Waves in Honeycomb Structures

Transcription

1 Waves in Honeycomb Structures Michael I. Weinstein Columbia University Nonlinear Schrödinger Equations: Theory and Applications Heraklion, Crete / Greece May 20-24, 2013

2 Joint work with C.L. Fefferman We are interested in novel properties of waves governed by: Schrödinger eqn: i tψ = ( + V h (x)) ψ Nonlinear Schrödinger / Gross-Pitaevskii eqn: i tψ = + V h (x) + g ψ 2 ψ ψ where V h (x) is a honeycomb lattice potential

3 The period lattice, Λ h Fix a > 0, the lattice constant. Period (triangular) lattice: Λ h = Zv 1 Zv 2 : 0 v 1 = A, v 2 = A Take 2 points within an elementary period cell of Λ h : «1 3 A = (0, 0) and B = a, 0, Generate two triangular lattices: Λ A = A + Λ h and Λ B = B + Λ h Honeycomb structure: H = Λ A Λ B

4 v_1 k_2 0 v_2 A B k_

5 Honeycomb lattice potentials, HLPs V C (R 2 ) is a honeycomb lattice potential if x 0 R 2 such that Ṽ (x) = V (x x 0) satisfies: 1. Ṽ is Λ h periodic, i.e. Ṽ (x + v) = Ṽ (x) for all x R2 and v Λ h. 2. Ṽ is even, i.e. Ṽ ( x) = Ṽ (x) (inversion symmetry w.r.t. x 0) 3. Ṽ is R- invariant, i.e. R[Ṽ ](x) Ṽ (R x) = Ṽ (x), where R is the 2π/3- counter-clockwise rotation matrix

6 Examples of honeycomb structures Example 1: Graphene: 2- dimensional honeycomb arrangement of C atoms 2010 Nobel prize: A. Geim, K. Novoselov

7 Generating an atomic graphene potential Honeycomb structure: H = Λ A Λ B Let V 0 (x) be real, smooth, radial and rapidly decreasing - atomic potential Then, V (x) = P a H V 0(x + a) is a honeycomb lattice potential v_1 k_2 0 v_2 A B k_

8 Example 2: Honeycomb arrays of optical waveguides Figures Rechtsman, Segev et. al. Figure 1. (a) Diagram of the honeycomb photonic lattice geometry. Lig through the structure along the axis of the waveguides (the z-axis) thro

9 Example 3: Optical graphene 2d refractive index profile, n(x 1, x 2 ), with honeycomb lattice symmetry Honeycomb pattern achieved via interference pattern of 3 plane wave beams in a nonlinear - crystal (optical induction) M. Segev et al., Z. Chen et al.

10 Optical honeycomb lattice potentials via interference of plane waves Period lattice, Λ h, and its dual, Λ h Λ h = Zv 1 Zv 2 : 0 v 1 = A, v 2 = A 0 Λ h = Zk 1 Zk 2 : k 1 = 4π A, k 2 = 4π a A v j k l = 2πδ jl, j,l=1,2

11 Interfere 3 plane waves: E(x) e ir 1 x + e ir 2 x + e ir 3 x, r = 2π/λ Nonlinear crystal with defocusing Kerr effect Kerr effect: n(x) E(x) 2 Recall Λ h = Zk 1 Zk 2 and choose r j (k 1, k 2 ), j = 1, 2, 3 so that V h (x) = E(x) 2 e ir1 x + e ir2 x + e ir 3 x 2 cos(k 1 x) + cos(k 2 x) + cos((k 1 + k 2 ) x) Claim: V h (x) is a honeycomb lattice potential (real, inversion-symmetric, R invariant).

12 Waves in periodic media Spectral theory of + V (x), where V (x), periodic w.r.t. a (Bravais) lattice, Λ = Zv 1 Zv 2 Floquet-Bloch EVP: For each k B ( + V (x)) u(x) = µ u(x), u(x + v; k) = e ik v u(x; k) B Λ, Brillouin zone; fundamental domain in k space K k_ K' k_

13 Floquet - Bloch review cont d - The Band Structure of V Equivalently, u(x; k) = e ik x p(x; k), where H(k) p(x; k) p(x + v; k) = p(x; k), ( + ik) 2 + V (x) p(x; k) = µ(k)p(x; k) v Λ k B, discrt sequence of e-values: µ 1 (k) µ 2 (k) µ 3 (k)... with corresponding Λ periodic eigenfunctions, p j (x; k), j 1.

14

15 The set { p b (x; k) } b 0, complete in L 2 per,λ. k {µ b (k)} b 1, k B intervals R: spectral bands L 2 - spec ( + V (x)) = b 1 µ b (B) = union of intervals Graphs of k µ b (k), b 1: dispersion surfaces Completeness of Floquet-Bloch modes of H V = + V = [ ] e ih V t f (x, t) = u b ( ; k), f ( ) L2 (R d ) e i(kx µ b(k)t) p b (x; k) dk b 1 B

16

17 Dynamics of wave packets Wave-packet initial data: ψ(x, 0) = α 0 (δx) e ik0x p(x; k 0 ) = ψ(x, t) α(δx, δt) e ik0x p(x; k 0 ) Transport (ballistic) dynamics if µ b (k 0 ) interior to a spectral band B 1 B 2 B 0 E (k) 2 E (k) 1 E (k) 0-1/2 0 1/2 k T α + µ b (k 0 ) X α = 0, µ b (k 0 ) 0, X = δx, T = δt

18 Dispersive dynamics if µ b (k 0 ) is at a spectral band edge ψ(x, t) α(δx, δ 2 t) e ik 0x p(x; k 0 ) B 1 B 2 B 0 E (k) 2 E (k) 1 E (k) 0-1/2 0 1/2 k i T α = X A (b) (k 0 ) X α, X = δx, T = δ 2 t (Schrödinger equation) A (b) lm (k 0) = 1 Dk 2 2 l k m µ b (k 0 ) (inverse effective mass tensor) lm

19 In general, dispersion surfaces are not smooth; they are at best Lipschitz continuous (min-max characterization of eigenvalues) Symmetries of V (x) = degeneracies in µb (k ), b 1, k B Dirac points arise when band dispersion surfaces touch conically at some quasi-momentum k B

20 In general, dispersion surfaces are not smooth; they are at best Lipschitz continuous (min-max characterization of eigenvalues) Symmetries of V (x) = degeneracies in µb (k ), b 1, k B Dirac points arise when band dispersion surfaces touch conically at some quasi-momentum k B Dirac aka diabolical points are well-known for homogeneous ani-isotropic Maxwell equations. = Conical diffraction: Hamilton (1837), Ludwig (1961), Uhlmann (1982), Berry (1983, 2007),...

21 Why are Dirac Points for periodic structures important?

22 (a) Design Bdry : Such conical intersections permit tuning of the physics i.e. by deforming the honeycomb structure, we can transition the material between states of (i) conduction (no gap) and insulation (gapped) (ii) non-dispersive (Dirac) and dispersive (Schro dinger) wave dynamics Simulations: Linear (a) (b) (c) (d) (e) (f) Simulations of linear Dirac eq IC: wavefronts a =unit Gaussian b=0 Figure: Left: Dirac Points, Right: Dirac eqn (Ablowitz-Zhu) (a)-(c) intensities of a (d)-(f) intensity of b

23 (b) Topologically protected edge states

24 This talk will emphasize (i) Existence of Dirac points for generic honeycomb lattice potential, V (x), e.g. no constraint on size of V (not weak, not tight-binding ) (ii) Dynamics of wave-packets and the 2D Dirac effective equations (iii) Remarks on NLS / GP nonlinear Dirac equation References: C.L. Fefferman & M.I.W. Honeycomb Lattice Potentials and Dirac Points, J. AMS, 2012 Wave packets in Honeycomb Structures & 2-dim. Dirac Eqns, to appear, Comm. Math. Physics, arxiv.org/abs/ Waves in Deformed Honeycomb Structures, in preparation

25 Dirac points K Bh is a Dirac point if in a neighborhood of K, two dispersion surfaces intersect conically at K, i.e. k 7 µb (k) = µ (k) and k 7 µb +1 (k) = µ+ (k) with µ µ+ (K ) = µ (K ) such that µ+ (k) µ + λ k K µ (k) µ λ k K, λ > 0

26 Theorem 1 (C. L. Fefferman - W, 12) Let V (x) be a honeycomb lattice potential which satisfies the non-degeneracy condition: Z V 1,1 = e i(k 1+k 2 ) y V (y) dy 0 Ω Consider H ε = + εv and let K be any vertex of B h. Then for generic values of ε 0, H ε has Dirac points at k = K, i.e. conical intersections of two band dispersion surfaces at vertices of B h

27 Theorem 1 (C. L. Fefferman - W, 12) Let V (x) be a honeycomb lattice potential which satisfies the non-degeneracy condition: Z V 1,1 = e i(k 1+k 2 ) y V (y) dy 0 Ω Consider H ε = + εv and let K be any vertex of B h. Then for generic values of ε 0, H ε has Dirac points at k = K, i.e. conical intersections of two band dispersion surfaces at vertices of B h Remarks: No assumptions on the size of εv At non-generic values of ε, there can in principle be higher order eigenvalue (band) crossings.

28 Theorem 1, more precisely a countable and closed set C R such that for all ε / C + εv h has Dirac points at the vertices, K, of B h That is, for all ε / C, there exist δ ε > 0, C ε > 0 and Lipschitz fns, E ±(k) :, k K < δ ε, E±(k) ε C ε k K, s.t. µ ε = µ ε +(K ) = µ ε (K ), and µ ε +(k) µ ε = + λ ε k K ( 1 + E ε +(k) ) and µ ε (k) µ ε = λ ε k K ( 1 + E ε (k) ), λ ε C, λ ε 0 for ε / C, λ ε = Fermi velocity n o nullspace (H(K )) = span Φ 1 (x; K ), Φ 1 ( x; K )

29 On λ, the Fermi velocity + εv h (x) has degenerate L 2 K eigenvalue µ ε = µ ε +(K ) = µ ε (K ) o L 2 K nullspace( + εv h ) = span nφ ε 1(x; K ), Φ ε1 ( x; K ) µ ε +(k) µ ε + λ ε k K and µ ε (k) µ ε λ ε k K λ ε X m S c ε (m, µ ε ) 2 1 i «K m {c(m, µ )} m S = L 2 K,τ - Fourier coefficients of Φ(x; K ) Remark: Obstructions to analyticity of ε λ ε

30 Theorem 1 - cont d: Small ε µ 1 (k) µ 2 (k) µ 3 (k)... ε < ε 0 : There exists ε 0 > 0, such that for all ε ( ε 0, ε 0 ) \ {0} (i) εv 1,1 > 0 = conical intersection of 1 st and 2 nd dispersion surfaces at K (ii) εv 1,1 < 0 = conical intersection of 2 nd and 3 rd dispersion surfaces at K. µ ε +(k) µ ε = + λ ε k K ( 1 + E ε +(k) ) and µ ε (k) µ ε = λ ε k K ( 1 + E ε (k) ), where λ ε 0 for ALL 0 < ε < ε 0

31 Deformations of honeycomb structures Theorem 2 : Consider H = + V, where V is a HLP. Take a small perturbation W : H 2 (R 2 /Λ h ) L 2 (R 2 /Λ h ) which is not necessarily R- invariant. BUT assume that H + W commutes with f (x) conj inv[f ](x) = f ( x) Conclude: Dirac points persist but move away from vertices of B h Remark: For large deformations, there are interesting phenomena associated with collision and annihilation of Dirac points conduction (gapless) insulation (gapped)

32 Examples where Dirac points persist [H + W, conj inv] = 0: Uniformly strained lattice: V M (x) = V (Mx), M real invertible, 0 < MM T I small Deformed optical lattice (via changing angles of input beams): V (x) V (x) + ηw (x), W Cper,Λ h, W (x) = W ( x), η small Theorem 3 : Break inversion symmetry = Conical points no longer persist Dispersion surfaces are locally smooth. Conjecture: Structure can be gapped. A rigorous proof requires global control of dispersion surfaces.

33 Previous work: 1. Wallace : Band structure of graphite Tight binding limit ( ε ) potential is concentrated at lattice points H = + W h (x) difference operator explicit dispersion relation 2. Weak potential limit ( ε 0 ) formal perturbation theory (e.g. Haldane-Raghu 07, Grushin 09, Ablowitz-Zhu 11) 3. Formal derivation of Dirac envelope equations in tight-binding and weak-potential limits: Ablowitz-Zhu, Ablowitz-Curtis-Zhu

34 Dynamics of Wave Packets and 2D Dirac equations Degenerate Floquet-Bloch subspace at the Dirac point, k = K : ψ(x, t) = e iµ t ( α 1 Φ 1 (x) + α 2 Φ 2 (x) ) = e iµ t e ik x ( α 1 p 1 (x; K ) + α 2 p 2 (x; K ) ) Wave packet initial condition: ψ δ (x, 0) = ψ δ 0(x) = δ e ik x ( α 10 (δx) p 1 (x) + α 20 (δx) p 2 (x) ) {z } modulation of degenerate Floquet-Bloch mode Ansatz: 0 2X ψ δ (x, t) = e iµ j=1 1 δ α j (δx, δt)φ j (x) + η δ (x, t) A. Seek effective ( homogenized ) equations for α j (X, T ).

35 Effective 2D Dirac dynamics Theorem 4 (Fefferman-W. CMP 13) V h (x), x R 2, generic honeycomb lattice potential ( = Dirac points) i tψ = ( + V h (x)) ψ Wave packet initial data: ψ0, δ ψ0 δ L 2 = O(1) spec loc about Dirac pt: ψ0(x) δ = δ ( α 10 (δx) Φ 1 (x) + α 20 (δx) Φ 2 (x) ), α 10, α 20 S(R 2 X) Solution: 0 2X ψ δ (x, t) = e iµ j=1 1 δ α j (δx, δt)φ j (x) + η δ (x, t) A, where α(x, T ) satisfies the 2D Dirac system ( massless ): sup 0 t ρ δ 2+τ T α 1 = λ ( X1 + i X2 ) α 2 T α 2 = λ ( X1 i X2 ) α 1. α x η δ (x, t) L = o(δ τ ) as δ 0. τ 2 (R 2 x ) > 0, all α N 2

36 Nonlinear waves in honeycomb structures i tψ = + V h (x) + g ψ 2 ψ, x R 2 ψ δ (x, t) = δ X α j (X, T )Φ j (x) + δη δ (x, t) A e iµ t j=1 Nonlinear Dirac Equations T α 1 = λ ( X1 + i X2 ) α 2 i g β 1 α β 2 α 2 2 α 1 T α 2 = λ ( X1 i X2 ) α 1 i g 2β 2 α β 1 α 2 2 α 2

37 Nonlinear Dirac Equations T α 1 = λ ( X1 + i X2 ) α 2 i g β 1 α β 2 α 2 2 α 1 T α 2 = λ ( X1 i X2 ) α 1 i g 2β 2 α β 1 α 2 2 α 2 Z N [α 1, α 2 ] α α 1 2 dx 1 dx 2, R» 2 Z H [α 1, α 2 ] Im λ α 2 ( X1 + i X2 ) α 1 dx 1 dx 2 R 2 g Z β 1 α β 2 α 1 2 α β 1 α 2 4 dx 1 dx 2. 4 R D NLS: g > 0 defocusing case = global well-posedness 2. 2D NLS: g < 0 focusing case = blow-up (singularity formation) Z H[ψ] = ψ 2 + V h (x) ψ 2 + g R 2 2 ψ 4 dx. 3. What s the role of g for nonlinear Dirac? Energy is always indefinite. 4. Blowup for nonlinear Dirac? (L 2 supercritical) 5. How does honeycomb structure effect blow up?