Abstract
A key concept proposed by Landau to explain superfluid liquid helium is the elementary excitation of quantum particles called rotons1,2,3,4,5,6,7,8. The irregular arrangement of atoms in a liquid leads to the aperiodic dispersion of rotons, which played a pivotal role in understanding fractional quantum Hall liquids (magneto-rotons)9,10 and the supersolidity of Bose–Einstein condensates11,12,13. Even for a two-dimensional electron or dipole liquid, in the absence of a magnetic field, the repulsive interactions have been predicted to form a roton minimum14,15,16,17,18,19, which can be used to trace the transition to Wigner crystals20,21,22,23,24 and superconductivity25,26,27, although this has not yet been observed. Here, we report the observation of such electronic rotons in a two-dimensional dipole liquid of alkali-metal ions donating electrons to surface layers of black phosphorus. Our data reveal the striking aperiodic dispersion of rotons, which is characterized by a local minimum of energy at finite momentum. As the density of dipoles decreases so that interactions dominate over the kinetic energy, the roton gap reduces to 0, as in a crystal, signalling Wigner crystallization. Our model shows the importance of short-range order arising from repulsion between dipoles, which can be viewed as the formation of Wigner crystallites (bubbles or stripes) floating in the sea of a Fermi liquid. Our results reveal that the primary origin of electronic rotons (and the pseudogap) is strong correlations.
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Acknowledgements
This work was supported by National Research Foundation of Korea, which was funded by the Ministry of Science and ICT (Grant Nos. NRF-2021R1A3B1077156, NRF-RS-2024-00416976 and NRF-RS-2022-00143178) and the Yonsei Signature Research Cluster Program (Grant No. 2024-22-0163). This research also used resources of the Advanced Light Source (Contract No. DE-AC02-05CH11231), which is a US Department of Energy Office of Science user facility.
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S.P. performed the ARPES experiments with help from C.J., E.R. and A.B. and carried out the data analysis with help from M.H. K.S.K. conceived and directed the project. S.P. and K.S.K. wrote the manuscript with contributions from all other co-authors.
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Extended data figures and tables
Extended Data Fig. 1 Raw ARPES data and curve-fitting.
a, Raw data of Fig. 1d, e shown in the form of EDCs over the k range from 0 to 1.005 Å–1 with the k interval of 0.067 Å–1. There are well-defined peaks even in the k region of weak intensity. For these well-defined peaks, it is straightforward by fitting each EDC with the Gaussian function (red lines overlaid) to reliably determine their peak positions as marked by red circles relative to E0 (dotted line). b-d, Peak position (b), width (c), and intensity (d) taken by curve-fitting in a and plotted as a function of ky. The error bars at maximum in b is smaller than the size of circled numbers. The light red line underlaid in b is the band dispersion calculated by our model30,31, which is in excellent agreement (quantitatively less than 4%). Even in the k range of weak intensity shaded in grey, the peak width in c is only 20% greater than that of strong intensity (this is related to the imaginary part of Δk in Extended Data Fig. 6a). On the other hand, the red curve overlaid in d shows the variation of intensity in conventional insulators across the zone boundary due to the well-known phase interference effect57,58. There is a small gap between peak intensity and the red curve, which means there remains relatively weak but finite intensity. e, Doping series of EDCs taken at the roton minimum or k2 and plotted relative to E0 (dotted line). Black lines overlaid are obtained by fitting each EDC with the Gaussian peak in colour, where their peak positions are marked by red ticks.
Extended Data Fig. 2 Direction dependence of electronic rotons.
a, Surface Brillouin zone of black phosphorus with high-symmetry points marked by open circles. Grey arrows show the three k directions indexed by kx, ks, and ky. b-d, ARPES data of black phosphorus doped by K at n = 3.8 x 1013 cm−2, taken along armchair (b), diagonal (c), and zigzag (d) directions corresponding to kx, ks, and ky, respectively. e-g, ARPES data obtained from those in b-d by normalizing the maximum intensity of EDCs at each k point. These normalized data reveal the clear signature of electronic rotons regardless of measurement directions.
Extended Data Fig. 3 Element dependence of electronic rotons.
a-d, ARPES data taken in ky for black phosphorus doped by Na (a), K (b), Rb (c), and Cs (d) at n = 3.8 ~ 5.3 × 1013 cm−2. Dotted lines show the magnitude of ΔPG, which is 106 meV for Na, 77 meV for K, 61 meV for Rb, and 48 meV for Cs as discussed in the previous report33. e-h, ARPES data obtained from those in a-d by normalizing the maximum intensity of EDCs at each k point. The normalized data show the clear signature of electronic rotons regardless of the kinds of alkali metals, which confirms that (i) it is not an artifact, and (ii) it is a generic property of alkali metals.
Extended Data Fig. 4 Comparison between ARPES data and simulations.
a-c, ARPES data (left panel) compared with simulations (right panel) for black phosphorus doped by K at n marked on top of each panel in units of 1013 cm−2 and rs. The band dispersion for simulations is obtained by fitting the peak position of ARPES data and the variation of spectral intensity as a function of k is obtained from the well-known phase interference effect57,58 (Methods). d-f, Corresponding ARPES data and spectral simulations after normalizing the intensity of peaks in each EDC. Our spectral simulations reproduce key aspects of not only raw data in a-c, but also normalized data in d-f.
Extended Data Fig. 5 Doping dependence of electronic rotons.
a-f, ARPES data taken for black phosphorus doped by K at n marked on top of each panel in units of 1013 cm−2 with rs. We found the pseudogap of 71−107 meV persistent in the range of n.
Extended Data Fig. 6 Complete electronic structure of liquid metals.
a,b Band dispersion (a) and probability density (b) of wavefunctions obtained by the theoretical model46,47,48,49 that was initially developed for liquid metals but can be generally applied to any non-crystalline system in the presence of the short-range order. There are two branches in terms of partial wave analysis46: One is the p-wave or d-wave states at resonance scattering predicted by Anderson and McMillan47 to show the back-bending band dispersion (due to the real part of Δk as shown by the black curve) and the pseudogap (due to the imaginary part of Δk as shown by the grey area), as shown in a. This is due to the formation of quasi-bound states (QBS), as shown by the black curve in b, within the scattering potential (dotted black line). The other is s-wave states49, for which resonance scattering is forbidden by the absence of a potential barrier as shown by the red dotted line in b. The presence of the unbound states represented by the red curve in b was predicted by Schwartz and Ehrenreich48 to be related to another aperiodic (damped oscillatory) branch in band dispersion that extends towards the zone boundary30,31 as shown by the red curve in a. The grey region surrounding the red curve shows the imaginary part of Δk that accounts for the small increase in the peak width of EDCs indicated by the red arrow in Extended Data Fig. 1c.
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Park, S., Huh, M., Jozwiak, C. et al. Electronic rotons and Wigner crystallites in a two-dimensional dipole liquid. Nature 634, 813–817 (2024). https://doi.org/10.1038/s41586-024-08045-0
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DOI: https://doi.org/10.1038/s41586-024-08045-0
