Self-Accelerating Topological Edge States

Abstract

Edge states emerging at the boundaries of materials with nontrivial topology are attractive for many practical applications due to their remarkable robustness to disorder and local boundary deformations, which cannot result in scattering of the energy of the edge states impinging on such defects into the bulk of material, as long as forbidden topological gap remains open in its spectrum. The velocity of the such states traveling along the edge of the topological insulator is typically determined by their Bloch momentum. In contrast, here, using valley Hall edge states forming at the domain wall between two honeycomb lattices with broken inversion symmetry, we show that by imposing Airy envelope on them one can construct edge states which, on the one hand, exhibit \textit{self-acceleration} along the boundary of the insulator despite their fixed Bloch momentum and, on the other hand, \textit{do not diffract} along the boundary despite the presence of localized features in their shapes. We construct both linear and nonlinear self-accelerating edge states, and show that nonlinearity considerably affects their envelopes. Such self-accelerating edge states exhibit self-healing properties typical for nondiffracting beams. Self-accelerating valley Hall edge states can circumvent sharp corners, provided the oscillating tail of the self-accelerating topological state is properly apodized by using an exponential function. Our findings open new prospects for control of propagation dynamics of edge excitations in topological insulators and allow to study rich phenomena that may occur upon interactions of nonlinear envelope topological states.

Figures (7)

• Figure 1: (a) Inversion-symmetry-broken honeycomb waveguide array with the domain wall indicated by the dashed rectangle. The depth of the red and blue waveguides is ${p+\delta}$ and ${p-\delta}$, respectively. (b) Band structure of the array from panel (a). The blue and gray lines represent propagation constants of the valley Hall edge state and of the bulk states, respectively. (c) First-order ($b'$, solid line) and second-order ($b"$, dashed line) derivatives of the propagation constant $b$ of the valley Hall edge state. (d) Field modulus distribution $|\psi|$ of the valley Hall edge state at $k_y=-0.3\textrm{K}_y$ corresponding to the blue dot in panel (b). Panels (a) and (d) correspond to ${-20\le x \le 20}$ and ${-3.5\le y \le 3.5}$ windows.
• Figure 2: (a) Cross-section ${|\psi(x=0,y)|}$ illustrating propagation dynamics of the valley Hall edge state with ${k_y=0}$ and superimposed Airy envelope with ${\mu=0.002}$. The parabolic dashed line is the theoretically predicted trajectory of the self-accelerating valley Hall edge state. The dynamics is shown within the window ${0\le z \le 200}$ and ${-80\le y \le 80}$. (b,c) Same as in (a), but for the valley Hall edge states with Bloch momenta ${k_y=-0.3{\rm K}_y}$ and ${k_y=0.3{\rm K}_y}$, respectively. (d) Self-healing of the self-accelerating valley Hall edge state from (a) with eliminated second lobe. (e) Field modulus distributions $|\psi(x,y)|$ at distances corresponding to the vertical dashed lines in (a) that clearly illustrate self-acceleration of the beam along the domain wall. Panels (e) are shown within the window ${-20 \le x \le 20}$ and ${-80\le y \le 80}$. (f) Field modulus distributions at different distances $z$ corresponding to the vertical dashed lines in (d).
• Figure 3: (a) Blue curve (ref. the left $y$ axis): Peak amplitude of the nonlinear self-accelerating beam versus energy shift $\beta=b_\textrm{nl}-b_0$. Red curve (ref. the right $y$ axis): FWHM of the main lobe in the intensity distribution of the nonlinear self-accelerating solution versus its peak amplitude with ${k_y=0}$. The "energy shift" corresponding to the dots labeled ${1\sim4}$ is given by $0.006$, $0.038$, $0.138$, and $0.516$, respectively. (b) Profiles of the self-accelerating solutions for different peak amplitudes $|w|_{\max}$, corresponding to the dots in (a). For all cases: ${\mu=0.002}$.
• Figure 4: (a) Evolution dynamics of the nonlinear self-accelerating valley Hall edge state at ${k_y=0}$, for ${\chi\approx0.1592}$, ${b"\approx-0.7763}$, ${\mu=0.002}$, and ${|\mathcal{A}|_{\max}=0.63}$. (b) Field modulus distributions $|\psi(x,y)|$ at selected propagation distances. (c,d) Same as in (a,b), but for ${k_y=-0.3{\rm K}_y}$, at ${\chi\approx0.1663}$, ${|b'|\approx0.5029}$, ${b"\approx-0.6584}$, and ${|\mathcal{A}|_{\max}=0.61}$. (e,f) Same as (c,d) but for and ${k_y=+0.3{\rm K}_y}$. Dashed lines in (a,c,e) stand for the predicted accelerating trajectories. Panels in (a,c,e) are shown in the window ${0\le z \le 200}$, ${-80\le y \le 80}$. Panels in (b,d,f) are shown in the window ${-20 \le x \le 20}$ and ${-80\le y \le 80}$. For all cases: ${\mu=0.002}$.
• Figure 5: (a) Composite photonic graphene lattice with an $Z$-path domain wall indicated by the blue color. The arrows indicate the propagation direction of the input beam. (b) Field modulus distributions of a finite-energy self-accelerating valley Hall edge state at different distances illustrating passage through the $Z$-shaped region. The insets show the spatial spectrum of the beam in the Fourier domain with hexagons representing Brillouin zone. All panels are shown within the window ${-20 \le x \le 20}$, ${-100\le y \le 100}$. The insets are shown within the window ${-5 \le k_{x,y} \le 5}$.