Energy localization in surface modes or edge states at the interface

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1 Topologically protected states in one-dimensional continuous systems and Dirac points Charles L. Fefferman a,, James P. Lee-Thorp b, and Michael I. Weinstein b a Department of Mathematics, Princeton University, Princeton, NJ 8544; and b Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 27 Contributed by Charles L. Fefferman, April 28, 4 (sent for review March 22, 4) We study a class of periodic Schrödinger operators on R that have Dirac points. The introduction of an edge via adiabatic modulation of a periodic potential by a domain wall results in the bifurcation of spatially localized edge states, associated with the topologically protected zero-energy mode of an asymptotic one-dimensional Dirac operator. The bound states we construct can be realized as highly robust transverse-magnetic electromagnetic modes for a class of photonic waveguides with a phase defect. Our model captures many aspects of the phenomenon of topologically protected edge states for 2D bulk structures such as the honeycomb structure of graphene. Floquet Bloch theory Hill s equation surface states multiple scale analysis wave-packets Energy localization in surface modes or edge states at the interface between dissimilar media has been eplored, going back to the 93s, as a vehicle for localization and transport of energy ( 8). These phenomena can be eploited in, for eample, quantum, electronic, or photonic device design. An essential property for applications is robustness; the localization properties of such surface states needs to be stable with respect to distortions of or imperfections along the interface. A class of structures, which has attracted great interest since about 5, is topological insulators (9, ). In certain energy ranges, such structures behave as insulators in their bulk (this is associated with an energy gap in the spectrum of the bulk Hamiltonian), but have boundary conducting states with energies in the bulk energy gap; these are states that propagate along the boundary and are localized transverse to the boundary. Some of these states may be topologically protected; they persist under deformations of the interface that preserve the bulk spectral gap, e.g., localized perturbations of the interface. In honeycomb structures, e.g., graphene, where a bulk gap is opened at a Dirac point by breaking time-reversal symmetry ( 4), protected edge states are unidirectional. Furthermore, these edge states do not backscatter in the presence of interface perturbations (3 5, 8). Chiral edge states, observed in the quantum Hall effect, are a well known instance of topological protected states in condensed matter physics. In tightbinding models, this property can be understood in terms of topological invariants associated with the band structure of the bulk periodic structure (5 ). In this article we introduce a one-dimensional continuum model, a Schrödinger equation with periodic potential modulated by a domain wall, in which we rigorously study the bifurcation of topologically protected edge states as a parameter lifts a Dirac point degeneracy (symmetry-protected linear band crossing). This model, which has many of the features of the above eamples, is motivated by the study of photonic edge states in honeycomb structures in ref. 3. The bifurcation we study is governed by the eistence of a topologically protected zero-energy eigenstate of a one-dimensional Dirac operator, D (Eq. 5). The zero mode of this operator plays a role in electronic ecitations in coupled scalar spinor fields (2) and polymer chains (22). There are numerous studies of edge states for tight-binding models (for eample, refs. 7, 9, and 8). The present work considers the far less-eplored setting of edge states in the underlying partial differential equations (3, 4). A version of this article with detailed rigorous proofs can be found in ref. 23. Finally, we remark that the topologically protected states we construct can be realized as highly robust transverse-magnetic (TM) electromagnetic guided modes of Mawell s equations for a class of photonic waveguides. Floquet Bloch Theory Let Q C denote a one-periodic real-valued potential, i.e., Qð + Þ = QðÞ; R, and consider the Schrödinger operator H Q = 2 + QðÞ: Definition : The space of k pseudoperiodic L 2 functions is given by L 2 k = f L 2 loc : f ð + ; kþ = eik fð; kþ : The Sobolev spaces Hk N ; N = ; ;...are analogously defined. Because the k pseudoperiodic boundary condition is invariant under k k + 2π, it is natural to work with a fundamental dualperiod cell or Brillouin zone, which we take to be B ½; 2πŠ. We net consider a one-parameter family of Floquet Bloch eigenvalue problems, parameterized by k B: H Q Φ = EΦ; Φð + ; kþ = e ik Φð; kþ: [] The eigenvalue problem [] is self-adjoint on L 2 k and has a discrete set of eigenvalues, listed with repetitions E ðkþ E 2 ðkþ E j ðkþ with corresponding L 2 k eigenfunctions Φ ð; kþ; Φ 2 ð; kþ;...,which, for fied k, can be taken to be a complete orthonormal set in Significance Topological insulators (TIs) have been a topic of intense study in recent years. When appropriately interfaced with other structures, TIs possess robust edge states, which persist in the presence of localized interface perturbations. Therefore, TIs are ideal for the transfer or storage of energy or information. The prevalent analyses of TIs involve idealized discrete tight-binding models. We present a rigorous study of a class of continuum models, for which we prove the emergence of topologically protected edge states. These states are bifurcations at linear band crossings (Dirac points) of localized modes. The bifurcation is induced by the -energy eigenmode of a class of one-dimensional Dirac equations. Author contributions: C.L.F., J.P.L.-T., and M.I.W. performed research and wrote the paper. The authors declare no conflict of interest. To whom correspondence should be addressed. cf@math.princeton.edu. PNAS June 7, 4 vol. no

2 L 2 ½; Š. Furthermore, the states Φ j ð; kþ; j ; k B are complete in L 2 ðrþ, f ðþ L 2 ðrþ f ðþ = X Z ~ f j ðkþφ j ð; kþdk; j B where ~ f j ðkþ = hφ j ð ; kþ; f i L 2 ðrþ. We make use of the decompositions L 2 k = L2 k;e L2 k;o and Hs k = Hk;e s Hs k;o, into subspaces defined in terms of even- and oddinde Fourier projections, introduced as follows: Definition 2: ) L 2 k;e is the subspace of L2 k consisting of functions of the form e ik P e ðþ = e ik P Pm 2Z pðmþe2πim ; m 2Z jpðmþj2 < ; i.e., P e ðþ is an even-inde periodic Fourier series. 2) L 2 k;o is the subspace of L2 k consisting of functions of the form e ik P o ðþ = e ik P Pm 2Z+ pðmþe2πim ; m 2Z+ jpðmþj2 < ; i.e., P o ðþ is an odd-inde periodic Fourier series. 3) Sobolev spaces, Hk;e M and HM k;o ; M = ; ; 2;...; are defined in the natural way. Motivating Eample Start with a smooth, real-valued, even, and periodic function, QðÞ. Define the one-parameter family of potentials Qð; sþ, a sum of translates of QðÞ: Qð; sþ = Q + s + Q s ; s : 2 2 Clearly, Qð; sþ is periodic. Moreover, Qð; s = =2Þ has minimal period equal to =2. That is, Qð; sþ has an additional translation symmetry at s = =2 (Fig. ). The function Qð; sþ may be epressed as a Fourier series Qð; sþ = X m Z + Q m cosðπmsþcosð2πmþ; which for s = =2 reduces to an even-inde cosine series: V e ðþ Qð; =2Þ = X Qm ~ cosð2πmþ: [2] m 2Z + Dirac Points The operator 2 + V eðþ can be shown to have distinguished ( symmetry-protected ) quasi-momentum/energy pairs, called Dirac points, at which neighboring spectral bands touch and at which dispersion loci cross linearly (Fig. 2). Theorem. Consider the Schrödinger operator H = 2 + V eðþ, where V e ðþ is a generic smooth even-inde cosine series. Then, H has Dirac points ðk * = π Þ in the following sense: ) There eists b * such that E * = E b* ðπþ = E b* +ðπþ. 2) E * is an L 2 k eigenvalue of multiplicity 2. * 3) Hk 2 = H 2 * k * ;e H2 k * ;o,where H : H2 k * ;e L2 k * ;e and H : H2 k * ;o L2 k * ;o. 4) The inversion S½fŠðÞ = f ð Þ, where S : Hk 2 * ;e H2 k * ;o and S : Hk 2 * ;o H2 k * ;e, is such that S commutes with Hðk * Þ = e ik * He ik * ; i.e., ½Hðk * Þ; SŠ Hðk * ÞS SHðk * Þ vanishes. 5) The L 2 k nullspace of H E * * I is spanned by functions Φ L 2 k * ;e, Φ 2 S½Φ ŠðÞ L 2 k * ;o, hφ a; Φ b i L 2 ½;Š = δ ab; a; b = ; 2. 6) The Fermi velocity satisfies λ = 2ihΦ ; Φ 2 i L 2 ½;Š [3] Fig.. Dimer periodic potential: Qð; sþ = Qð + s=2þ + Qð s=2þ. For< s < =2, the potential Qð; sþ has minimal period and the period cell contains a double well. For s = =2, Qð; =2Þ has minimal period =2. (for generic V e ), and there eist ζ > and Floquet Bloch eigenpairs ðφ + ð; kþ; E + ðkþþ and ðφ ð; kþ; E ðkþþ and smooth functions η ± ðkþ, with η ± ðþ =, defined for k k* < ζ and such that Φ ± ð; kþ have the same span as the functions Φ j ð; kþ; j = ; 2 and E ± ðkþ E * =±λ k k* + η ± k k* : Fig. 2, Left illustrates the eistence of Dirac points for a periodic potential, V e of the type plotted in Fig. 3, Top. Superimposed on V e is the mode Φ L 2 k * ;e. In ref. 24, Dirac points play a central role in the construction of one-dimensional almost periodic potentials for which the Schrödinger operator has nowhere a dense spectrum. Dimers For s =2, Qð; sþ is a periodic potential consisting of dimers, double-well potentials in each period cell (Fig. ). Set s δ = =2 + δκ,with< δ and κ R fied. Then, Qð; s δ Þ is of the form Q ; s δ = X Q δ m cosð2πmþ + δκ X W δ m cosð2πmþ; m 2Z + m 2Z ++ where Q δ m ; Wδ m = OðÞ as δ. The operator H 2 + δκ = 2 + Q ; s δ is a Hill s operator (25). The character of its spectrum is well known (Fig. 4, Upper). For δ fied, the gaps that open at the Dirac points have widths of order OðδÞ. Now fi a constant κ > and let s δ ðþ = =2 + δκðδþ; where κðxþ is the domain wall function, κðxþ ± κ as X ± : We assume that κðxþ is sufficiently smooth and approaches its asymptotic values sufficiently rapidly. The operator Hðs δ ðþþ = Fefferman et al.

3 Fig. 2, Center demonstrates the opening of gaps at the Dirac points in the essential spectrum for the perturbations ± δκ W o ðþ. Topologically Protected States Our main result is the following: Theorem 3. Consider the eigenvalue problem for the Schrödinger operator H δ. Let ðk * = π Þ denote a Dirac point of 2 + V eðþ in the sense of Theorem. Furthermore, assume the condition ϑ = hφ ; W o Φ 2 i L 2 ð½;šþ ; [4] which holds for generic V e and W o. Assume that κðxþ ± κ sufficiently rapidly as X ±. Then, the following holds: ) There eists δ > and a branch of solutions to the eigenvalue problem H δ Ψ δ = E δ Ψ δ, δ E δ ; Ψ δ ; < δ < δ ; Fig. 2. Spectra (blue) for H δ=5 = 2 + cosð2ð2πþþ + 3κðXÞcosð2πÞ for different choices of κðxþ. (Left) κðxþ. Plotted are the first four Floquet Bloch dispersion curves k E b ðkþ (gray) and the Dirac points ðk * = π,e * Þ approimately at E * = π 2,ð3πÞ 2. (Center) κðxþ ± κ, a nonzero constant. Shown are open gaps about Dirac points of the unperturbed potential and smooth dispersion curves (gray). (Right) Midgap eigenvalues (red) are shown for periodic potential modulated by domain wall: κðxþ = tanhðxþ. bifurcating from energy E * at δ =, into the gap I δ (Proposition 2), with E δ E * K δ 2, and with corresponding spatially localized (eponentially decaying) eigenstate, Ψ δ. 2) Ψ δ ðþ is approimated by a slowly varying and spatially decaying modulation of the degenerate Floquet Bloch modes Φ and Φ 2 : Ψ δ ðþ δ =2 α * ;ðδþφ ðþ + α * ;2ðδÞΦ ðþ H ðr K δ: Þ 2 +Qð; sδ ðþþ is the Hamiltonian for a structure that adiabatically transitions between dimer periodic structures at ± through a domain wall. For δ small, Q ; s δ ðþ V e ðþ + δκðδþw o ðþ; where V e ðþ denotes an even-inde cosine series, a structure with a Dirac point, and W o ðþ, an odd-inde cosine series, is a noncompact perturbation induced by the domain wall (phase defect). This motivates our study of the family of operators: H δ = 2 + V eðþ + δκðδþw o ðþ: 3) The amplitudes, α * ðxþ = ðα * ;ðxþ; α * ;2ðXÞÞ, are governed by the topologically protected zero eigenmode of the 2 2 matri Dirac operator, where σ = and 4). D iλ σ 3 X + ϑ κðxþσ ; [5] ; σ 3 =, and λ ϑ (Eqs. 3 Theorem 3 is illustrated in Figs Fig. 2, Right displays the midgap eigenvalues, E δ. Fig. 3 displays the unperturbed The operator H δ interpolates adiabatically, through a domain wall, between the operators H H δ; H δκ W o at = ; 2 κ δ H 2 + κ δ H δ;+ H 2 + δκ W o at =+ : 2 The noncompact perturbations δκðδþw o ðþ and ± δκ W o ðþ, which break the symmetries of V e ðþ, open a gap in the essential spectrum. Proposition 2. Let ðk * Þ denote a Dirac point of 2 + V eðþ. Assume the condition ϑ = hφ ; W o Φ 2 i L 2 ½;Š, which holds for generic W o and V e. Fi c less than but arbitrarily close to and define the real interval I δ E * cδκ jϑ j + cδκ jϑ j : Then, there eists δ >, such that for all < δ < δ, I δ σ ess ðh δ Þ; I δ σ ess Hδ; ; and I δ σ ess Hδ;+ are all empty sets. 4 4 Fig. 3. Potentials and modes for H δ = 2 + U δðþ, where U δ ðþ = cosð2ð2πþþ + 2δ tanhðδþcosð2ð2πþþ. (Top) Floquet Bloch eigenmode Φ ðþ L 2 π,e (blue) corresponding to the degenerate eigenvalue E * at the Dirac point of unperturbed Hamiltonian, H δ=, superimposed on plot of U ðþ. Here, Φ 2 ðþ = Φ ð Þ L 2 π,o.(middle) Domain-wall modulated periodic structure, U δ ðþ, δ = 5. (Bottom) Localized midgap eigenmode Ψ δ ðþ of H δ. Fefferman et al. PNAS June 7, 4 vol. no

4 Assume the following conditions: I) E * is a simple L 2 k * ;e eigenvalue of H with one-dimensional eigenspace spanfφ ðþg L 2 k * ;e : II) E * is a simple L 2 k * ;o eigenspace eigenvalue of H with one-dimensional spanfφ 2 ðþ = Φ ð Þg L 2 k * ;o : III) Nondegeneracy condition ( λ 2ihΦ ; Φ i = 2π 2 X ) mjc ðmþj 2 + : [6] m 2Z Fig. 4. Energy spectra (E) ofh δ for different choices of κðxþ over a range of δ. (Upper) Spectrum of Hill s operator: 2 + cosð2ð2πþþ + 2δκ P j=,3,5 cosð2πjþ. For δ, spectral gaps of width OðδÞ open about Dirac points of H, approimately located at π 2,ð3πÞ 2,ð5πÞ 2,...(Lower) Spectrum of the domain-wall modulated periodic potential: 2 + cosð2ð2πþþ + 2δκðδÞ P j=,3,5cosð2πjþ with κðxþ = tanhðxþ. Solid (red) curves are the first three bifurcation curves of topologically protected states. potential, V e ðþ, and the mode Φ ðþ (Fig. 3, Top), the domainwall modulated potential V e ðþ + δκðδþw o ðþ (Fig. 3, Middle), and the midgap mode Ψ δ ðþ (Fig. 3, Bottom). Fig. 4, Lower displays the bifurcation curves of topologically protected states from the first several Dirac points of 2 + V eðþ. Remark : ) The zero mode of a one-dimensional Dirac operator, D, plays an important role in refs. 3, 2, and 22. 2) The bifurcation discussed in Theorem 3 is associated with a noncompact perturbation of H = 2 + V eðþ, namely a phase defect across the structure, which at once changes the essential spectrum and spawns a bound state. This is in contrast to bifurcations from the edge of continuous spectra, arising from localized perturbations (for eample, refs. 26 3). A class of edge bifurcations due to a noncompact perturbation is studied in ref ) Our model captures many aspects of the phenomenon of topologically protected edge states for 2D bulk structures such as the honeycomb structure of graphene (for eample, refs. 3 and 3). Proof of Theorem To establish that ðk * Þ is a Dirac point, we verify the sufficient conditions of the following: Theorem 4. Consider H = 2 + V e, where V e, given by Eq. 2, is sufficiently smooth. Let k * = π and assume that E * is a double eigenvalue, lying at the intersection of the b th * and ðb * + Þ st spectral bands: E * = E b* k* = Eb* + k* : Here, fc ðmþg m Z denote the L 2 k * ;e Fourier coefficients of Φ ðþ. Then, ðk * Þ is a Dirac point in the sense of Theorem. The proof follows that of theorem 4. of ref. 3 for the case of honeycomb lattice potentials in R 2.ToestablishTheorem, that 2 + V e has Dirac points for generic V e, we consider the family of operators H ð«þ = 2 + «V e. The conditions of Theorem 4 can be verified for all «½; «Þ,forsome«ðE * Þ > byaperturbative/ Lyapunov Schmidt reduction argument about quasi-momentum energy pairs for «= : ðk * ; E m; * Þ = ðπ; ð2m + Þ2 π 2 Þ; m = ; ;...A continuation argument to «of arbitrary size is implemented following the strategy of ref. 3. This shows the persistence of the conditions of Theorem 4 and therefore the eistence of a Dirac point, ðk * ; E ð«þ * Þ,for all «C,where ~ C ~ denotes a countable and closed subset of R. Proof of Theorem 3 A formal multiple-scale epansion anticipates the form of the solution at any finite order in δ. To establish the validity of this epansion, the corrector is decomposed into its near-energy and far-energy components, using the spectral decomposition of the unperturbed operator: 2 + V e. The near-energy regime corresponds to energies within the intersecting spectral bands, which are near E * and quasi-momentum, k, satisfying k k* δ τ, where < τ <. The far-energy components correspond to all other energies (within the intersecting bands and all other bands). The eigenvalue problem is equivalent to a coupled infinite system for the near- and far-energy spectral components. To solve this system we first solve for the far-energy components as a functional of the near-energy components. This leads to a (Lyapunov Schmidt) reduction to a closed nonlocal system, which determines the near-energy components of the corrector and the correction to the degenerate eigenvalue E *. Under rescaling, the latter equation may be written as a Dirac-type equation with Dirac operator D (Eq. 5) band limited to rescaled momenta jξj δ τ.we then solve this system for all δ positive and sufficiently small. ACKNOWLEDGMENTS. The authors thank P. Deift, J. B. Keller, J. Lu, A. Millis, M. Rechtsman, M. Segev, and C. W. Wong for stimulating discussions. This work was supported in part by National Science Foundation (NSF) Grants DMS (to C.L.F.) and DMS and Columbia University Optics and Quantum Electronics Integrative Graduate Education and Research Traineeship (IGERT) NSF Grant DGE-694 (to M.I.W. and J.P.L.-T.).. Tamm IE (932) A possible kind of electron binding on crystal surfaces. Phys Z Sowjetunion : Shockley W (939) On the surface states associated with a periodic potential. Phys Rev 56: Raghu S, Haldane FDM (8) Analogs of quantum-hall-effect edge states in photonic crystals. Phys Rev A 78: Yu Z, Veronis G, Wang Z, Fan S (8) One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal. Phys Rev Lett (2): Wang Z, Chong YD, Joannopoulos JD, Soljacic M (8) Reflection-free one-way edge modes in a gyromagnetic photonic crystal. Phys Rev Lett (): Malkova N, Hromada I, Wang X, Bryant G, Chen Z (9) Observation of optical Shockley-like surface states in photonic superlattices. Opt Lett 34(): Rechtsman MC, et al. (3) Topological creation and destruction of edge states in photonic graphene. Phys Rev Lett : Rechtsman MC, et al. (3) Photonic Floquet topological insulators. Nature 496(7444): Fefferman et al.

5 9. Kane CL, Mele EJ (5) Z 2 topological order and the quantum spin Hall effect. Phys Rev Lett 95(4): Hasan MZ, Kane CL () Colloquium: Topological insulators. Rev Mod Phys 82:345.. Castro Neto AH, Guinea F, Peres NMR, Novoselov KS, Geim AK (9) The electronic properties of graphene. Rev Mod Phys 8: Katsnelson M (2) Graphene: Carbon in Two Dimensions (Cambridge Univ Press, Cambridge, UK). 3. Fefferman CL, Weinstein MI (2) Honeycomb lattice potentials and Dirac points. J Am Math Soc 25: Fefferman CL, Weinstein MI (4) Wave packets in honeycomb lattice structures and two-dimensional Dirac equations. Commun Math Phys 326: Wen X-G (99) Gapless boundary ecitations in the quantum Hall states and in the chiral spin states. Phys Rev B Condens Matter 43(3): Thouless DJ, Kohmoto M, Nightingale MP, Den Nijs M (982) Quantized hall conductance in a two-dimensional periodic potential. Phys Rev Lett 49: Halperin BI (982) Quantized Hall conductance, current-carrying edge states, and the eistence of etended states in a two-dimensional disordered potential. Phys Rev B 25: Hatsugai Y (993) Chern number and edge states in the integer quantum Hall effect. Phys Rev Lett 7(22): Graf J-M, Porta M (2) Bulk-edge correspondence for two-dimensional topological insulators. Commun Math Phys 324: Avila JC, Schulz-Baldes H, Villegas-Blas C (3) Topological invariants of edge states for periodic two-dimensional models. Math Phys Anal Geom 6: Jackiw R, Rebbi C (976) Solitons with Fermion number /2. Phys Rev D Part Fields 3: Su WP, Schrieffer JR, Heeger AJ (979) Solitons in polyacetylene. Phys Rev Lett 42: Fefferman CL, Lee-Thorp JP, Weinstein MI (4) Topologically protected states in one-dimensional systems. arxiv: Moser J (98) An eample of a Schrödinger equation with almost periodic potential and nowhere dense spectrum. Comment Math Helvetici 56: Magnus W, Winkler S (979) Hill s Equation. (Dover, New York). 26. Simon B (976) The bound state of weakly coupled Schrödinger operators in one and two dimensions. Ann Phys 97: Deift PA, Hempel R (986) On the eistence of eigenvalues of the Schrödinger operator h-λw in a gap of σ(h). Commun Math Phys 3: Figotin A, Klein A (997) Localized classical waves created by defects. JStatPhys86( 2): Borisov DI, Gadyl shin RR (8) On the spectrum of a periodic operator with a small localized perturbation. Izv Math 72: Duchêne V, Vukicevic I, Weinstein MI (4) Scattering and localization properties of highly oscillatory potentials. Commun Pure Appl Math 67(): Duchêne V, Vukicevic I, Weinstein MI Homogenized description of defect modes in periodic structures with localized defects. Commun Math Sci, in press. 32. Bronski J, Rapti Z () Counting defect modes in periodic eigenvalue problems. SIAM J Math Anal 43: Fefferman et al. PNAS June 7, 4 vol. no