Quantum electron liquid and its possible phase transition

Purely quantum electron systems exhibit intriguing correlated electronic phases by virtue of quantum fluctuations in addition to electron-electron interactions. To realize such quantum electron systems, a key ingredient is dense electrons decoupled from other degrees of freedom. Here, we report the discovery of a pure quantum electron liquid, which spreads up to ~ 3 {\AA} in the vacuum on the surface of electride crystal. An extremely high electron density and its weak hybridisation with buried atomic orbitals evidence the quantum and pure nature of electrons, that exhibit a polarized liquid phase as demonstrated by our spin-dependent measurement. Further, upon enhancing the electron correlation strength, the dynamics of quantum electrons changes to that of non-Fermi liquid along with an anomalous band deformation, suggestive of a transition to a hexatic liquid crystal phase. Our findings cultivate the frontier of quantum electron systems, and serve as a platform for exploring correlated electronic phases in a pure fashion.

vicinity of the Wigner crystal phase, which are not expected in the classical regime. In addition to the various liquid crystalline, stripe, and bubble phases 15 , even the superconducting and supersolid phasesthe prime problems of condensed matter physicsare possible intermediate phases in the quantum regime 16 which remain to be substantiated experimentally.
To realize pure 2D quantum electron systems, the prerequisite is decoupling the highdensity 2D confined electrons from other degrees of freedom. Decoupling is possible when electrons are captured on the positively charged surface of solid matter. Such quantum electrons can be found at the surface of 2D electride crystals [17][18][19] , which consist of positively charged cationic layers and counteranionic electrons located in the interlayer space. Interstitial anionic electrons (IAEs) are localized at the interstitial space between cationic layers and are not bound to the atomic orbitals of neighbouring cations. When the positively charged cationic layer is terminated at the surface of 2D electrides, the IAEs are inevitably detained on the terminated cationic layers to maintain charge neutrality as the distinct surface state from IAEs.
It is notable from the geometrical aspect that the theoretical maximum density of the electrons on the surface can be as high as ~ 9.5 × 10 14 cm −2 , which is expected from the concentration of IAEs in a single interlayer space of the 2D electrides with a bulk electron density of 1.4 × 10 22 ~ 2.9 × 10 22 cm −3 (ref. [17][18][19]. Through the experiments, the uncharted high-density pure 2D electron system with a concentration of ~ 2.0 × 10 14 cm −2 is confirmed to be realized on the surface of 2D [Gd2C] 2+ •2e − electride at high temperatures around 10 K, extending the phase diagram of pure 2D electrons to the quantum regime (Fig. 1a).
What is more surprising is that the pure 2D quantum electrons guarantee a weak which is more apparent in the intensity ratio of ISurface/IIAE (black in Fig. 2g), confirming a very weak hybridisation between their wave functions and the outermost Gd orbitals, a pure nature of 2D surface electrons.
Having evidence that these surface electrons are successfully isolated from other degrees of freedom of the electride solid, the next questions to be addressed are whether the pure electron system is within the desired quantum regime and which quantum phase evolves by electron-electron interactions. Fitting the parabolic band gives an effective mass (m * ) of ~ 2.1 me and extremely high density (n) of ~ 2.0 × 10 14 cm −2 (Fig. 3a,b), in contrast to other 2D electron gases bound to the atomic nuclei with a small m * value in the range of 0.5-1.4 me (ref. [20][21][22] and to the surface electrons on the LHe with a low density of < 10 10 cm −2 in the classical regime 7--9,14 . The extracted scattering rate of the surface electrons obtained by measuring the peak width in the spectra reveals the liquid nature with a clear quadratic energy dependence, which corresponds to the behaviour of a Fermi liquid 23,24 (Fig. 3c). The high density that surpasses the critical boundary (the solid red line in Fig. 1a) verifies that the electron liquid is in the quantum regime. Furthermore, our quantum electron liquid is spin-polarized, as verified by the spin-resolved measurements that support the spin-polarized band of the surface electrons, which is also revealed by DFT calculations (Fig. 3d). Four sets of spin-dependent spectra are obtained for the surface electrons and the trapped IAEs along the out-of-plane and in-plane directions ( Fig. 3f-i). It is evident that spin polarisation of the quantum electron liquid occurs along the out-of-plane direction. Meanwhile, the IAEs have spin polarisation along the in-plane direction, corresponding to the magnetic easy-axis of ferromagnetic bulk [Gd2C] 2+ •2e − electride, which is induced by the exchange interaction of magnetic quasi-atomic IAEs with Gd atoms of cationic layers 19 . We note that the different direction of spin polarisation between the quantum electron liquid and trapped IAEs implies that the polarized nature of surface electrons is not relevant to the magnetic moment of the underlying bulk but is likely induced by the intrinsic electron-electron interaction, as it is one of the predicted ground states of pure 2D quantum electron phases [25][26][27] . Therefore, in short, an unprecedented pure 2D quantum electron liquid is demonstrated, which substantiates that the polarized Fermi liquid can emerge between the paramagnetic Fermi liquid and crystalline electron phases in the quantum regime [25][26][27] (Fig. 1a).
Realising the pure 2D quantum electron liquid assures that the exotic phases expected in the quantum regime with lower electron density and thus stronger electron correlation, are ready to be explored. As a trigger for enhancing the electron correlation, potassium (K) atoms were deposited to the system which usually change the electron density at the surface. Due to the ultralow work function of the electride system, particularly for the loosely bound surface state to the topmost atomic layer, K deposition would decrease the electron density of the quantum liquid. Indeed, the signatures of enhanced electron correlation strength were captured as discussed below, suggesting the reduction of electron density by K deposition. The complete disappearance of the surface electron state below EF in the measured bands at higher K coverages (Extended Data Fig. 4), as well as the consumption of the surface electrons by deposited K atoms verified by DFT calculations with different coverages of K overlayer (Supplementary Figure 2) also demonstrate the reduction in density. Upon progressive K deposition, we observed an anomalous band deformation of the quantum electrons from the initial parabolic dispersion (Fig. 4a) with an isotropic circular Fermi surface topology (Fig. 4b) to a W-shaped dispersion ( Fig. 4a) with an anisotropic hexagonal topology ( Fig. 4c and Extended Data Fig. 5).
Next, we discuss the origin of the anomalous W-shaped band dispersion compared to the conventional band renormalisations. The deformed hexagonal topology of the quantum electrons indicates the breaking of continuous rotational symmetry. Meanwhile, when the spatial ordering of deposited K atoms occurs on the surface, band renormalisation, such as band folding, can take place, inducing the deformation of the parabolic band. However, we rule out the band folding effect on the W-shaped band dispersion because the replica bands do not appear in the higher orders of the reduced Brillouin zone (BZ), which should occur if the deposited K atoms order and the quantum electrons are strongly affected by the K ordering (Extended Data Fig. 6). The intact V-shaped IAE band also supports that the ordering of deposited K atoms is unlikely.
Another possibility is that the K deposition can initiate the effect of the underlying lattice, which inevitably induces the band deformation of the quantum electrons into the Vshape around zero momentum, mimicking the IAE band (Extended Data Fig. 7). However, we observed a W-shaped band dispersion around zero momentum, precluding the lattice potential effect of the underlying lattice. In addition, the nontrivial angular warping of the band that cannot be accounted for by the rotational symmetry of the underlying lattice provides further evidence against the lattice potential effect (Extended Data Fig. 5). Indeed, the absence of resonance behaviour of quantum electrons even after K deposition (Extended Data Fig. 8) strongly supports the fact that the surface electrons are persistently in the loosely bound state and thus that the lattice potential effect on the band deformation is negligible. Therefore, the origin of the W-shaped band is not attributed to the spatially modulating potentials of deposited K atoms or the terminated cationic layer. It should be noted that the observed deformation of the band dispersion, predominantly at zero momentum, occurs hardly from the change of translational symmetry.
Instead, a phase transition could be responsible for the band deformation induced by the decreased electron density of the quantum electron liquid. The gradual increase in m * up to ~ 3.9 me indicates that the electron correlation becomes stronger upon K deposition (Fig. 4a), which is expected to be accomplished by lowering the density and can trigger the phase transition 15,16,28 . A series of scattering rates in the sequence of K deposition shows the drastic change of energy dependence from quadratic (#0 and #1) to linear (#2, #3, and #4) dependence ( Fig. 4d,e) together with the reduction of cut-off energy, which consistently reflects the enhancement of electron correlation strength 29,30 . Linear energy dependence is a well-known behaviour of non-Fermi liquids 23,24,31 , strongly suggesting that the quantum electrons transit into a distinct phase, departing from the Fermi liquid phase. Further, the deformation is temperature dependent, indicating that the observed band deformation is due to the phase transition (Extended Data Fig. 9).
According to the phase diagram of pure 2D electrons (Fig. 1a), a possible phase, accessed by reducing the density along with the strengthened electron-electron interaction, is a hexatic phase or Wigner crystal. The remaining metallic band crossing EF and the lack of a signature of translational symmetry change in the W-shaped band dispersion preclude the formation of the Wigner crystal. It is reasonable to speculate that the W-shaped band dispersion is derived from a hexatic phase as the aforementioned symmetry characteristics of the Wshaped band are coincident with those of the hexatic phase, which is a liquid crystal phase with marginally broken rotational symmetry and preserved continuous translational symmetry 28,32 .
Our experimental observations thus suggest the emergence of the hexatic phase in the quantum regime and extend the phase diagram of pure 2D electrons, which can provide a further understanding of the transition process in the quantum regime.
In summary, we discovered an unprecedented 2D quantum electron liquid on the surface of a 2D electride crystal. The clear identification of the quantum electron liquid and its spin-polarized nature provide a step towards experimental accessibility of correlated electronic phases in the quantum regime. Indeed, the possible phase transition from the initial liquid phase to the phase exhibiting non-Fermi liquid behaviour, which mimics the symmetry characteristics of hexatic phase of liquid crystalline phase, was demonstrated upon the enhancement of electron correlation. We believe our results will stimulate the exploratory study of exotic phases in the quantum regime, such as the long-lasting solid phase of pure electrons that may be realized by cooling the present system and reducing the density. Furthermore, the practical feasibility of this new pure electron systemelectrons on a rigid crystal surface at relatively high temperatureallows various technical approaches, not only the transport measurements but also various forms of microscopy or other spectroscopy. Indeed, as a preliminary example, we succeeded in imaging pure 2D quantum electrons in real space as well as obtaining spectroscopic evidence by scanning tunneling microscopy/spectroscopy (STM/S, Extended Data Fig. 10). Our system thus calls broad-ranging further study, which will provide new insights into intriguing quantum phenomena and extend the field to future applications such as quantum computation 33

Competing interests
The authors declare no competing interests. [Gd2C] 2+ •2e − was cleaved at 20 K in a UHV with a pressure better than 4 × 10 −11 Torr and immediately plugged into the STM head. A mechanically sharpened PtIr wire was used for an STM tip. To acquire the differential conductance (dI/dV) spectra and dI/dV maps, we used a standard lock-in technique with a modulation frequency f = 718 Hz and root-mean-square amplitude Vmod = 5 mV. Measurements were performed at a temperature of 4.3 K.

Self-energy analysis.
To obtain the information of the imaginary part of the self-energy, EDCs were fitted as the ARPES intensity, which is nothing but the spectral function with the following form: where Re is the real part of the self-energy and is the bare band dispersion. The obtained half-width at half-maximum of the Lorentzian curve is the imaginary part of the self-energy , and m * of each step was estimated from the above equation using the obtained vF. In particular, for the pristine sample (deposition step #0 of Fig. 4a), m * was double-checked using parabolic band fitting (Fig. 3b), which matches well with the estimated m * value from vF. Fig.   5e) was calculated by substituting the symmetrized and filtered CECs into the following formula 34,35 :

Autocorrelation analysis. Autocorrelation of ARPES intensity AC ( , ) (Extended Data
where ( , ) is the ARPES intensity at momentum and binding energy extracted by a given CEC, and is the momentum transfer. The three-fold symmetrisation was applied to avoid the momentum-dependent modulation of intensity due to the matrix element effect. It was ensured that the symmetrisation does not deform the original shape of CECs from the surface band (Extended Data Fig. 5c,d). The signal from the outer bands was filtered out to include only the signal from the surface electron band in the analysis (Extended Data Fig. 5d).

Data Availability
All data supporting the findings of this work are included in the main text, extended data, and supplementary information. These are available from the corresponding authors upon reasonable request.    Extended Data Figure 4 | Determination of K coverage. a, Core-level spectra of K 3p with different K coverage on the cleaved [Gd2C] 2+ •2e − surface. The K 3p core-level peak starts to grow with K deposition near the binding energy of 19 eV and eventually saturates, where we estimate the coverage as 1 ML (thick green curve). Above 1 ML, chemically shifted additional peaks emerge at lower binding energy close to 18 eV. b, A 3D representation of the band evolution with increasing K coverage. c, Extracted band dispersion at several different K coverages indicated by red arrows in b. Surface 2D electron band (1) evolves via K deposition (2) and disappears (3). The complete disappearance of the electron band implies that the surface electron density is actually reduced by K deposition. In the rightmost panel (4), corresponding to K 1 ML, which we estimate with core-level spectra in a, K band appears near Fermi level, which well agrees with the core level estimation. Absence of W-shape band at higher momentum above the assumed new zone boundaries, which can be induced by band folding, evidences that W-shaped band deformation is not due to the ordering of deposited K. i,j, Schematic drawings of conceivable scenarios for band deformation (hybridisation) by K deposition with assumed zone folded band (i) and upward convex band (j), respectively. Solid and dashed bands show the original and assumed bands, respectively. Both scenarios may exhibit band deformation; however, they should form the fragment of the original electron band at higher binding energy near BZ centre, which is absent in ARPES results (Fig. 4a). Extended Data Figure 9 | Phase transition via decreasing temperature. a, Core-level spectra of K 3p with different K coverage at 40 K. b, Surface electron band dispersion with respect to K deposition. Corresponding deposition steps to the core-level spectra in a are indicated in the panel (#0-#2). Despite the broadening of the spectra and the slight reduction of the band minimum energy after the K deposition, the entire band dispersion still preserves nearly parabolic close to the pristine case. c, Enlarged phase diagram of pure 2D electron phase taken from Fig. 1a. ARPES measurements were performed by following the process marked in the phase diagram by black and blue arrows. d, Core-level spectra of K 3p observed at different temperatures. The preserved intensity of the K 3p core level indicates the absence of K desorption during the cooling process. e, Temperature-dependent surface electron band dispersion observed after K deposition (deposition step #2). Temperatures were set to 40, 25, and 10 K. f, Peak positions (markers) obtained from e by fitting EDCs. Overlaid solid lines are guides to the eyes for the band dispersion. This exhibits the band deformation from parabola (red, 40 K) to W-shape (blue, 10 K) with decreasing the temperature.