Self-induced topological transitions and edge states supported by nonlinear staggered potentials

Yakir Hadad, Alexander B. Khanikaev, and Andrea Alù

Phys. Rev. B 93, 155112 – Published 7 April 2016

Abstract

The canonical Su-Schrieffer-Heeger (SSH) array is one of the basic geometries that have spurred significant interest in topological band-gap modes. Here, we show that the judicious inclusion of third-order Kerr nonlinearities in SSH arrays opens rich physics in topological insulators, including the possibility of supporting self-induced topological transitions, as a function of the applied intensity. We highlight the emergence of a class of topological solutions in nonlinear SSH arrays localized at the array edges and with unusual properties. As opposed to their linear counterparts, these nonlinear states decay to a plateau of nonzero amplitude inside the array, highlighting the local nature of topologically nontrivial band gaps in nonlinear systems. We study the conditions under which these states can be excited and their temporal dynamics as a function of the applied excitation, paving the way to interesting directions in the physics of topological edge states with robust propagation properties based on nonlinear interactions in suitably designed periodic arrays.

Received 17 December 2015
Revised 15 March 2016

DOI:https://doi.org/10.1103/PhysRevB.93.155112

Physics Subject Headings (PhySH)

Topological insulators Topological phase transition

1-dimensional systems

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Yakir Hadad^1,*, Alexander B. Khanikaev^2,3,†, and Andrea Alù^1,‡

¹Department of Electrical and Computer Engineering, The University of Texas at Austin, 1616 Guadalupe St., UTA 7.215, Austin, Texas 78701, USA
²Department of Physics, Queens College of The City University of New York, Queens, New York 11367, USA
³Department of Physics, The Graduate Center of The City University of New York, New York, New York 10016, USA

^*yakhadad@gmail.com
^†akhanikaev@qc.cuny.edu
^‡To whom correspondence should be addressed: alu@mail.utexas.edu

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Vol. 93, Iss. 15 — 15 April 2016

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Figure 1
Geometry of interest: nonlinear SSH model. Each unit cell consists of two resonators, red and blue, with same resonance frequency $ω = ω_{0}$ $(\bar{ω} = 0)$ . The intracell coupling coefficient is $ν$ , kept constant, whereas the intercell coupling coefficient $κ$ depends on the intensities in the two resonators connected by the bond.
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Figure 2
Infinite nonlinear SSH chain. (a) For negligible intensity $I \to 0$ , the system reduces to the linear problem. Here $ν = 2 \times 10^{- 3}$ and $κ_{0} = 2.3 \times 10^{- 3}$ , such that the dispersion has a topologically trivial band gap. (b) The band-gap width is tuned by intensity. With nonlinear Kerr coefficient $α = 5 \times 10^{- 5}$ the band gap is closed and reopened at modal intensity $I_{t h} \approx 2.45$ . (c) The edge of the eigenvector of the operator $H (φ; Ψ_{0})$ is plotted for different intensities; each is represented by a different color shown in the color bar. The winding number $W = 0$ when the vector does not encircle the origin (indicated by the bold black + sign) and $W = 1$ otherwise. The threshold intensity is $I_{t h} \approx 2.45$ . (d) $W$ plotted versus intensity $I$ , illustrating the topological transition at $I_{t h} \approx 2.45$ for which the band gap closes and reopens.
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Figure 3
Frequency-domain analysis of a finite nonlinear SSH chain. (a) Topologically trivial linear chain of $N = 40$ dimers, with 40 pairs of eigenfrequencies; all eigenfrequencies are located inside the passband of the infinite chain. (b) The trajectory of the smallest eigenfrequencies [marked by bold red + signs in (a)] versus intensity. The band gap versus intensity is plotted in dashed lines. At the threshold intensity, for which the band gap closes and reopens, an abrupt change in eigenfrequency trajectory is clearly observed and the eigenfrequencies steadily converge to $\bar{ω} = 0$ . (c) The evolution of the corresponding mode profile versus intensity demonstrating a transition from standing wave to exponentially decaying solutions localized at the edges. (d) Snapshot of the mode profile at zero intensity. (e) Snapshot of the localized edge state at the highest mode intensity. The analytical solution for the semi-infinite nonlinear SSH chain is plotted as a red dashed curve.
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Figure 4
Finite-chain response versus array length. (a) Evolution of the lowest-order eigenfrequency versus intensity for different chain lengths. The threshold at which the dynamics of the eigenfrequency evolution changes depends on the infinite-chain band-gap crossing point and takes place at $I_{t h} \approx 2.45$ . (b) The same as (a) but for high intensities, deep in the edge-state regime. Like in (a), as the chain length increases the eigenfrequency trajectory tends to converge, as expected in an edge-state response. (c) Maximal intensity versus chain length at a fixed eigenfrequency $\bar{ω} = 2 \times 10^{- 6}$ . In the finite chain, the eigenfrequencies converge to $\bar{ω} = 0$ , but never reach that limit due to the unavoidable coupling between the two edges of the finite chain. If the coupling coefficient is kept constant, i.e., the frequency splitting $\bar{ω}$ is constant, while the chain length is reduced, the modal intensity increases in order to increase the attenuation rate.
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Figure 5
Time-domain analysis of a finite $N = 40$ chain. Left column: excitation with a modulated Gaussian pulse with amplitude $ξ S_{0} = 5 \times 10^{- 3}$ ; right column: same but with higher excitation amplitude $ξ S_{0} = 10^{- 1}$ . (a) Time-domain response of the first resonator in three dimers, located at the left edge, center, and right edge of the chain. The amplitudes are nearly the same, and multiple reflections of the bulk mode are observed. (b) The same as (a) but for $ξ S_{0} = 10^{- 1}$ . Excitation of the dimers in the center and at the right edge of the chain is negligible. No reflections are found, indicating the emergence of edge states. (c) The Fourier (power) spectrum of the signals in (a). At low input intensity, the chain response is nearly linear, exhibiting a clear passband and a band gap in the center. (d) Same as (c), but for $ξ S_{0} = 10^{- 1}$ . A dominant excitation of an edge state in the center of the band gap is evident. (e) Amplitude distribution along the chain, indicating excitation of a bulk mode. (f) Same as (e), indicating the excitation of an edge state. (g) Local topological number calculated from the amplitudes in (e), indicating uniform topologically trivial chain. (h) Same as (g), indicating uniformly topologically nontrivial chain.
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Figure 6
Evolution of an edge state through time-domain study. (a) Local winding number versus input intensity and dimer index. (b) Frequency of the strongest excitation per given intensity. A sharp transition is observed at about $ξ S_{0} = 0.02$ , which is in agreement with the transition seen in the winding number. The spectrum peak is located on the $\bar{ω} = 0$ axis if an edge state is excited. (c) Spectrum density at peak amplitude normalized to the input signal amplitude. The graph indicates increasing power concentration in the edge state as the input intensity is increased.
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