Emergence of caustics in dynamics of the Kitaev model

Abstract

We study quasiparticle dynamics in two-dimensional (2D) integrable Kitaev honeycomb model both without and in the presence of an external periodic drive. We identify light-cones in wavefunction propagation as a signature of quantum caustic, i.e. bright structures formed during quantum dynamics analogous to that of imperfect focusing in geometrical optics. We show that this dynamics follows an angle in spatial direction and it is anisotropic with respect to model parameters. Using coalescence of critical points, we provide an exact solution to the envelope of caustic, which corresponds to the Lieb-Robinson bound in 2D. Further, consedering the system to be periodically driven, we point out that the caustic structure completely changes in presence of external time dependent drive.

Figures (7)

• Figure 1: (Color online) Kitaev honeycomb lattice with couplings $J_{1}$, $J_{2}$ and $J_{3}$. $\mathbf{M}_{1}$ and $\mathbf{M}_{2}$ are the spanning vectors of the lattice, and $a$ and $b$ represent two inequivalent sites of a unit cell.
• Figure 2: (Color online) Plots of $|\psi(\mathbf{r},t)|$ as a function of $\mathbf{r}$ for several representative values of $J_{2}/J$ for $J_{1}=J$ and $J_{3}=5$ at $t=3$. The plot displays the change in wavefunction amplitude as a function of $J_{2}/J_{1}$.
• Figure 3: (Color online) Plot of $|\psi(\mathbf{r},t)|$ at points $(-10, 10), (-20, 20), (-30, 30)$ along $-45^{\circ}$ in the $n_{1}-n_{2}$ plane as a function of $\phi=\tan^{-1}(J_{2}/J_{1})$ keeping $J^{2}=1$ fixed.
• Figure 4: (Color online) Relation between $x_{1}$, $x_{2}$ and $t$ representing the caustics trajectories in (a) gapped (b) gapless phase
• Figure 5: (Color online) Plot of $|\psi(\mathbf{r},t)|$ in Eq. (\ref{['Eq.2.9']}) depict light-cone like dynamics for $J_{1}=J_{2}=1, J_{3}=3$ and $N=100$. White lines show the envelope representing Lieb-Robinson bound.