Universal behavior of the bosonic metallic ground state in a two-dimensional superconductor

Anomalous metallic behavior, marked by a saturating finite resistivity much lower than the Drude estimate, has been observed in a wide range of two-dimensional superconductors. Utilizing the electrostatically gated LaAlO3/SrTiO3 interface as a versatile platform for superconductor-metal quantum phase transitions, we probe variations in the gate, magnetic field, and temperature to construct a phase diagram crossing from superconductor, anomalous metal, vortex liquid, to Drude metal states, combining longitudinal and Hall resistivity measurements. We find that the anomalous metal phases induced by gating and magnetic field, although differing in symmetry, are connected in the phase diagram and exhibit similar magnetic field response approaching zero temperature. Namely, within a finite regime of the anomalous metal state, the longitudinal resistivity linearly depends on field while the Hall resistivity diminishes, indicating an emergent particle-hole symmetry. The universal behavior highlights the uniqueness of the quantum bosonic metallic state, distinct from bosonic insulators and vortex liquids.


Introduction
The anomalous metallic state observed in various two-dimensional (2D) superconductors 1 , including the LaAlO3/SrTiO3 interface [2][3][4][5] , has attracted attention recently. In the standard paradigm for weakly interacting, disordered electronic systems, a 2D system without spin-orbit coupling cannot have zero temperature metallic phases 6,7 . The implication for thin film superconductors is that the only admissible ground states are superconductors and insulators: metallic phases and their associated transitions are prohibited, and any experimental observations to the contrary have been deemed anomalous. This viewpoint has persisted for over four decades, despite the extended history of the observation of metallic behavior approaching zero temperature in 2D superconductors [8][9][10][11][12][13] . Recent observations of the metallic state in layered materials like ZrNCl (ref. 14), MoS2 (ref. 15), TiSe2 (ref. 16), WTe2 (ref. 17,18), the oxide interface LaAlO3/SrTiO3 (ref. [2][3][4][5], cuprate thin films 19,20 , and artificial composite systems [21][22][23] , have substantially expanded the family of materials hosting such anomalous metallic ground states. These studies, conducted in different ranges of temperatures (from ~ 20 mK to > 10 K), also suggest the existence of a 2D quantum superconductor-metal phase transition (QSMT) 1 . For instance, magnetoresistance oscillations in nanoporous YBa2Cu3O7 (YBCO) thin films unambiguously show that conduction in the anomalous metal phase occurs with 2e charge carriers (where e is the electron charge) 20 , consistent with vanishing Hall resistivity indicating particle-hole symmetry 24 . In the previous studies, the anomalous metals were observed either by applying magnetic field or by varying the carrier density/disorder of the system. Since the underlying symmetries are distinct in these cases, a key open issue is the extent to which the anomalous metals in both cases (with/without time-reversal symmetry) are similar. Here, using 4 the gated LaAlO3/SrTiO3 superconducting interface, we control and investigate the QSMT combining three separate parameters: gate, magnetic field, and temperature.

Sample preparation
We utilize gold top-gated LaAlO3/SrTiO3 Hall bar devices, with structure shown in Figure 1a.  Figure 1c shows the top gate modulation of carrier density and mobility at temperature T = 5 K. With increasing VG, carrier density increases, and mobility decreases due to the enhanced scattering by stronger confinement of the electron wavefunctions against the interface. Therefore, VG simultaneously modulates carrier density and disorder, both of which are relevant for 2D superconductivity. Based on the mobility and density values, we estimate that kFl is in the range of 80 ~ 200 ⨠ 1 in our system (here kF and l are the Fermi momentum and electron mean free path, respectively), and thus we ignore localization effects and treat the normal state of our sample as a Drude metal (i.e. the sample dimension is far below the localization length) 1 .

Gate voltage and temperature dependence
Upon further decreasing T below 0.3 K, the interface turns superconducting. Figure 1d shows the resistivity-versus-temperature (R-T) measurements with VG being the tuning parameter. We We further note that for some gate voltages there are kinks in the R-T curves lower than TP and before resistivity vanishes/saturates. We denote the characteristic temperature scale at which the kinks occur as TF, and discuss their physical interpretation below. TP and TF are quantitatively defined by the peaks in the second derivative of R-T curves. Figure 1e shows TC, TF, and TP as a function of VG. Note that TC vanishes at VG = VC while TP is still finite, suggesting the existence of a QSMT critical point, on both sides of which the conduction is dominated by Cooper pairs.

Gate voltage and magnetic field dependence
We measure the low-field magnetoresistance (field-induced resistivity difference) at base temperature near the critical point of the gate-tuned QSMT. Figure 2a shows resistivity versus magnetic field near zero field while tuning VG. We observe at the low field limit 1) zero magnetoresistance within the superconducting regime, and 2) positive and linear magnetoresistance in the metallic regime. Importantly, upon leaving the superconducting phase, 6 the onset of positive magnetoresistance is much more pronounced than the onset of zero-field resistivity near the critical point. At zero field, the slope for magnetoresistance dR/dB is discontinuous within our measurement resolution down to ~ 0.4 G, more than 10 3 times lower than the typical upper critical field HC2 of the system. The discontinuous slope likely indicates the existence of a singular point in magnetoresistance at zero field in the anomalous metal phase, distinct from the quadratic behavior for a normal Drude metal. As a comparison, we fix the VG at 1.8 V and measure magnetoresistance with varying temperature shown in Figure 2b. We observe qualitatively the same onset of positive linear magnetoresistance at a certain temperature, and the low-field magnetoresistance vanishes again at higher temperatures. Specifically, the singular behavior is still pronounced for 180 mK and 200 mK, but it becomes rounded for 220 mK. To quantify these observations, dR/dB as functions of VG and T are plotted in Figures 2c and 2d, respectively. For the gate-tuned case, dR/dB rises sharply from zero to a finite value at a critical voltage, which matches the critical voltage VC extracted from R-T analysis within our data resolution. For the temperature-tuned case, dR/dB is zero at low temperatures and becomes positive at around TC (extracted from R-T analysis). After it peaks and drops, dR/dB returns to zero at around TP (obtained from R-T analysis). These results demonstrate that the positive linear low-field magnetoresistance is a sensitive indicator for both gate-tuned QSMT and temperaturetuned SMT.

Magnetic field and temperature dependence
The QSMT can also be induced by magnetic field.

Data Availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.   Color scale in the background shows the interpolated resistivity. All data in this figure are from Sample A.