Zhang, F. C. & Sarma, S. D. Excitation gap in the fractional quantum Hall effect: finite layer thickness corrections. Phys. Rev. B 33, 2903–2906 (1986).

Hirjibehedin, C. F. et al. Resonant enhancement of inelastic light scattering in the fractional quantum Hall regime at v = 1/3. Solid State Commun. 127, 799–803 (2003).

Du, L. et al. Observation of new plasmons in the fractional quantum Hall effect: interplay of topological and nematic orders. Sci. Adv. 5, eaav3407 (2019).

Bergshoeff, E. A., Rosseel, J. & Townsend, P. K. Gravity and the spin-2 planar Schrödinger equation. Phys. Rev. Lett. 120, 141601 (2018).

Rhone, T. D. et al. Higher-energy composite fermion levels in the fractional quantum Hall effect. Phys. Rev. Lett. 106, 096803 (2011).

Platzman, P. M. & He, S. Resonant Raman scattering from mobile electrons in the fractional quantum Hall regime. Phys. Rev. B 49, 13674–13679 (1994).

Haldane, F. D. M., Rezayi, E. H. & Yang, K. Graviton chirality and topological order in the half-filled landau level. Phys. Rev. B 104, L121106 (2021).

Gallais, Y., Yan, J., Pinczuk, A., Pfeiffer, L. N. & West, K. W. Soft spin wave near v = 1: evidence for a magnetic instability in Skyrmion Systems. Phys. Rev. Lett. 100, 086806 (2008).

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Girvin, S. M., MacDonald, A. H. & Platzman, P. M. Collective-excitation gap in the fractional quantum Hall effect. Phys. Rev. Lett. 54, 581–583 (1985).

Golkar, S., Nguyen, D. X. & Son, D. T. Spectral sum rules and magneto-roton as emergent graviton in fractional quantum Hall effect. J. High Energy Phys. 2016, 21 (2016).

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/lB, confirming its long-wavelength nature. PL, photoluminescence.

Liou, S.-F., Haldane, F. D. M., Yang, K. & Rezayi, E. H. Chiral gravitons in fractional quantum hall liquids. Phys. Rev. Lett. 123, 146801 (2019).

Trico’s services open the door for many new costs saving strategies, including a full list of recommendations on improving oil sampling procedures, assessing fluid storage and handling procedures, and setting appropriate target and alarm limits.

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.

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.

Kang, M. et al. Inelastic light scattering by gap excitations of fractional quantum Hall states at 1/3 ≤ v ≤ 2/3. Phys. Rev. Lett. 84, 546–549 (2000).

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.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Fierz, M. & Pauli, W. On relativistic wave equations for particles of arbitrary spin in an electromagnetic field. Proc. R. Soc. London, Ser. A 173, 211–232 (1939).

Simon, S. H. & Halperin, B. I. Finite-wave-vector electromagnetic response of fractional quantized Hall states. Phys. Rev. B 48, 17368–17387 (1993).

This comprehensive information is presented in a detailed written report along with a visual scorecard, where we summarize opportunities for improvement in key areas versus current efforts and conditions. Each assessment report concludes with an Executive Summary that specifies your target areas for most rapid ROI and lays out detailed next steps that (if implemented) will reenergize your lubrication management program to peak-performance. This provides plant personnel with a roadmap to address plant status and provide direction where to focus resources to ensure improvements in equipment reliability and reduction in overall maintenance costs.

Luo, X., Wu, Y.-S. & Yu, Y. Noncommutative Chern–Simons theory and exotic geometry emerging from the lowest Landau level. Phys. Rev. D 93, 125005 (2016).

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.

Yang, B., Hu, Z.-X., Papić, Z. & Haldane, F. D. M. Model wave functions for the collective modes and the magnetoroton theory of the fractional quantum hall effect. Phys. Rev. Lett. 108, 256807 (2012).

Johri, S., Papić, Z., Schmitteckert, P., Bhatt, R. N. & Haldane, F. D. M. Probing the geometry of the Laughlin state. New J. Phys. 18, 025011 (2016).

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

Exotic physics could emerge from interplay between geometry and correlation. In fractional quantum Hall (FQH) states1, novel collective excitations called chiral graviton modes (CGMs) are proposed as quanta of fluctuations of an internal quantum metric under a quantum geometry description2,3,4,5. Such modes are condensed-matter analogues of gravitons that are hypothetical spin-2 bosons. They are characterized by polarized states with chirality6,7,8 of +2 or −2, and energy gaps coinciding with the fundamental neutral collective excitations (namely, magnetorotons9,10) 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.

Maciejko, J., Hsu, B., Kivelson, S. A., Park, Y. & Sondhi, S. L. Field theory of the quantum Hall nematic transition. Phys. Rev. B 88, 125137 (2013).

Wang, Y. & Yang, B. Geometric fluctuation of conformal Hilbert spaces and multiple graviton modes in fractional quantum Hall effect. Nat. Commun. 14, 2317 (2023).

Davies, H. D. M., Harris, J. C., Ryan, J. F. & Turberfield, A. J. Spin and charge density excitations and the collapse of the fractional quantum Hall state at v = 1/3. Phys. Rev. Lett. 78, 4095–4098 (1997).

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.

Tsui, D. C., Stormer, H. L. & Gossard, A. C. Two-dimensional magnetotransport in the extreme quantum limit. Phys. Rev. Lett. 48, 1559–1562 (1982).

Kang, M., Pinczuk, A., Dennis, B. S., Pfeiffer, L. N. & West, K. W. Observation of multiple magnetorotons in the fractional quantum Hall effect. Phys. Rev. Lett. 86, 2637–2640 (2001).

Yang, K. Geometry of compressible and incompressible quantum Hall states: application to anisotropic composite-fermion liquids. Phys. Rev. B 88, 241105 (2013).

Gao, A. et al. Quantum metric nonlinear Hall effect in a topological antiferromagnetic heterostructure. Science 381, 181–186 (2023).

Park, K. & Jain, J. K. Two-roton bound state in the fractional quantum Hall effect. Phys. Rev. Lett. 84, 5576–5579 (2000).

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 Ec 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 (e2/εlB)/|2p + 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.

Farjami, A., Horner, M. D., Self, C. N., Papić, Z. & Pachos, J. K. Geometric description of the Kitaev honeycomb lattice model. Phys. Rev. B 101, 245116 (2020).

Pinczuk, A., Dennis, B. S., Pfeiffer, L. N. & West, K. Observation of collective excitations in the fractional quantum Hall effect. Phys. Rev. Lett. 70, 3983–3986 (1993).

Goldberg, B. B. et al. Optical transmission spectroscopy of the two-dimensional electron gas in GaAs in the quantum hall regime. Phys. Rev. B 38, 10131–10134 (1988).

Ghosh, T. K. & Baskaran, G. Modeling two-roton bound state formation in the fractional quantum Hall system. Phys. Rev. Lett. 87, 186803 (2001).

Nguyen, D. X., Haldane, F. D. M., Rezayi, E. H., Son, D. T. & Yang, K. Multiple magnetorotons and spectral sum rules in fractional quantum hall systems. Phys. Rev. Lett. 128, 246402 (2022).

Gianfrate, A. et al. Measurement of the quantum geometric tensor and of the anomalous Hall drift. Nature 578, 381–385 (2020).

Wang, R., Sedrakyan, T. A., Wang, B., Du, L. & Du, R.-R. Excitonic topological order in imbalanced electron–hole bilayers. Nature 619, 57–62 (2023).

School of Physics, National Laboratory of Solid State Microstructures, and Collaborative Innovation Center for Advanced Microstructures, Nanjing University, Nanjing, China

Balram, A. C., Liu, Z., Gromov, A. & Papić, Z. Very-high-energy collective states of partons in fractional quantum hall liquids. Phys. Rev. X 12, 021008 (2022).

Kirmani, A. et al. Probing geometric excitations of fractional quantum hall states on quantum computers. Phys. Rev. Lett. 129, 056801 (2022).

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

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.

Nguyen, D. X. & Son, D. T. Probing the spin structure of the fractional quantum Hall magnetoroton with polarized Raman scattering. Phys. Rev. Res. 3, 023040 (2021).

Wurstbauer, U., West, K. W., Pfeiffer, L. N. & Pinczuk, A. Resonant inelastic light scattering investigation of low-lying gapped excitations in the quantum fluid at v = 5/2. Phys. Rev. Lett. 110, 026801 (2013).

Hirjibehedin, C. F. et al. Splitting of long-wavelength modes of the fractional quantum Hall liquid at v = 1/3. Phys. Rev. Lett. 95, 066803 (2005).

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

The Lubrication Management Assessment is a critical step in improving the overall lubrication program within the plant. Trico’s services team gathers data on critical equipment, maintenance practices, procedures, and technologies currently deployed at your facility. This is done through personnel interviews, record and document reviews and a detailed plant-wide walk-through. With this information, we assess and rate your current program, and create a report that compares our findings to the industry’s best practices standards. This will help you pinpoint the strengths and weaknesses of your existing lubrication management program.

a, RILS spectra at v = 1/3 in the unpolarized geometry as a function of ωL. 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 Ez, 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 result29 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 ωL. 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 klB 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.

In RILS experiments, the wavevector k = (2ωL/c)sinθ 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 klB 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 klB ≈ 0.02 (Fig. 3a and Extended Data Fig. 3) and decreases by 15% as klB 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 klB ≈ 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.

Scarola, V. W., Park, K. & Jain, J. K. Rotons of composite fermions: comparison between theory and experiment. Phys. Rev. B 61, 13064–13072 (2000).

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.