Laughlin anyon complexes with Bose properties

The study addresses fundamental questions about neutral excitations in fractional quantum Hall (FQHE) systems, specifically in the Laughlin state at filling factor ν = 1/3. While anyonic statistics of charged quasiparticles have been theoretically predicted since the early days of the FQHE and recently observed experimentally, neutral excitations remain less understood because they are inactive in transport. Prior work identified magneto-rotons and predicted rich neutral excitation spectra, but short lifetimes and experimental challenges hinder understanding of their statistics and thermodynamics. This work aims to determine whether spin-one neutral excitations can form the lowest-energy excitation branch at ν = 1/3, create a macroscopic quasi-equilibrium ensemble of such excitations in GaAs/AlGaAs quantum wells, and probe their statistical properties, with an emphasis on identifying Bose-like behavior.

Foundational predictions of anyons in the FQHE (Laughlin; Arovas, Schrieffer, Wilczek; Halperin) and applications in non-Abelian anyons and topological quantum computation are cited. Neutral excitations such as magneto-rotons were described in the single-mode approximation (SMA) by Girvin, MacDonald, and Platzman, with experimental confirmation via microwave absorption in certain FQHE states. Theoretical developments link magneto-rotons/magneto-gravitons to emergent geometric/gravitational modes (Haldane; Yang et al.; Golkar et al.; Gromov & Son; Can et al.). Earlier exact diagonalization suggested low-energy spin excitations near inverse magnetic length but did not fully include Zeeman energy. Methods to create quasi-equilibrium roton ensembles were established in integer quantum Hall insulators, and prior observations of condensate-like behavior in spin-triplet magneto-rotons (ν = 2) with Bose properties are referenced. The composite fermion framework (Jain), spin exciton physics in ν = 1 Hall ferromagnets, and slow spin relaxation phenomena are also discussed.

Theory and computation: The authors develop a numerical exact diagonalization scheme for finite electron systems in high magnetic fields. They validate the approach by reproducing the known analytical spin-exciton dispersion for the ν = 1 Hall ferromagnet with systems of 28–30 electrons and by matching the SMA magneto-roton dispersion for ν = 1/3 with 7–9 electrons at small momenta. They identify spin-zero magneto-gravitons (quadrupole density oscillations) and confirm the roton minimum, including the magneto-roton branch dipping into the continuum of low-momentum magneto-gravitons. Crucially, they compute spin-one excitation spectra showing multiple branches: a spin exciton (spin flip within the lowest Landau level) and combined spin–charge modes (spin flip with induced charge-density oscillations). Anti-crossing between branches yields a lowest-energy spin-one magneto-roton branch with negative dispersion near zero momentum that can lie below spin-zero magneto-rotons, depending on parameters. They incorporate realistic quantum well form factors (reducing Coulomb interactions) and Zeeman energy to determine regimes where spin-one branches are the lowest-energy excitations over all momenta. Experimental setup: The sample is a GaAs/AlGaAs quantum well of width 18 nm with electron density 8.4 × 10^10 cm^−2 and mobility 3.5 × 10^6 cm^2/V·s. The Laughlin ν = 1/3 state occurs at B ≈ 10.2 T. Experiments are conducted in a pumped 4He cryostat with base temperature ≈0.45 K and magnetic fields up to 14 T. Optical access uses a dual-fiber geometry with 400 µm core multimode fibers (NA 0.39). Two tunable continuous-wave lasers (linewidths 20 and 5 MHz) are employed: one as a pump to create neutral excitations via interband excitation to the upper spin sublevel of the zeroth Landau level; the other as a weak probe for resonant reflection. Pump power is kept below 0.1 mW to minimize heating; the probe is an order of magnitude weaker. Specular reflection is collected at −10° incidence with crossed linear polarizers to suppress surface reflection. Photoinduced resonant reflection (PRR) spectra are obtained as the difference between reflectance with pump on and off. Optical processes: Continuous-wave photoluminescence (PL) both forms and diagnoses a quasi-equilibrium ensemble of neutral excitations through interband excitation followed by rapid hole relaxation to the lowest heavy-hole sublevel and recombination with an equilibrium electron, leaving behind spin-one neutral excitations (spin excitons and spin magneto-gravitons). Only spin-flip channels are active. Photoinduced resonant reflection uses the pump to set excitation density and the probe to interrogate elastic scattering channels, including a standard phase-space filling signal and a higher-energy channel associated with creation/absorption of a spin-one magneto-graviton with zero momentum. Relaxation times are measured by shuttering the pump (10 µs trigger) and recording the decay of the PRR signal at fixed probe energy.

• Numerical exact diagonalization reveals that, contrary to SMA expectations, spin-one branches (combined spin–charge modes) can become the lowest-energy neutral excitations at ν = 1/3. Anti-crossing between the spin exciton and spin-one magneto-roton leads to a lowest-energy branch with negative dispersion near zero momentum that under realistic well form factor and Zeeman energy can lie below spin-zero magneto-rotons across momenta.
• Photoluminescence exhibits a strong, universal enhancement from the upper spin sublevel when the ν = 1/3 Laughlin state forms: at 0.5 K the PL intensity from the upper spin sublevel is about an order of magnitude larger than from the lower sublevel; at T > 1 K (when ν = 1/3 is destroyed), the lower sublevel dominates, consistent with transition probabilities. The effect is independent of excitation photon energy.
• Photoinduced resonant reflection shows a standard negative signal indicating phase-space filling and an additional elastic scattering channel at an energy higher than the principal interband transition by approximately the calculated zero-momentum spin-one magneto-graviton energy (~1.4 meV). The amplitude of this nonstandard channel grows with the number of neutral excitations (pump power), increasing by nearly an order of magnitude as pump is raised (e.g., from 20 to 300 µW) while the probe remains fixed, consistent with Bose enhancement.
• Direct relaxation measurements using pump shuttering show ultra-long decay of the PRR signal, with relaxation time exceeding 10 s (approximately 15 s inferred from logarithmic decay), far longer than known neutral excitation lifetimes in quantum Hall systems.
• The ensemble of neutral spin-one excitations displays Bose-like properties, notably enhanced elastic backscattering probability proportional to occupation number and collective optical response.

The combination of theory and experiment demonstrates that in realistic GaAs/AlGaAs quantum wells at ν = 1/3, spin-one neutral excitations can constitute the lowest-energy branch, enabling their accumulation under continuous optical pumping. The PL enhancements indicate the formation of a dense quasi-equilibrium ensemble of spin-flip neutral excitations. The appearance of a photoinduced resonant reflection channel offset by the spin-one magneto-graviton energy and its strong, excitation-number-dependent amplification supports Bose statistics via stimulated scattering (Bose enhancement). The ultra-long relaxation time (>10 s) establishes the formation of a macroscopic, long-lived, quasi-equilibrium state of neutral anyon complexes. These findings address the central question of neutral excitation statistics in FQHE systems, showing that neutral anyon complexes in the Laughlin liquid can exhibit Bose properties, which is crucial for understanding thermodynamics and opens pathways for controlled manipulation of anyon matter.

The study identifies and realizes a macroscopic quasi-equilibrium ensemble of neutral spin-one anyon complexes in the ν = 1/3 Laughlin state, demonstrating Bose-like properties through enhanced elastic light scattering and an exceptionally long relaxation time (~15 s). Numerically, the work establishes that spin-one neutral branches can be energetically favored under realistic conditions, explaining the experimental feasibility of creating dense ensembles. The results offer a platform for real-time manipulation and optical interrogation of neutral anyon matter, providing new insights into the thermodynamics and dynamics of FQHE excitations. Future research could quantify the occupation dependence of scattering rates, map dispersion via momentum-resolved probes, explore other filling factors and material platforms, and seek direct interferometric or correlation-based signatures of anyonic statistics in neutral excitations.

The interpretation of the additional PRR channel, while consistent with spin-one magneto-graviton processes and Bose enhancement, may admit alternative descriptions, and the complex valence band structure can influence scattering efficiencies. Numerical results rely on finite-size exact diagonalization and model form factors; extrapolation to the thermodynamic limit and to different sample parameters requires caution. Experimentally, only spin-flip optical channels are accessible in the chosen geometry and narrow (≤20 nm) wells, and conclusions are drawn for specific GaAs/AlGaAs structures at B ≈ 10.2 T and T ≈ 0.5 K. Direct measurement of exchange statistics for neutral modes is not performed; Bose properties are inferred from optical scattering behavior.