Nonlinear Quantum Shock Waves in Fractional Quantum Hall Edge States

Using the Calogero model as an example, we show that the transport in interacting nondissipative electronic systems is essentially nonlinear and unstable. Nonlinear effects are due to the curvature of the electronic spectrum near the Fermi energy. As is typical for nonlinear systems, a propagating semiclassical wave packet develops a shock wave at a finite time. A wave packet collapses into oscillatory features which further evolve into regularly structured localized pulses carrying a fractionally quantized charge. The Calogero model can be used to describe fractional quantum Hall edge states. We discuss perspectives of observation of quantum shock waves and a direct measurement of the fractional charge in fractional quantum Hall edge states.

Figure 1

Numerical solution of BO Eq. (8). (a) The initial Lorentzian profile containing 7 solitons is shown at time $t=0.9t_c$. (b) The wave after a shock is shown together with the overhanging solution of Riemann Eq. (5) (red, dashed) at $t=2.7t_c$. Vertical lines mark the trailing and the leading edges. The moduli $v$, $\tilde{v}$, $K$ of the one-phase solution (9) are assigned to different branches of the multivalued solution of (5). (d) The solutions of BO and Riemann’s equations are shown at $t=11t_c$. (c) Whitham modulated wave represents an approximate soliton train behavior for an initial Lorentzian shape of a larger area.