Solitons and Propagating Domain Walls in Topological Resonator

Jun 28, 2017 - An array of linear resonators with alternating linear–nonlinear bonds can transition from a topologically trivial to a nontrivial ban...
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Solitons and Propagating Domain Walls in Topological Resonator Arrays Yakir Hadad,†,‡ Vincenzo Vitelli,§ and Andrea Alu*,† †

Department of Electrical and Computer Engineering, The University of Texas at Austin, 1616 Guadalupe Street, Austin, Texas 78701, United States ‡ School of Electrical Engineering, Tel-Aviv University, Ramat-Aviv, Tel-Aviv, 69978, Israel § Instituut-Lorentz, Universiteit Leiden, 2300 RA Leiden, The Netherlands S Supporting Information *

ABSTRACT: An array of linear resonators with alternating linear− nonlinear bonds can transition from a topologically trivial to a nontrivial band structure as a function of the applied intensity. Here, we demonstrate that edge modes supported by the topology of the band diagram in such arrays can evolve into solitons, sustained by propagating domain walls induced by the local intensity distribution. The character of the nonlinear response is related to intensity-induced topological transitions in band theory, and it can be successfully captured by a continuum model. Our findings open up opportunities for self-reconfigurable topologically protected signal transport in various wave systems from optics to acoustics. KEYWORDS: nonlinear optics, photonic topological insulators

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alternating, linear−nonlinear bonds, has been preliminarily discussed in.11 Interestingly, these edge-modes were shown to decay to a nonzero plateau in the bulk, as opposed to edge modes in a linear topologically nontrivial array. In this Letter, we demonstrate a dynamic analog to the mechanical linkage chain studied in ref 4 and show that the topological edge modes can evolve into topologically robust propagating solitons sustained by moving domain walls, and the dynamics of these two nonlinear modes can, interestingly, be explained based on the band topology of linear excitations. These results illustrate the deep connection between band theory, within which topological invariants are determined, and the nonlinear dynamics of topological solitary waves.

he Su−Schrieffer−Heeger (SSH) model is among the simplest settings to study topological excitations of the organic molecule polyacetylene where electrons coupled to domain walls propagate as charged solitons.1,2 An analogous phenomenon was recently demonstrated in a fully mechanical implementation, formed by a chain-like arrangement of masses and springs (or rigid beams) that behaves as a topological mechanical insulator.3,4 In this purely mechanical example, topological edge modes that are localized within linear elastic theory can propagate freely into the bulk as domain walls (or solitons) upon applying an arbitrarily small external force and without the need of coupling phononic and electronic degrees of freedom.4,5 Unlike in polyacetaline, the domain wall propagation through the array can take place without distorting the elastic bonds (i.e., at zero energy) only when it is started from one of the two edges, reflecting the breaking of inversion symmetry of the undistorted chain. The same interplay between nonlinearities and spatial symmetry breaking is responsible for the spatially inhomogeneous distribution of topologically robust mechanical states in a variety of artificial structures, such as linkages,6 origamis,7 cellular,8 and geared metamaterials,9 which are easily deformed in some selected locations and rigid elsewhere. In all of these cases the presence of topological invariants protecting the response is a purely structural property that emerges from the unit cell anisotropy. The extension of these concepts to fully dynamic systems may find diverse applications in acoustics, microwave, optics, and electronic systems, including the realization of reconfigurable waveguides and broadband isolation.10 Toward this goal, the possibility of self-induced topological edge states in an electrodynamic version of the SSH chain with © 2017 American Chemical Society

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SSH MODEL WITH NONLINEAR BONDS Here, we study the tight-binding SSH model of a chain of optical resonator dimers with alternating linear−nonlinear bonds, as illustrated in Figure 1a. The chain dynamics is given by the nonlinear differential system −jaṅ(1) = ω0an(1) + ν(A n)an(2) + κan(2) −1 −jaṅ(2) = ω0an(2) + ν(A n)an(1) + κan(1) +1

(1)

The resonators in the array are assumed to be identical and lossless, with self-resonance frequency ω0, excited with (2) amplitudes a(1) n , an for the first and second resonator in the nth dimer, κ being the linear interdimer coupling, and v(An) Received: March 24, 2017 Published: June 28, 2017 1974

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supports a topological transition from trivial to nontrivial and topological edge modes in the high-intensity regime.11 These edge modes exist at the boundary, or domain wall, between the topologically nontrivial array in the high-intensity regime and free-space due to the difference in topological (winding) number. Different from a linear topological array for which the order is static and determined by the array geometry, here the local intensity controls and can reconfigure in real-time the topology. This unique feature enables the possibility of creating propagating domain walls, accompanied by a plethora of selfinduced discrete vector envelope, kink, antikink, and hole solitons,26 which we study below.

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Figure 1. (a) SSH model with identical resonators and alternating linear−nonlinear bond couplings. (b) Circuit model of a unit-cell of the proposed nonlinear SSH array. (c) Nonlinear bond coupling has a realistic saturable nonlinearity form. The linear coupling lies within the coupling range of the nonlinear bond.

CONTINUUM LIMIT We focus on nonlinear solutions that reduce in the lowintensity regime to linear eigenmodes close or inside the bandgap, which were shown in10 to possess topological invariants within band theory. These solutions are highly oscillatory in time t and expected to be dimerized, namely, a(1,2) n ≈ −a(1,2) n+1 . Consistent with ref 27, we use the ansatz

being the nonlinear intradimer coupling, modeled as a saturable optical nonlinearity ν0 − ν∞ v (A n ) = + ν∞ 1 + A n /A 0 (2) |a(1) n

an(1) = j( −1)n bn(1)e jω0t , an(2) = ( −1)n bn(2)e jω0t

Here, = − is the field intensity in the nth dimer and v0 (v∞) is the coupling coefficient in the low (high) intensity limit, and A0 is a reference level associated with the nonlinear coupling v(An). A realistic13 circuit implementation of this array is introduced in Figure 1b, and a similar optical implementation using arrays of nanoparticles or thin plasmonic layers connected by alternating linear−nonlinear dielectrics can be envisioned14−16 and modeled using the theory of nanocircuits for light.17,18 In addition, linear plasmonic particle arrays have already been proposed in the context of topological insulators and nontrivial states.19−21 Considering such a realization, we envision a packed chain of plasmonic nanoparticles nearly touching each other, with nonlinear material placed in between the particles, every second particle. Each plasmonic particle, close to its resonance frequency, behaves as an LC tank.17,18 The coupling mechanism due to the small gap between particles is capacitive. The presence of the nonlinear material in the gap, where the field intensities are maximized, gives rise to an enhanced nonlinear capacitive coupling response that well fits the model in eqs 1 and 2. Retardation effects in this system are important, meaning that the coupling coefficients depend on the electrical distance between neighboring particles and, hence, will be dispersive. Yet, the coupling terms in the coupled mode theory in eq 1 are necessarily real due to energy conservation considerations within the array,22 consistent with our model. We expect our coupled mode tight-binding model to deviate from the exact particle array dispersion only close to the edges of the dispersion diagram, where interaction between far particles becomes significant.23 Other possible implementations in the microwave and terahertz frequency range are, in principle, possible, directly implementing the circuit model in Figure 1b through nonlinear circuit elements, such as voltage varying capacitors or transistors. At terahertz frequencies, the quantum capacitance of carbon nanotubes may also be used to achieve the required nonlinear coupling.24 For low excitation intensities, the system is linear with v(An) ≈ v0, supporting bulk modes and possibly edge modes depending on whether its bandgap is topologically trivial (v0 > κ) or nontrivial (v0 < κ).25 For sufficiently high intensities, if κ is chosen such that v∞ < κ < v0, as shown in Figure 1c, the array A2n

2 a(2) n |

(3)

to recast eq 1 into a form with unknowns slowly varying with respect to the dimer index n and time t. The imaginary unit in (1,2) a(1) n (t) ensures that bn (t) are real, as shown below, implying (1,2) that an (t) can be considered the two components of an (1) elliptically polarized state vector Ψn = b(2) n + jbn . We now move to the continuum limit b(1,2) → b (x,t) and, using a n 1,2 Taylor expansion to approximate the n ± 1 terms in eq 1, b(1,2) n±1 → b1,2(x,t) ± Δ∂b1,2(x,t)/∂x, where Δ is the dimer length. Anticipating soliton solutions, we define a moving coordinate variable ξ = x − ct, which represents a soliton frame moving at velocity c. In this new coordinate we obtain a nonlinear system of ordinary differential equations. To that end, we define the local intensity q(ξ) = b12(ξ) + b22(ξ) /A 0 and the local polarization angle α(ξ) = tan−1(b1(ξ)/b2(ξ)), which is the angle toward which the state vector Ψ = b2 + jb1 = qejα points in the complex plane, and obtain ⎛ vgq cos 2α ⎞ d ⎛⎜ q ⎞⎟ ⎟ = F(q)⎜⎜ ⎟ dξ ⎝ α ⎠ ⎝−c − vg sin 2α ⎠

(4)

where vg = κΔ is the maximal group velocity in the lowintensity limit and F(q) is defined by F(q) =

(ν0 − κ ) − (κ − ν∞)q 1 1+q (κ Δ)2 − c 2

(5)

Bound solutions of this nonlinear system in the continuum limit are vector solitons represented in (q,α) coordinates.

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PHASE SPACE ANALYSIS The continuum limit solutions can be found through a fixed point analysis of the nonlinear system (eq 4). The analysis reveals three critical points: x1*: q1* = (ν0 − κ )/(κ − ν∞), ∀ α1* x 2*: q2* = 0, α2* = −0.5 sin−1[c /vg ], x3*: q3* = 0, α3* = −π /2 − α2* 1975

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similar to Figure 2a, but symmetric around α = α*2 = 0. This family of solutions is therefore a generalization of the edge states found in ref 11, which stem from the topological order of the array. They evolve along the center blue trajectory with α = 0 or red trajectory with α = ±π/2 and can be derived from eq 1, (2) assuming that either a(1) n = 0 or an = 0 for all n or, equivalently, in the continuum model α = 0 or α = ±π/2. In the fast-frame scenario, for which the frame motion is faster than vg, the phase portrait (shown in Figure 3a for the

The last two points x*2 , x*3 are isolated saddle nodes, while x*1 is a continuum of critical points located on the line q = q*1 , with sink (stable) or repel (unstable) sections depending on the value of α1* and on the velocity ratio c/vg. The critical line q = q*1 coincides with the threshold intensity for which the topological transition arises in the array, that is, the intensity level at which the bandgap vanishes and its topology locally transitions from trivial to nontrivial.11 As discussed in the following, this is a critical quantity for soliton formation, and it represents the intensity plateau level toward which edge and soliton modes asymptotically decay. In the slow-frame (c < vg) regime, the sections |α*| < π/4, π/4 < |α*| < π/2 are sinks and repel, respectively, whereas in the fast-frame (c > vg), the situation reverses. The saddle nodes x2*, x3* exist only for c < vg, as seen in Figure 2a, which shows the “phase portrait” of the system,

Figure 3. (a) Phase portrait in the fast-frame case with trajectories that correspond to two types of solitons, as discussed in the text. (b) A: Hole [B: Envelope] solitons of intensity with (c) kink [antikink] of polarization angle. (d) Evolution of the state vector for fast solitons A and B with respect to ξ as ξ = −∞ → ∞.

same parameters as in Figure 2, but with c = 6 × 10−3 > vg) change compared to the slow frame, allowing only two types of solitons, holes and envelopes in the intensity with kink and antikink in the polarization angle, shown in Figure 3b,c. Clearly no edge states are possible in this case, since those are physical only for c = 0. Interestingly, no soliton waves can exist if κ < v∞ or κ < v0, that is, if the chain does not admit a nonlinearity-triggered topological transition from trivial to nontrivial. In other words, if the array is topologically trivial for all intensities or, more surprisingly, even if the array is topologically nontrivial at all intensities including in the low-intensity limit, soliton waves will not be supported, since domain walls cannot exist. In each of these cases, the critical point q*1 < 0, which is not physical. In this model, a moving domain wall in the sense of linear topological theory is necessary to transport a soliton wavepacket. Using the first of the two equations in eq 4 it is easy to show that the extremes of intensity of solitons A and B in the slow as well as in the fast frames take place at α/π = ±0.25, where dq/ dξ = 0 @ ξ = ξp. Integrating the phase parameters (q,α) with respect to ξ can be done only numerically. However, it is possible to analytically integrate q with respect to α, which will help us to find the q value at the soliton’s peak or deep. Note that dq/dξ = qξ/αξ, where subscript ξ denotes the derivative with respect to ξ. Then, by dividing the two equations in eq 4 and rearranging the expression we obtain

Figure 2. (a) Phase portrait in the slow-frame regime with trajectories that correspond to four types of solitons and edge-states, as discussed in the text. (b) Holes of intensity and (c) corresponding kink (antikink) of the polarization angle; (d) kink (antikink) of intensity, but (e) uniform polarization angle.

mapping the solution trajectories in the (q,α) plane as ξ evolves in the range (−∞,∞). This representation gives a complete description of the chain nonlinear dynamics, and the expected soliton solutions for the array. Figure 2 is calculated for κ = 5 × 10−3, A0 = 1, Δ = 1, v0 = 0.01, and v∞ = 2.5 × 10−3, and c = 1 × 10−3 < vg, using normalized units with respect to ω0 and unit length. These parameters correspond to q*1 = 2, α*1 = −0.205 and α1* = −1.364. For the sake of clarity, in the figure we show only the trajectories that repel-from or sink-in the critical points. The stable [unstable] region of x*1 is marked by a solid [dashed] line, while the saddle nodes x*2 ,x*3 are marked by bold diamonds. Solitons are bounded solutions and, as such, are solutions that repel from an unstable critical point and sink into a stable one. In this slow-frame regime there are two continuum families of solitons, marked by A and B in Figure 2a, with hole in the excitation intensity and kink and antikink in the polarization angle (trajectories in purple), shown in Figure 2b,c. In addition, there are two embedded solitons in the continuum that evolve from the saddle point q*2 to q*1 along trajectory C, and from q1* to q3* along trajectory D. These have, respectively, kink and antikink in intensity, shown in Figure 2d,e. Finally, the blue and red trajectories represent solutions that diverge at either ξ → ±∞, and decay to a constant in the other limit. Therefore, these solutions are not physical in the moving frame but become physical edge states in the zero velocity limit (c = 0 and, hence, ξ = x), evolving along the blue [red] trajectories, and representing stable [unstable] solutions that are finite at the edge at x = 0 and decay to a nonzero plateau as x → ∞[x → −∞] . In the zero velocity limit, the phase portrait is indeed

dq κ Δcos 2α =− dα q c + κ Δsin 2α

(7)

Hence, the intensity maxima/minima of the envelopes and holes can be obtained as qp = q1* 1976

c + vg sin 2α0 c + vg sin 2αp

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In addition, the phase difference in the polarization angle Δα = α(ξ = ∞) − α(ξ = −∞) is Δα = 2 × (αp − α0)

(9)

where α0 = α(ξ = −∞) is a soliton parameter and αp/π = ∓0.25 for solitons A and B in both slow and fast frames. This implies that soliton A in both frames will have a sharper, steeper shape as the soliton velocity approaches vg. The soliton width Δξ scales as Δξ ∼ v2g − c2 for solitons A and B in the slow and fast frames and as Δξ ∼ (v2g − c2)0.5 for solitons C and D in the slow frame,12 resembling the Lorentz contraction of solitons of the Sine-Gordon equation and of the ϕ4 theory,26 and implying that the solitons become narrow in time and space as their velocity approaches the maximum group velocity of the mode supported by the chain in the low-intensity limit.

Figure 4. Numerical solutions of eq 1 with initial condition at t = 0 (shown in black in all panels) calculated using the continuum theory. (a) Intensity and (b) polarization angle of a soliton with initial condition that corresponds to velocity c = 9 × 10−3. (c) and (d) as (b) and (c) but for a narrower soliton, with initial condition obtained with velocity c = 6 × 10−3.

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DOMAIN WALLS AND TOPOLOGICALLY NONTRIVIAL BAND STRUCTURE A soliton solution in our model can be interpreted as a propagating domain wall, induced by the nonlinearity, between two asymptotic responses at Ψ(ξ = −∞) and Ψ(ξ = ∞), with a topological charge proportional to their difference. Figure 3d shows the state vector evolution for fast solitons A and B. The two state vectors originate from Ψ = 2ej0.1π and evolve in opposite directions to Ψ = 2e−j0.6π and Ψ = 2e−j0.4π, thus, having opposite phases after the transition. The transition region is locally topologically trivial for the fast soliton B and nontrivial for fast soliton A, depending on whether q < q*1 or q > q*1 , that is, whether effectively the local nonlinear intracell coupling v satisfies κ < v or κ > v. This topological property is used to describe the soliton dynamics. In light of eq 4, the relaxation rates of a soliton, dq/dξ and dα/dξ, depend through the nonlinear term in eq 4 on how much the local intensity level deviates from the plateau value given by the critical point q1*, which is also the plateau value to which the edge states decay in the static frame c = 0. Intuitively speaking, for a large deviation from the plateau level, a topologically nontrivial bandgap is locally widely open, implying that the edge state or soliton decay rate will be necessarily faster, and the soliton intensity profile more concentrated. Moreover, the width of the nonlinear excitations is directly related to the localization length of the topological edge modes, which diverges when the bandgap closes.12 A relevant difference with their mechanical counterparts is that these electrodynamic domain walls can propagate from both edges, since the chiral symmetry of the linear bulk modes is not broken upon going from a closed ring to an open chain. The continuum analysis solutions discussed so far are fully validated below with full “molecular dynamics” simulations of the discrete system in eq 1.

distribution (q,α) of fast soliton A obtained from the continuum limit with c = 9 × 10−3, α0 = 0.1π, qp = 3.455, is used as initial condition (black curve marked with dashed arrow). The numerical solution then evolves in time and shows a soliton with velocity c = 8.8 × 10−3 (calculated as c = Δn/Δt with Δn = 22 and Δt = 2.5k), very close to the continuum limit. The polarization angle is shown in Figure 4b, again with very good agreement with Δα = −2.2 predicted by eq 9 in the main text. The results in Figure 4c and d are similar to Figure 4a and b, but with the initial condition obtained using c = 6 × 10−3, qp = 5.91, Δα = −2.2, and the numerical solution of eq 1 in the main text stabilizes at c = 5.6 × 10−3. Remarkably, the continuum limit agrees extremely well with the numerical results for discrete solitons, even for confined solitons living over just a few dimers. This property is quite exceptional for discrete solitons, whose stability and mobility is known to be an issue of significant importance.28−30 In general, discrete solitons that are found through a continuum limit analysis are typically poorly stable when excited on the discrete lattice. The effect of discretization yields the so-called Peierls Nabarro (PN) barrier that causes scattering of the soliton into dispersion waves, hence, slows the soliton down and eventually causes soliton pinning around a certain site. Discussion in the context of the discrete nonlinear Schrodinger equation, and estimation of the PN barrier can be found in refs 31 and 32. An exception of this role can be found in ref 33, where topological mechanical chains are discussed and it is shown that the moving kink experiences no PN barrier. The solitons found here are unaffected by lattice discretization and robust against scattering by defects along the lattice, as demonstrated and discussed in the following section. The immunity against the discretization effect is discussed in ref 12, where we show that solutions of the discrete system in eq 1, that satisfy the solution ansatz in eq 3, conserve energy. Namely, dE/dt = 0, where E(t) = Σn[q2n(t) − 2 (2) 2 q*1 2], with qn = |a(1) n | + |an | . We conclude that the soliton solutions are not affected by discretization effects.

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COMPARISON TO “MOLECULAR DYNAMICS” SIMULATIONS The continuum analysis solutions discussed in the main text are validated here with full “molecular dynamics” simulations of the discrete system in eq 1. The simulations show that discrete solitons are supported by the array and are not a mere artifact of the continuum approximation. In our simulations, we plug the analytical solution stemming from the continuum analysis as the initial condition at t = 0, and then let the solution evolve in time according to eq 1. Figure 4 shows calculations for the fast soliton A, and similar results have been obtained for the other soliton types. In Figure 4a and b, the complex amplitude

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ROBUSTNESS AGAINST DEFECTS In Figure 5 we demonstrate the robustness of the topologically protected moving domain walls against defects. As discussed above, we use the continuum model to find an initial solution, marked in Figure 5 by black-thick lines, that we plug into the “molecular dynamics” solver. At t = 0 we introduce abruptly a 1977

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this complex defect distribution, as shown in Figure S6 in ref 12, the impinging fast soliton A completely preserves its shape after interacting with the defected region. Similar results are found also for fast soliton B. As opposed to slow solitons C and D and to fast solitons A and B, slow solitons A and B are much less resistant against the presence of defects. This is due to two reasons: first, slow solitons A and B live in a parameter domain close to the one of the edge states and of the linear spectra. Therefore, coupling between these waves is relatively easy. Second, slow solitons A and B propagate slower than the linear spectra waves (c < vg), whereas, as opposed to slow solitons C and D, these solitons are localized with respect to their intensity. Hence, shortly after a slow soliton A or B hits a defect, the scattered linear spectral waves propagate faster and overwhelm the soliton field. Thus, these solitons are not robust, as opposed to slow solitons C and D and to the fast solitons A and B.

Figure 5. Protection against defects of the moving nonlinearly induced topological domain walls. In the inset (top-left): the defect at the unitcell N is marked by the red arrow with the cross (a) local intensity of slow soliton C with c = 1 × 10−3. (b) As (a) but for slow soliton D. (c, d) Local intensity and polarization angle of fast soliton A with c = 6 × 10−3.

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CONCLUSIONS In this work, we have discussed a rich family of self-induced nonlinear optical solitary waves supported by a variation of the SSH model that are associated with nonlinearity-induced topological transitions and their dynamics can be explained using linear band structure topological considerations. While localized boundary states and solitons were studied in diatomic nonlinear chains,35−37 and solitons have been studied in topological insulators,38−40 the model studied here is unique in that, due to the alternating linear−nonlinear nature of the bonds, it allows controllable, either local or global, topological transitions induced by the level of applied intensity. Our analysis shows that dynamic solitons can be self-induced in the absence of interaction between different types of waves (originally electron−phonon interaction), an extension of the static mechanical case4 to electrodynamic systems. These solitons are propagating topological domain walls induced at certain intensity levels by the nonlinear response of the array. We envision the extension of these concepts to twodimensional nonlinear arrays supporting self-induced solitons in the bulk along moving domain walls that can be reconfigured in real-time as a function of the applied intensity. These topologically protected reconfigurable nonlinear waveguides are also robust against disorder. As discussed here and in ref 12, this robustness is striking, as it applies to strong isolated defects as well as to many random defects distributed over a large region. Our results may be of special relevance in the design of nanophotonic devices, for which disorder in fabrication is inherent due to imperfection tolerances.

defect in the structure by replacing one of the linear intercell bonds κ to unit-cell number N with a weaker bond κD = 0.75κ, as illustrated in the inset. We plot the resulting solutions at equal time steps, and the final time-step solution is marked by red-thick line for comparison. In Figure 5a we show qn of the slow-frame soliton C (kink) with c = 1 × 10−3 (αn is a constant and not shown). The presence of a defect at N = 60 creates minor reflection but interestingly has no effect on transmission, consistent with the results in ref 34 related to perturbations to the scalar Sine-Gordon equation. In Figure 5b we show the results for the antikink counterpart of the kink in (a). Here, on the contrary, there is no reflection, while the transmitted wave is distorted and becomes essentially a superposition solution of the linear spectrum (low intensity limit) of the chain, as evident from the dispersive nature of the transmitted wave and from the ∼vg velocity of its wavefront, marked by brown arrows. The improved stability of the high intensity side of these solitons can be attributed to the fact that in the phase space portrait in Figure 2a at these regions the solution is very close to the equilibrium line q*1 . In Figure 5c,d, qn and αn are shown for the fast soliton A, with c = 6 × 10−3. Upon hitting the defect at N = 75, the transmitted soliton loses some of its velocity and, therefore, become sharper. Robust propagation is observed also when the defect is much stronger. In ref 12, we show results similar to Figure 5c,d, but with κD = 0.2κ. Very similar considerations apply to the fast soliton B, as shown in the SM file. The transmitted soliton, after the defect, completely preserves its shape, but loses some of its kinetic energy, and it becomes slower. Since the defect is very large in this case, an edge state is also excited at the defect location. An even stronger defect, above the threshold κD = 0.2κ, would yield new dynamics. For instance, an impinging fast soliton A will result in a transmitted fast soliton B and an additional edge state excitation at the defect location. This is demonstrated in Figure S4 in ref 12 for κD = 0.05κ. The robustness of fast solitons A and B against disorder is not limited to scattering by a single defect, but it also applies when we consider many defects in a range of cells, even wider than the soliton width. We assume that the at the defected bond the linear coupling is random in space and time, through the use of time-dependent linear bonds, and that they are distributed uniformly in the range κD(n,t) ∼ U[0.2κ,κ]. Despite

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsphotonics.7b00303. Analysis and discussions regarding soliton localization length, energy conservation of solitons in the discrete system, further numerical study with examples of soliton robustness against defects (PDF).

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. 1978

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ORCID

(23) Hadad, Y.; Steinberg, B. Z. Green’s function theory for infinite and semi-infinite particle chains. Phys. Rev. B: Condens. Matter Mater. Phys. 2011, 84, 125402. (24) Akinwande, D.; Nishi, Y.; Philip Wong, S. H. Carbon nanotube quantum capacitance for nonlinear terahertz circuits. IEEE Trans. Nanotechnol. 2009, 8, 31−36. (25) Rice, M. J.; Mele, E. J. Elementary excitations of a linearly conjugated diatomic polymer. Phys. Rev. Lett. 1982, 49, 1455−1459. (26) Dauxois, T.; Peyrard, M. Physics of Solitons; Cambridge University Press, 2006. (27) Kivshar, Y. S. Creation of nonlinear localized modes in discrete lattices. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 1993, 48, 4132−4135. (28) Pelinovsky, D. E.; Kevrekidis, P. G.; Frantzeskakis, D. J. Stability of discrete solitons in nonlinear Schrodinger lattices. Phys. D 2005, 212, 1−19. (29) Oxtoby, O. F.; Barashenkov, I. V. Moving solitons in the discrete nonlinear Schrodinger equation. Phys. Rev. E 2007, 76, 036603. (30) Blit, R.; Malomed, B. A. Propagation and collisions of semidiscrete solitons in arrayed and stacked waveguides. Phys. Rev. A: At., Mol., Opt. Phys. 2012, 86, 043841. (31) Jenkinson, M.; Weinstein, M. I. Onsite and offsite bound states of the discrete nonlinear Schrodinger equation and the PeierlsNabarro barrier. Nonlinearity 2016, 29, 27−86. (32) Jenkinson, M.; Weinstein, M. I. Discrete solitary waves in systems with nonlocal interactions and the Peierls-Nabarro barrier. Commun. Math. Phys. 2017, 351, 45−94. (33) Zhou, Y.; Chen, G. G.; Upadhyaya, N.; Vitelli, V. Kink-antikink asymmetry and impurity interactions in topological mechanical chains. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2017, 95, 022202. (34) Fogel, M. B.; Trullinger, S. E.; Bishop, A. R.; Krumhandl, J. A. Dynamics of Sine-Gordon solitons in the presence of perturbations. Phys. Rev. B 1977, 15, 1578−1592. (35) Sievers, J.; Takeno, S. Intrinsic localized modes in anharmonic crystals. Phys. Rev. Lett. 1988, 61, 970−973. (36) Kivshar, Y. S.; Flytzanis, N. Gap solitons in diatomic lattices. Phys. Rev. A: At., Mol., Opt. Phys. 1992, 46 (12), 7972−7978. (37) Chubykalo, O. A.; Kivshar, Y. S. Strongly localized gap solitons in diatomic lattices. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 1993, 48, 4128−4131. (38) Leykam, D.; Chong, Y. D. Edge solitons in nonlinear photonic topological insulators. Phys. Rev. Lett. 2016, 117, 143901. (39) Kartashov, Y. V.; Skryabin, D. V. Modulation instability and solitary waves in polariton topological insulators. Optica. 2016, 3, 1228−1236. (40) Lumer, Y.; Plotnik, Y.; Rechtsman, M. C.; Segev, M. Selflocalized states in photonic topological insulators. Phys. Rev. Lett. 2013, 111, 243905.

Andrea Alu: 0000-0002-4297-5274 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Science Foundation, the Air Force Office of Scientific Research, and the Simons Foundation.

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REFERENCES

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DOI: 10.1021/acsphotonics.7b00303 ACS Photonics 2017, 4, 1974−1979