for two-dimensional bulk structures such as the honeycomb structure of graphene. The states we construct can be realized as highly robust TM-electromagnetic modes for a class of photonic waveguides with a phase-defect.
- [1] J. C. Avila, H. Schulz-Baldes, and C. Villegas-Blas, Topological invariants of edge states for periodic two-dimensional models, Mathematical Physics, Analysis and Geometry 16 (2013), 136\ndash170.
- [2] D. I. Borisov and R. R. Gadyl’shin, On the spectrum of a periodic operator with a small localized perturbation, Izvestiya: Mathematics 72 (2008), no. 4, 659\ndash688.
- [3] J. Bronski and Z. Rapti, Counting defect modes in periodic eigenvalue problems, SIAM J. Math. Anal. 43 (2011), no. 2, 803\ndash827.
- [4] D. D’Ancona and Fanelli L., $L^p$-boundedness of the wave operator for the one dimensional Schrödinger operator, Commun. Math. Sci. 268 (2006), no. 415.
- [5] P. A. Deift and R. Hempel, On the existence of eigenvalues of the schrödinger operator h-$\lambda $w in a gap of $\sigma $(h), Commun. Math. Phys. 103 (1986), no. 3, 461\ndash490.
- [6] V. Duchêne, J. L. Marzuloa, and M. I. Weinstein, Wave operator bounds for one-dimensional Schrödinger operators with singular potentials and applications, J. Math. Phys. 52 (2011), no. 013505.
- [7] V. Duchêne, I. Vukićević, and M. I. Weinstein, Scattering and localization properties of highly oscillatory potentials, Comm. Pure Appl. Math. 1 (2014), 83\ndash128.
- [8] V. Duchêne, I. Vukićević, and M. I. Weinstein, Homogenized description of defect modes in periodic structures with localized defects, Commun. Math. Sci. (to appear).
- [9] M. Eastham, Spectral theory of periodic differential equations, Hafner Press, 1974.
- [10] C. L. Fefferman, J. P. Lee-Thorp, and M. I. Weinstein, Topologically protected states in one-dimensional continuous systems and dirac points, Proceedings of the National Academy of Sciences (2014), 201407391.
- [11] C. L. Fefferman and M. I. Weinstein, Honeycomb lattice potentials and Dirac points, J. Amer. Math. Soc. 25 (2012), no. 4, 1169\ndash1220.
- [12] C. L. Fefferman and M. I. Weinstein, Wave packets in honeycomb lattice structures and two-dimensional Dirac equations, Commun. Math. Phys. 326 (2014), 251\ndash286.
- [13] A. Figotin and A. Klein, Localized classical waves created by defects, J. Statistical Phys. 86 (1997), no. 1/2, 165\ndash177.
- [14] J.-M. Graf and M. Porta, Bulk-edge correspondence for two-dimensional topological insulators, Comm. Math. Phys. 324 (2012), no. 3, 851\ndash895.
- [15] F. D. M. Haldane and S. Raghu, Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry, Phys. Rev. Lett. 100 (2008), no. 1, 013904.
- [16] B. I. Halperin, Quantized Hall conductance, current-carrying edge states, and the existence of extended states in a two-dimensional disordered potential, Phys. Rev. B 25 (1982), no. 4, 2185\ndash2190.
- [17] M. Z. Hasan and C. L. Kane, Colloquium: Topological Insulators, Reviews of Modern Physics 82 (2010), 3045.
- [18] Y. Hatsugai, The Chern number and edge states in the integer quantum hall effect, Phys. Rev. Lett. 71 (1993), 3697\ndash3700.
- [19] R. Jackiw and C. Rebbi, Solitons with fermion number $1/2$, Phys. Rev. D 13 (1976), no. 12, 3398\ndash3409.
- [20] R. P. Boas Jr., Poisson’s summation formula in $L^2$, J. London Math. Soc. 21 (1946), 102\ndash105.
- [21] C. L. Kane and E. J. Mele, $\mathbb {Z}_2$ topological order and the quantum spin Hall efect, Phys. Rev. Lett. 95 (2005), 146802.
- [22] M. Katsnelson, Graphene: Carbon in two dimensions, Cambridge University Press, 2012.
- [23] K. Knopp, Theory of functions - part 2; applications and further development of the general theory, Dover, 1947.
- [24] S. Krantz, Function theory of several complex variables,, AMS Chelsea Publishing, Providence, Rhode Island, 1992.
- [25] J. P. Lee-Thorp, I. Vukićević, J. Yang, X. Xu, C. L. Fefferman, C. W. Wong, and M. I. Weinstein, Photonic realization of topologically protected bound states in waveguide arrays, submitted.
- [26] W. Magnus and S. Winkler, Hill’s equation, Courier Dover Publications, 2013.
- [27] N. Malkova, I. Hromada, X. Wang, G. Bryant, and Z. Chen, Observation of optical shockley-like surface states in photonic superlattices, Opt. Express 34 (2009), no. 11, 1633\ndash1635.
- [28] J. Meixner and R.W. Schäfke, Mathieusche funktionen und sphäroidfunktionen, Springer-Verlag, Berline-Göttingen-Heidelberg, 1954.
- [29] J. Moser, An example of a Schrödinger equation with almost periodic potential and nowhere dense spectrum, Comment. Math. Helvetici 56 (1981), 198\ndash224.
- [30] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, The electronic properties of graphene, Reviews of Modern Physics 81 (2009), 109\ndash162.
- [31] S. Raghu and F. D. M. Haldane, Analogs of quantum-Hall-effect edge states in photonic crystals, Phys. Rev. A 78 (2008), no. 3, 033834.
- [32] M. C. Rechstman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, Photonic Floquet topological insulators, Nature 496 (2013), 196.
- [33] M. C. Rechtsman, Y. Plotnik, J. M. Zeuner, D. Song, Z. Chen, A. Szameit, and M. Segev, Topological creation and destruction of edge states in photonic graphene, Phys. Rev. Lett. 111 (2013), 103901.
- [34] M. Reed and B. Simon, Methods of modern mathematical physics: Analysis of operators, volume iv, Academic Press, 1978.
- [35] F. S. Rofe-Beketov and A. M. Kholkin, Spectral analysis of differential operators - interplay between spectral and oscillatory properties, World Scientific Monograph Series in Mathematics, vol. 7, World Scientific, Singapore, 2005.
- [36] W. Shockley, On the surface states associated with a periodic potential, Phys. Rev. 56 (1939).
- [37] B. Simon, The bound state of weakly coupled Schrödinger operators in one and two dimensions, Annals of Physics 97 (1976), no. 2, 279\ndash288.
- [38] W. P. Su, J. R. Schrieffer, and A. J. Heeger, Solitons in polyacetylene, Phys. Rev. Lett. 42 (1979), 1698\ndash1701.
- [39] I. E. Tamm, A possible kind of electron binding on crystal surfaces, Phys. Z. Sowjetunion 1 (1932), no. 733.
- [40] D. J. Thouless, M. Kohmoto, M. P. Nightgale, and M. Den Nijs, Quantized hall conductance in a two-dimensional periodic potential, Phys. Rev. Lett. 49 (1982), 405.
- [41] Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, Reflection-free one-way edge modes in a gyromagnetic photonic crystal, Phys. Rev. Lett. 100 (2008), 013905.
- [42] R. Weder, The $W_{k,p}$-continuity of the Schrödinger wave operators on the line, Commun. Math. Phys. 208 (1999), no. 507.
- [43] M. I. Weinstein and J. B. Keller, Asymptotic behavior of stability regions for Hill’s equation, SIAM J. Appl. Math. 47 (1987), no. 5, 941\ndash958.
- [44] X.-G. Wen, Gapless boundary excitations in the quantum Hall states and in the chiral spin states, Phys. Rev. B 43 (1991), no. 13, 11025.
- [45] K. Yajima, The $W_{k,p}$-continuity of wave operators for Schrödinger operators, J. Math. Soc. Jpn. 47 (1995), no. 551.
- [46] Z. Yu, G. Veronis, Z. Wang, and S. Fan, One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal, Phys. Rev. Lett. 100 (2008), 023902.
