Higher-Energy Composite Fermion Levels in the Fractional Quantum Hall Effect

Even though composite fermions in the fractional quantum Hall liquid are well established, it is not yet known up to what energies they remain intact. We probe the high-energy spectrum of the $1/3$ liquid directly by resonant inelastic light scattering, and report the observation of a large number of new collective modes. Supported by our theoretical calculations, we associate these with transitions across two or more composite fermions levels. The formation of quasiparticle levels up to high energies is direct evidence for the robustness of topological order in the fractional quantum Hall effect.

Figure 1

(a) Schematic description of the light scattering setup. ${\omega }_L$ and ${\omega }_S$ are the frequencies of the incident laser photons ($L$) and of the scattered photons ($S$). (b) Optical emission at $\nu =1/3$ ($B_T=8.0\mathrm{T}$, 70 mK). The two peaks, labeled $S_1$ and $T_B$, are two of the fundamental optical excitons of the GaAs quantum well; tuning the incident photon energy close to 1 of these peaks produces a resonance enhancement of ILS intensities.

Figure 2

ILS spectra of excitations at $\nu =1/3$ as a function of the energy shift (with total magnetic field $B_T=8.0\mathrm{T}$, and a tilt of 30°). The energy is shown in units of $e^2/ϵl$ on the top scale, where $l$ is magnetic length and $ϵ$, the dielectric constant of GaAs. The upper panels show peaks of several modes for certain selected incident photon energies. The lower panel contains a color plot of the intensities of both (a) “low energy” and (b),(c) the novel high-energy modes. The vertical lines mark the positions of the collective modes. The symbols, explained in the text, identify the modes with excitations of CFs across several $\Lambda$ levels, both with and without spin reversal.

Figure 3

Schematic diagram of CF excitons accompanied by theoretical calculations of their dispersions. (a) The right panel shows pictorially the SC excitations $|0,\uparrow ⟩\to |K,\uparrow ⟩$ across $K$ $\Lambda$ levels. The left panel shows the spin-flip modes $|0,\uparrow ⟩\to |K,\downarrow ⟩$ (b) Calculated dispersions of CF excitons for a 35 nm wide GaAs quantum well with an electron density of $5.0\times {10}^{10}{\mathrm{cm}}^{-2}$. The right (left) panel shows the dispersions for SC (SF) modes. The error bar at the end of each curve represents the typical statistical uncertainty in the energy determined by Monte Carlo method. Critical points in the dispersion are labeled.

Figure 4

Comparison of CF excitons with exact diagonalization results (in spherical geometry) for eight particles at $\nu =1/3$. The (red) stars show the CF exciton dispersions for the lowest three SC branches for this system as a function of the total orbital angular momentum $L$. The exact spectra are taken from Ref. [12]. The area of each black rectangle is proportional to the normalized spectral weight under the state; larger spectral weight implies greater intensity in ILS. The level-1 and level-2 CF excitons closely trace lines of high spectral weight; it is possible that still higher modes will become identifiable in the exact spectra for larger systems. The other states in the exact spectrum are interpreted as made up of multiple excitons, which are expected to couple less strongly to light.

Figure 5

Comparison between experimental energies [from Fig. 2, (red) circles] with theoretical CF exciton energies [from Fig. 3, (blue) stars], organized according to the level of the excitation. The identification of experimental modes is explained in the text. The discrepancy between theory and experiment, less than 0.2–0.3 meV, is presumably due to disorder. Estimated error bars for the experimental values are shown, unless smaller than the symbol size.