Lévy Flights and Hydrodynamic Superdiffusion on the Dirac Cone of Graphene

We show that the hydrodynamic collision processes of graphene electrons at the neutrality point can be described in terms of a Fokker-Planck equation with a fractional derivative, corresponding to a Lévy flight in momentum space. Thus, electron-electron collisions give rise to frequent small-angle scattering processes that are interrupted by rare large-angle events. The latter give rise to superdiffusive dynamics of collective excitations. We argue that such superdiffusive dynamics is of more general importance to the out-of-equilibrium dynamics of quantum-critical systems.

Figure 1

(a) A wrapped Gaussian flight (upper circle) and a wrapped Cauchy flight (lower circle) with rare large momentum-transfer processes. (b) Illustration of Lévy flight in momentum space for graphene at the Dirac point. Electrons and holes that are thermally excited collide with each other. Most of the time, the momentum transfer due to the electron-electron Coulomb interaction leads to small-angle scattering. However, those processes are interrupted by rare processes with large momentum transfer. The latter change the dynamics of the system qualitatively, leading to an accelerated or superdiffusive dynamics.

Figure 2

Upper panel: Angular momentum dependence of matrix elements of the collision operator $⟨m,s|\mathcal{C}|m,s^{\prime }⟩$, where $s\in \{1,2,3\}$ refers to the collinear eigenmodes of Eq. (9). In the text, we discuss (for simplicity) only $⟨m|\mathcal{C}|m⟩\equiv ⟨m,1|\mathcal{C}|m,1⟩$. Lower panel: log-log plot of matrix element demonstrating that we can distinguish the $|m|$ dependence from, e.g., $|m|\log |m|$.

Figure 3

Upper panel: Postinjection distribution function that follows from the fractional Fokker-Planck equation [Eq. (2)] with the external perturbation of Eq. (15). Notice the superdiffusive dynamics at short times. Lower panel: Comparison of superdiffusive and diffusive dynamics at short times. At angles away from the peak at $\theta =0$, superdiffusion leads to a faster growth of the distribution function. Inset: The initial peak at $\theta =0$ decays as $1/t$ for superdiffusion and $1/\sqrt{t}$ for ordinary diffusion. This behavior dominates the heating of the system (see main text).