## Abstract

Exotic physics could emerge from interplay between geometry and correlation. In fractional quantum Hall (FQH) states^{1}, novel collective excitations called chiral graviton modes (CGMs) are proposed as quanta of fluctuations of an internal quantum metric under a quantum geometry description^{2,3,4,5}. Such modes are condensed-matter analogues of gravitons that are hypothetical spin-2 bosons. They are characterized by polarized states with chirality^{6,7,8} of +2 or −2, and energy gaps coinciding with the fundamental neutral collective excitations (namely, magnetorotons^{9,10}) in the long-wavelength limit. However, CGMs remain experimentally inaccessible. Here we observe chiral spin-2 long-wavelength magnetorotons using inelastic scattering of circularly polarized lights, providing strong evidence for CGMs in FQH liquids. At filling factor *v* = 1/3, a gapped mode identified as the long-wavelength magnetoroton emerges under a specific polarization scheme corresponding to angular momentum *S* = −2, which persists at extremely long wavelength. Remarkably, the mode chirality remains −2 at *v* = 2/5 but becomes the opposite at *v* = 2/3 and 3/5. The modes have characteristic energies and sharp peaks with marked temperature and filling-factor dependence, corroborating the assignment of long-wavelength magnetorotons. The observations capture the essentials of CGMs and support the FQH geometrical description, paving the way to unveil rich physics of quantum metric effects in topological correlated systems.

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## Data availability

All data needed to evaluate the conclusions in the paper are included in this paper. Additional data that support the plots and other analysis in this work are available from the corresponding author upon request.

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## Acknowledgements

We gratefully acknowledge illuminating discussions with B. Yang, D. X. Nguyen, D. T. Son, K. Yang, J. K. Jain and R.-R. Du. We thank B. Yang for comments on the manuscript. We thank Y. F. Wang and X. Y. Lu for assistance in low-temperature measurements. This work is supported by the National Natural Science Foundation of China (Grant No. 12074177), Innovation Program for Quantum Science and Technology (Grant No. 2021ZD0302600), Program for Innovative Talents and Entrepreneur in Jiangsu and the start-up funding of Nanjing University. The work at Columbia University is funded by the National Science Foundation, Division of Materials Research under Grant DMR-2103965. The Princeton University portion of this research is funded in part by the Gordon and Betty Moore Foundation’s EPiQS Initiative, Grant GBMF9615.01 to Loren Pfeiffer. U.W. acknowledges support from German Science Foundation under Grants WU 637/7−1 and 7-2.

## Author information

### Authors and Affiliations

### Contributions

L.D. supervised the project. L.D. and J.L. designed and set up the low-temperature optical facility. L.D. and Z.L. conceived the experiments. K.W.W. and L.N.P. grew the heterostructure. J.L., Z.L., Z.Y., Y.H. and L.D. performed the optical measurements. L.D., J.L., Z.L. and Z.Y. analysed the data. A.P., Z.L., U.W. and L.D. discussed the scientific objectives. L.D., Z.L. and J.L. wrote the paper. J.L., Z.L., Z.Y., U.W., C.R.D. and L.D. commented on the paper during the writing process.

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## Extended data figures and tables

### Extended Data Fig. 1 Filling factor and temperature dependence of magnetoroton modes at *v* = 1/3 in the unpolarized geometry with *θ* = 25°.

Spectra of \({\varDelta }_{{\rm{m}}}^{0}\), \({\varDelta }_{{\rm{m}}}^{{\rm{R}}}\) and \({\varDelta }_{{\rm{m}}}^{\infty }\) at filling factors around *v* = 1/3 are shown in **a**,**c** and **e**, respectively. The mode intensities reach their maxima at *v* = 1/3, and rapidly decrease as filling factors deviate from *v* = 1/3. The observations suggest that as the system becomes more compressible, the quantum liquid supporting magnetoroton excitations appears to vanish. Temperature dependence of \({\varDelta }_{{\rm{m}}}^{0}\), \({\varDelta }_{{\rm{m}}}^{{\rm{R}}}\) and \({\varDelta }_{{\rm{m}}}^{\infty }\) at *v* = 1/3 is shown in **b,d** and **f**, respectively. With increased temperatures, the intensities of the magnetoroton modes decrease and vanish at temperatures below 800 mK. The behaviors indicate that the magnetoroton modes are highly temperature-sensitive collective excitations, further highlighting their roles in characterizing the properties of the FQH states. RILS peaks are marked by vertical black arrows.

### Extended Data Fig. 2 Peak fitting of the \({{\boldsymbol{\Delta }}}_{{\bf{m}}}^{0}\) mode at *v* = 1/3 in the RR geometry with *θ* = 25°.

The measured \({\varDelta }_{{\rm{m}}}^{0}\) mode at resonance (black open dots) includes contribution from photoluminescence background. The red open dots show the \({\varDelta }_{{\rm{m}}}^{0}\) mode after subtracting smoothed photoluminescence background (the grey dashed line), which are fitted by a Lorentzian peak (the black line) with FWHM of 30 μeV. The combination (the red dashed line) of the fitted Lorentzian peak and photoluminescence background gives a remarkable match with the measured signals in the RR geometry. The relatively narrow peak width of this mode suggests wavevector conservation in the scattering process with *q* = *k* ≪ 1/*l*_{B}, confirming its long-wavelength nature. PL, photoluminescence.

### Extended Data Fig. 3 RILS measurements at *v* = 1/3 with *θ* = 10°.

**a**, RILS spectra at *v* = 1/3 in the unpolarized geometry as a function of *ω*_{L}. Similar to those in Fig. 1e, the red and blue dashed lines indicate magnetoroton and spin-wave excitations, respectively. Compared with the result at *θ* = 25°, \({\varDelta }_{{\rm{s}}}^{0}\) at *θ* = 10° has a lower energy but remains at *E*_{z}, confirming its assignment. **b**, Calculated dispersions of collective excitations at *v* = 1/3 that support the assignment of the modes. The red dashed line is scaled down from the ideal zero-width result^{29} by a factor of 0.305, accounting for the finite-thickness effect. The blue dashed line represents a generic dispersion for the spin-wave excitations. **c**, RILS spectra of the \({\varDelta }_{{\rm{m}}}^{{\rm{R}}}\) excitation at *v* = 1/3 in the unpolarized geometry at different *ω*_{L}. The well-resolved peaks are marked by the vertical red dashed line. We mention that the \({\varDelta }_{{\rm{m}}}^{{\rm{R}}}\) mode energy at 25° is larger than that at 10°, since a larger tilted angle induces a higher in-plane magnetic field, causing the electrons to behave in a more two-dimensional manner. On the other hand, the \({\varDelta }_{{\rm{m}}}^{0}\) energies at two tilted angles are closed. It is because a smaller tilted angle also gives a reduced *kl*_{B} in the magnetoroton dispersion, which corresponds to an increased \({\varDelta }_{{\rm{m}}}^{0}\) energy, as shown in the red dashed line in **b**. The two factors interplay in the case of \({\varDelta }_{{\rm{m}}}^{0}\). **d**, At *v* = 1/3, magnetoroton modes could be understood as excitations of composite fermions from the topmost (the lowest) occupied composite-fermion Landau level to the next unoccupied one.

### Extended Data Fig. 4 Optical spectra at *v* = 1/3 measured at different *ω*_{L} in the RR and LR geometries with *θ* = 10°.

**a**, Resonant enhancement of RILS signals of the \({\varDelta }_{{\rm{m}}}^{0}\) mode in the RR geometry. The RILS peaks maintain a consistent energy shift at different *ω*_{L}. The resonant enhancement of \({\varDelta }_{{\rm{m}}}^{0}\) is clearly demonstrated by the marked intensity dependence on *ω*_{L}. RILS peaks are marked by the dashed red line. **b**, Optical spectra measured in the LR geometry. The feature of the spectrum measured at *ω*_{L} = 1520.89 meV (that also appears in the LR geometry in Fig. 3a) shifts as *ω*_{L} varies, which is identified as photoluminescence signals. No RILS signals are found in the spectra in the LR geometry. PL, photoluminescence.

### Extended Data Fig. 5 Energy ratios of the measured spin-2 modes to \({{\boldsymbol{\Delta }}}_{{\bf{m}}}^{{\bf{R}}}\) in the *v* = 1/3 and 2/3 states.

In RILS experiments, the wavevector *k* = (2*ω*_{L}/*c*)sin*θ* transferred to the system can be adjusted by altering *θ*. At *v* = 1/3, a reduction of *θ* from 25° to 10° results in a decrease of *kl*_{B} from ≈ 0.05 to an extremely small value ≈ 0.02, effectively approaching the long-wavelength limit (*q* = *k* = 0). At *v* = 1/3, the energy ratio of the spin−2 mode to \({\varDelta }_{{\rm{m}}}^{{\rm{R}}}\) reaches 2.07 at *kl*_{B} ≈ 0.02 (Fig. 3a and Extended Data Fig. 3) and decreases by 15% as *kl*_{B} increases to ≈ 0.05 (Figs. 1e and 2c), as guided in the red dashed line. At *v* = 2/3, the energy ratio reaches 2.2 at *kl*_{B} ≈ 0.03 with *θ* = 10° (Fig. 3b and Supplementary Fig. 3). The error bars originate from the uncertainty in determining the energy positions of these two modes in RILS spectra. Notably, at extremely small wavevectors, the measured energy ratios at *v* = 1/3 and 2/3 are larger than the value (1.8 at zero wavevector) expected for a two-roton bound state (the black dashed arrow). The ratio for the two-roton bound state would increase with wavevectors but have to be lower than two because of its two-roton characteristic. We would like to mention that the large energy ratio at *v* = 2/3 indicates that \({\varDelta }_{{\rm{m}}}^{0}\) could be in the continuum of excitations. Interestingly, in CP-RILS measurements, \({\varDelta }_{{\rm{m}}}^{0}\) is well resolved in the LL geometry, which indicates that the continuum does not have a large contribution in this geometry.

### Extended Data Fig. 6 Peak fitting of the \({{\boldsymbol{\Delta }}}_{{\bf{m}}}^{0}\) modes at *v* = 1/3, *v* = 2/3, *v* = 2/5 and *v* = 3/5 with *θ* = 10°.

The black open dots represent the experimental signals of the \({\varDelta }_{{\rm{m}}}^{0}\) modes in CP geometries (RR for *v* = 1/3 and *v* = 2/5, LL for *v* = 2/3 and *v* = 3/5). The grey dash lines indicate smoothed photoluminescence background signals. The black lines are the fitted Lorentzian peaks with small FWHM (29 μeV for *v* = 1/3, 33 μeV for *v* = 2/3, 30 μeV for *v* = 2/5 and 27 μeV for *v* = 3/5). The combination of these fitted Lorentzian peaks and photoluminescence background signals (the red dashed lines) gives a remarkable agreement to the measured RILS spectra. The sharpness of these peaks is noteworthy, as it indicates wavevector conservation in the scattering. PL, photoluminescence.

### Extended Data Fig. 7 Comparison of the measured \({{\boldsymbol{\Delta }}}_{{\bf{m}}}^{0}\) energies to theoretical calculations.

The yellow dots represent theoretical calculations of the \({\varDelta }_{{\rm{m}}}^{0}\) energies at *v* = 1/3 (*p* = 1) and *v* = 2/5 (*p* = 2), obtained from ref. ^{29} for zero-width two-dimensional systems. Theoretical values given in the reference in the unit of *E*_{c} are converted to meV scale using the density of our sample. The black dots represent experimental results obtained in our RILS measurements. These experimental results are taken at *θ* = 10° and correspond to filling factors *v* = 1/3 (*p* = 1), 2/3 (*p* = −2), 2/5 (*p* = 2) and 3/5 (*p* = −3). The error bars indicate the uncertainty in determining the energy positions in the RILS spectra. Both theoretical (yellow dots) and experimental (black dots) gap energies are found proportional to (*e*^{2}/*εl*_{B})/|2*p* + 1|, characteristic of composite fermions moving under effective magnetic fields in the orbits, which determine the magnetoroton gaps. The dashed line represents an excellent linear fit of the experimental data, yielding a slope of 0.142 and y-intercept of 0.009 meV. The solid line is the guide to the eye.

### Extended Data Fig. 8 Filling factor dependence of the \({{\boldsymbol{\Delta }}}_{{\bf{m}}}^{0}\) modes at *v* = 2/3 and 3/5 in the LL geometry with *θ* = 10°.

**a**, RILS spectra of the \({\varDelta }_{{\rm{m}}}^{0}\) mode at filling factors around *v* = 2/3. The mode intensity rapidly decreases as the filling factor deviates from *v* = 2/3. **b**, RILS spectra of the \({\varDelta }_{{\rm{m}}}^{0}\) mode at filling factors around *v* = 3/5. A similar rapid decline in the mode intensity is observed as the filling factor moves away from *v* = 3/5. The FQH effect is known for its incompressible behavior at specific fractional filling factors, and deviations from these filling factors make the system more compressible. The observed pronounced sensitivity to filling factors is characteristic of the FQH effect. RILS peaks are marked by vertical black arrows.

### Extended Data Fig. 9 Temperature dependence of the \({{\boldsymbol{\Delta }}}_{{\bf{m}}}^{0}\) modes at FQH states with *θ* = 10°.

**a,b** and **c** present temperature dependence of the \({\varDelta }_{{\rm{m}}}^{0}\) modes at *v* = 2/3 (in the LL geometry), *v* = 2/5 (in the RR geometry) and *v* = 3/5 (in the LL geometry), respectively. As the temperature increases, the mode intensities are suppressed in all the three cases and the modes eventually vanish at 800 mK. In the FQH states, the formation of incompressible liquids results from strong electron-electron interactions with the presence of energy gaps. However, as the temperature rises, thermal excitations could disrupt the delicate correlated ground states, leading to the observed reduction in the mode intensity. RILS peaks are marked by vertical black arrows.

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Liang, J., Liu, Z., Yang, Z. *et al.* Evidence for chiral graviton modes in fractional quantum Hall liquids.
*Nature* **628**, 78–83 (2024). https://doi.org/10.1038/s41586-024-07201-w

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DOI: https://doi.org/10.1038/s41586-024-07201-w

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