Refermionized theory of the edge modes of a fractional quantum Hall cloud

TL;DR

The paper shows that the nonlinear chiral Luttinger liquid description of fractional quantum Hall edge modes can be mapped exactly to a one-dimensional system of massive, interacting chiral fermions via refermionization. This framework clarifies the mechanisms behind dynamic structure factor broadening, universal power-law exponents at spectral edges, and the detailed structure of edge excitations, achieving quantitative agreement with full two-dimensional simulations. By leveraging Tomonaga–Luttinger liquid techniques, the work reveals how confinement-induced dispersion and interactions control edge dynamics and decay times, while uncoupling complexity to accessible fermionic degrees of freedom. The results have broad implications for experiments in electronic, cold-atom, and photonic platforms and point toward nonlinear edge phenomena such as solitons in FQH fluids.

Abstract

Making use of refermionization techniques, we map the nonlinear chiral Luttinger liquid model of the edge modes of a spatially confined fractional quantum Hall cloud developed in our recent work [Phys. Rev. A 107, 033320 (2023)] onto a one-dimensional system of massive and interacting chiral fermions, whose mass and interactions are set by the filling factor of the quantum Hall fluid and the shape of the external confining potential at the position of the edge. As an example of the predictive power of the refermionized theory, we report a detailed study of the dynamic structure factor and of the spectral function of a fractional quantum Hall cloud. Among other features, our refermionized theory provides a physical understanding of the effective decay of the edge excitations and of the universal power-law exponents at the thresholds of the dynamic structure factor. The quantitative accuracy of the refermionized theory is validated against a full two-dimensional calculation based on a combination of exact diagonalization and Monte Carlo sampling.

Figures (14)

• Figure 1: Numerical results for the energy spectrum $E_{l,n}-E_0$ of the bosonic NL-$\chi$LL model Eq. \ref{['eq:NLXLL_hamiltonian']} (green lines) and its fermionic counterpart Eq. \ref{['eq:fermion_hamiltonian']} (red lines). Eigenstates are plotted against their angular momentum $l$. We here used $N=25$ bosons at $\nu=1/2$ filling in a quartic trap; the model free parameters, $\Omega$ and $c$ have been calculated from $N$ and $\nu$ using Eq. \ref{['eq:angular_velocity']} and Eq. \ref{['eq:curvature']}.
• Figure 2: (a) Numerical results for the energy spectrum $E_{l,n}-E_0$ of a FQH cloud confined by a quartic $V_{\rm conf}(r)=\lambda r^4$ trap. Eigenstates are plotted against their angular momentum $l$ and are coloured according to their DSF weight $|\bra{0}\delta\hat{\rho}_l\ket{l,n}|^2$. The red dotted line is the linear dispersion of the edge modes, $\Omega l$; the black line includes the cubic correction, $\Omega l-\alpha l^3$. Black circles with crosses indicate the center-of-mass \ref{['eq:DSFFirstMoment']} of the DSF. (b) Plot of the DSF (on the $x$-axis) against the excitation energies $E_{l,n}-E_0$ (on the $y$-axis). The black circles are the result of full 2D numerical calculations, the red crosses are the predictions of the bosonic NL-$\chi$LL model of Eq. \ref{['eq:NLXLL_hamiltonian']} and the green plus symbols show the result of the exact diagonalization of the refermionized description Eq. \ref{['eq:fermion_hamiltonian']}. For each angular momentum $l$, the main emerging structures have been joined with black-dashed lines as a guide for the eye. Both panels are for $N=25$ bosons at $\nu=1/2$ filling.
• Figure 3: NL-$\chi$LL numerical results for the DSF broadening $\Delta E_l$ (points connected by dashes), as a function of $l(l-1)$ - $l$ being the angular momentum. The numerically computed broadening $\Delta E_l$ is compared to its free fermion approximation (solid black line), Eq. \ref{['eq:broadening1']}, for different values of the filling fraction $\nu$. Analogously to Fig. \ref{['fig:comparison']}, the model free parameters $\Omega$ and $c$ have been calculated from $N=25$ and $\nu$ using Eq. \ref{['eq:angular_velocity']} and Eq. \ref{['eq:curvature']}.
• Figure 4: (a) Plot of the (normalized) total width $E_+(l)-E_-(l)$ defined in Eq. \ref{['eq:broadening1']} as a function of the (rescaled) angular momentum $l(l-1)$. The results of full 2D numerical calculations for various values of $\nu$ and $\delta$ accurately follow the behaviour $\sqrt{\nu}\Delta E_l/(\lambda \delta R_{cl}^{\delta-4})=(\delta-2)l(l-1)$ predicted by Eq. \ref{['eq:broadening1']} together with Eq. \ref{['eq:curvature']}, shown here as black-dashed lines. (b) Plot of the second moment $\omega_2(l)$ of the DSF as a function of the (rescaled) angular momentum $l^2(l^2-1)$. The results of the full 2D numerical calculations for various values of $\nu$ and $\delta$ accurately follow the behaviour $\nu\omega_2(l) / (\lambda\delta R_{cl}^{\delta-4})^2 = (\delta-2)^2l^2(l^2-1)/12$ predicted by Eq. \ref{['eq:broadening2']} together with Eq. \ref{['eq:curvature']}, shown here as black-dashed lines. (c) Time-evolution of the square modulus of the fundamental component of the edge-density response in response of a weak pulsed excitation of strength $u_0$ and angular momentum $l=3$. The black curve is the result of the full 2D numerical calculation, while the brown curve is the the short-time decay prediction of Eq. \ref{['eq:density_decay']}. In all panels, a FQH cloud of $N=25$ particles is considered.
• Figure 5: Schematic diagram of the different particle-hole excitations (black-arrows) across the Fermi point (vertical-dashed lines) at a fixed angular momentum $l=4$. For each state, the corresponding partition of $l=4$ are indicated on the right-hand side of the plot.