Linear and nonlinear edge dynamics of trapped fractional quantum Hall droplets

We report numerical studies of the linear and nonlinear edge dynamics of a non-harmonically-confined macroscopic fractional quantum Hall fluid. In the long-wavelength and weak excitation limit, observable consequences of the fractional transverse conductivity are recovered. The first nonuniversal corrections to the chiral Luttinger liquid theory are then characterized: for a weak excitation in the linear response regime, cubic corrections to the linear wave dispersion and a broadening of the dynamical structure factor of the edge excitations are identified; for stronger excitations, sizable nonlinear effects are found in the dynamics. The numerically observed features are quantitatively captured by a nonlinear chiral Luttinger liquid quantum Hamiltonian that reduces to a driven Korteweg–de Vries equation in the semiclassical limit. Experimental observability of our predictions is finally discussed.

Figure 1

(a) Radial profile of the GS density. (b),(c) Excitation spectra for (b) $N=9$ and (c) $N=25$ particles (red crosses), compared to ED [black dots in (b)] and the nonlinear $\chi \mathrm{LL}$ theory (18) [black dots in (c)]. Anharmonic $\delta =4$ trap with $\lambda ={10}^{-6}$; filling factor $\nu =1/2$.

Figure 2

(a) Amplitude of the edge density response after the weak $l=2$ external potential has been switched off, for different filling factors $\nu$, normalized to the one of a large IQH system. (b) DSF weights plotted against the excitation energy of each eigenstate. Within each $l$ sector, the dashed lines are guides to the eye. MC data (black dots) are compared to the nonlinear $\chi \mathrm{LL}$ theory (red crosses). (c) SSF $S_l$ as a function of $l$ for the same values of $\nu$ as in (a). Dashed lines indicate the $\chi \mathrm{LL}$ prediction $S_l=\nu l$. (d) Normalized edge-mode dispersion for different $N$. Same trap potential as in Fig. 1. In (b) and (d), the filling factor is fixed to $\nu =1/2$.

Figure 3

(a),(b) Normalized angular velocity $\mathrm{\Omega }$ and group velocity dispersion parameter $\alpha$ as a function of $N$ for different trap exponents $\delta$ at a constant $\nu =1/2$. (c),(d) Normalized $\alpha$ as a function of inverse filling $1/\nu$ for (c) $\delta =4$ and different $N$, and (d) as a function of trap curvature $\propto \delta (\delta -2)$ for different fillings $\nu$ at given $N$. All points are extracted from low-$l$ fits to the numerical MC predictions for ${\omega }_l$ as a function of $l$.

Figure 4

(a) Color plot of the density near the edge at $c_{0}t\simeq 0.1$ after an excitation pulse of the form (5) with an intensity large enough to induce a significant nonlinear dynamics on this temporal scale. White (black) lines are isodensity contours for the excited (unexcited) system. (b),(c) Time evolution of the fundamental and second harmonic spatial Fourier components of the edge density variation of $N=30$ (red) and $N=9$ (yellow) clouds. ED data for $N=9$ are shown as brown dashed lines as a benchmark. Dotted black lines and black dots indicate, respectively, the solution of the semiclassical equation (16) and of the quantum model ${\widehat{H}}_{\chi LL}^{NL}$. Insets show a magnified view of the dynamics at early times. Same trap parameters as in Fig. 1; filling factor $\nu =1/2$.

Figure 5

(a) Eigenenergy spectrum (with error bars) for a $N=25, \nu =1/2$ FQH cloud confined by a $\delta =4$ quartic potential. (b) Magnified view of the statistical errors on the eigenenergies. The panels on the right show histograms for the $M=250$ Monte Carlo realizations of the energy spectrum in each $l$ sector. Each point is obtained by an independent run.

Figure 6

(a) DSF weights $|\langle 0|\delta {\widehat{\rho }}_l{|l,n\rangle |}^2$ (with error bars) for a $N=25, \nu =1/2$ FQH cloud confined by a $\delta =4$ quartic potential. (b) Suitably normalized first moment ${\omega }_l$ of the DSF (with error bars). The panels on the right show histograms for the $M=250$ Monte Carlo realizations of the DSF weights in each $l$ sector.

Figure 7

Density variation $\delta \rho (x,y)$ at a time $c_{0}t\simeq 0.1$ after the $\nu =1/2$ cloud has been excited by means of (a) an excitation with a nontrivial radial profile described by Eq. (B3) or (b) by a radially flat profile. (c),(d) Time evolution of the fundamental and second harmonic spatial Fourier components of the edge-density variations in the same two cases (black and red).

Figure 8

Suitably normalized width $\mathrm{\Delta }{E}_l$ of the DSF of a $\nu =1/2$ FQH cloud as a function of the particle number $N$ for different values of (a) $l=2$, (b) 3, (c) 4, and (d) 5. The dashed lines are power-law fits to the data and highlight the scaling with $N$ at fixed $l$, for different confinements $\delta$. The fitted exponents are in close agreement with the expected ones indicated in the legends.

Figure 9

Time evolution of the (a) fundamental, (b) second, and (c) third harmonic of the spatial Fourier transform of the edge-density variation of a $N=30$ cloud at filling factor $\nu =1/2$ in a $\delta =4$ quartic trap. The results of the microscopic MC calculations (red lines) are compared with those of the nonlinear $\chi \mathrm{LL}$ model (black circles). For comparison, the result of a nonlinear $\chi \mathrm{LL}$ model without the dispersive contribution is shown as yellow triangles.