## 2 Results
### Metal Behaviors at Large \(r_{s}\) and a Critical Behavior  
Though most 2D metallic behaviors are well described by the Fermi liquid (FL) [18], the breakdown of the FL is inevitable in both disorder- and interaction-dominated regimes. Indications of non-FL states are mostly documented for the metallic states in the proximity of \(n_{c}\) corresponds to a \(r_{s}\sim 10\)-20. Evidence includes \(T\)-dependence of the resistivity [19; 20; 21; 22], the magnetoresistance (\(\rho_{xx}\)) in a parallel \(B_{||}\) field [22; 23], and the Coulomb drags [24], etc. The suppression of disorder reduces \(r_{s}^{c}\) and opens up a range for the non-FL regime. Verification of a metallic state persisting up to the anticipated \(r_{s}^{WC}\) is crucial to establish a LSPT. The results below confirm a metal state up to a \(r_{s}\) that is in fact approximately the \(r_{s}^{WC}\). The metallic state is determined by the \(T\) dependence of the resistivity \(\rho(T)\) measured with a four-probe AC lock-in technique. Figure 1a shows the \(\rho(T)\) for \(p=5.9\times 10^{9}\) cm\({}^{-2}\) (or \(r_{s}=37\)) which unambiguously confirms a metal behavior: \(\rho(T)\) is below the quantum of conductance \(h/e^{2}\) and \(d\rho/dT\sim\)3 k\(\Omega\)/100 mK has a positive sign below a characteristic temperature \(T_{max}\) where \(\rho(T)\) peaks. \(T_{max}\sim 125\) mK is \(\ll T_{F}=E_{F}/k_{B}\sim 400\) mK. Figure 1c confirms the same metal behavior in a different sample with a carrier density of \(p=5.2\times 10^{9}\) cm\({}^{-2}\) (or \(r_{s}=40\)) down to \(T=4\) mK. Note that the ac current excitation is kept under 0.5 nA so that Joule heating at \(<1\times 10^{-17}\) W is negligible. \(\rho(T)\) is studied for multiple carrier densities and the characteristics related to a critical behavior are found through the \(r_{s}\) dependence of \(T_{max}\). \(T_{max}\) is sometimes associated with a renormalized Fermi energy \(E_{F}^{*}(p)\)[11]. As shown in Figure 1c where the \(p\) or (\(r_{s}\)) dependence of \(T_{max}\) is shown, \(T_{max}\) is indeed below \(T_{F}=E_{F}/k_{B}\) which is qualitatively correct from the stand point of a strong renormalization [10]. However, the nonlinear fall of \(T_{max}\) with decreasing \(p\) is far too rapid to be accounted for by FL renormalization: a nearly three-fold drop from \(p=8.5\) to \(5.8\times 10^{9}\) cm\({}^{-2}\). In fact, the trend of diminishing \(T_{max}\) suggests a critical behavior. A nonlinear three-free-parameter fit to a scaling \(T_{max}(r_{s})=T_{0}(r_{s}^{\epsilon}-r_{s})^{\alpha}\) produces \(T_{0}=116.4\pm 5.3\) mK, \(r_{s}^{\epsilon}=41.08\pm 0.6\), and \(\alpha=0.488\pm 0.25\), with a \(R^{2}\approx 0.9987\). \(r_{s}^{\epsilon}\sim 41\) corresponds to a critical density \(p_{c}\sim 5\times 10^{9}\) cm\({}^{-2}\). Carrier density dependent scaling has been reported in Si-MOSFETs where a similar exponent is found [25]. Note that the value of \(m^{*}=0.35m_{0}\) used for extracting \(r_{s}\) values is based on the cyclotron resonant studies [26; 27] obtained for similar carrier densities. In addition, the DC-IV reveals a nonlinear response incompatible with a FL. As shown in Figure 1d, the DC-IV for \(r_{s}=40\) (or \(p=5.2\times 10^{9}\) cm\({}^{-2}\)) is nonlinear within a current window of just \(\pm 0.4\) nA. The derivative \(dV/dI\) shown in the inset figure varies continuously from 6.3 k\(\Omega\) at \(I=0\) to 3.5 k\(\Omega\) at \(I=\pm 0.4\) nA. The nonlinearity is easily resolved down to tiny drives below \(I\sim 0.1\) nA where the Joule heating dissipates negligibly at \(<5\times 10^{-17}\) W. Nonlinear DC response associated with interaction has been extensively reported for both the insulating regime, in connection to the finite pinning effects [28], as well as in the metallic regime, as identified in mesoscopic systems in a \(B\)-field [29]. As shown later for larger \(r_{s}\), the nonlinear DC-IV becomes increasingly prominent and eventually manifests as a threshold similar to the pinned charge density waves (CDWs) [30].