Linear and nonlinear edge dynamics of trapped fractional quantum Hall droplets

TL;DR

This work analyzes linear and nonlinear edge dynamics of a trapped Laughlin fractional quantum Hall liquid using a novel Monte Carlo-based projection method onto the Laughlin edge subspace. It demonstrates the quantization of transverse conductivity in the linear regime and reveals first non-universal corrections to χLL, including a cubic dispersion and DSF broadening caused by anharmonic confinement. Building on these insights, the authors formulate a nonlinear chiral Luttinger liquid Hamiltonian whose semiclassical limit yields a driven Korteweg–de Vries equation, and they show quantitative agreement with full numerics across dispersion, broadening, and nonlinear edge evolution. The results provide a scalable, one-dimensional framework to study large FQH edge systems and offer concrete guidance for experimental observations in ultracold atoms and related platforms. Overall, the paper bridges microscopic simulations and effective edge theories to capture essential nonlinear and dispersive edge physics in fractional quantum Hall droplets.

Abstract

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 non-universal 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.

Figures (9)

• 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 $\chi$LL theory \ref{['eq_NonLinearXLL']} [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 $\chi$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 $\chi$LL prediction $S_l=\nu l$. (d) Normalized edge-mode dispersion for different $N$. Same trap potential as in Fig.\ref{['fig_GSDensity_Spectrum']}. In panels (b,d) the filling factor is fixed to $\nu=1/2$.
• Figure 3: (a,b) Normalized angular velocity $\Omega$ 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) Colorplot of the density near the edge at $c_0 t\simeq0.1$ after an excitation pulse of the form \ref{['eq:Ul']} with an intensity large enough to induce a significant non-linear dynamics on this temporal scale. White (black) lines are iso-density 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 semi-classical equation \ref{['eq_NonLinearClassicalHydrodinamicalEquation']} and of the quantum model $\hat{H}_{\chi LL}^{NL}$. Insets show a magnified view of the dynamics at early times. Same trap parameters as in Fig.\ref{['fig_GSDensity_Spectrum']}, filling factor $\nu=1/2$.
• Figure 5: (a) Eigenenergy spectrum (with errorbars) for a $N=25$, $\nu=1/2$ FQH cloud confined by a $\delta=4$ quartic potential. (b) Magnified view on 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.