## 2 Results
### Nonlinear Responses and the Liquid-Solid Phase Transition  
Detection of collective modes of possible WCs has been carried out in both the quantum Hall regime, with respect to pinning [28; 33; 34; 35], resonant absorption (via RF (Radio frequency), microwaves, acoustic waves [36; 37; 38; 39], and tunneling [40]), as well as in a zero magnetic field (\(B\)) [14; 41; 42; 43]. Nevertheless, a WC phase is not yet well supported for a number of reasons. First, the correlation length \(\xi\) is rather small (\(\leq\)1 um) and decays exponentially with increasing \(T\). As the pinning strength scales with \(\xi\)[44; 45], it explains why the observed differential resistance is not as robust as expected. These findings favor an intermediate or a mixture of liquid and solid phases. Second, the reported melting [37; 38; 40] is reflected via a smooth crossover [28; 37; 38; 40], lacking a singularity that reflects the changes in the free energies across the phase boundary [46]. Some of the experimental challenges can be understood by the sensitive nature of the WC on the disorder, \(T\), the magnetic field (for WC studies in the fractional quantum Hall regime), and even the experimental conditions. Suppression of disorder is key to not only reducing the chances of glass [47; 48; 49] or mixed phases, but also to maintaining an accessible critical temperature \(T_{c}\) which can be suppressed substantially by disorder fluctuations [50; 51; 52]. The results presented below are based on the DC response as a probe to both pinning and melting effects. Figure 3a shows the DC-IV obtained for \(r_{s}\sim 50\) where a contrast is found between two close-by temperatures: 32 and 43 mK. The DC response at 43 mK or higher \(T\) (not shown) is approximately linear up to 50 \(pA\) and is invariant when switching the leads to different orientations. This finding  
Figure 2: (a) \(T\) dependence of the resistivity \(\rho(T)\) for a carrier density range from \(7\times 10^{8}\) cm\({}^{-2}\) to \(1.25\times 10^{10}\) cm\({}^{-2}\). (b) Comparison of the nonactivated conductance (in blue) of a clean system to the activated hopping conductance (in red) of a more disordered system for \(p\sim 1.5\times 10^{9}\) cm\({}^{-2}\). (c) The nonactivated behavior is turned into an activated behavior as a result of increase of disorders. \(p\) is from \(7\times 10^{10}\) cm\({}^{-2}\) to \(5\times 10^{9}\) cm\({}^{-2}\). Solid lines are the AC results and the scattered points are the DC results. Dotted lines are a guide to the eye. Inset: comparison between an Arrhenius hopping and the Efros-Shklovskii variable range hopping. is consistent with the previous results [28; 34] interpreting an isotropic liquid. On the other hand, the DC-IV exhibits a nonlinear response at 32 mK, indicating the presence of pinning. The differential resistance \(r_{d}\) measured with a current \(I\) below 5 \(p\)A is \(\sim\) 18 M\(\Omega\). \(r_{d}\) approaches a linear behavior with increasing \(I\) and reaches the same \(r_{d}\) as for the 43 mK case. Assuming collective pinning of \(N=p\xi^{2}\) carriers by a periodic potential after the CDW model, previous studies [28; 34] adopted the following equation to estimate \(\xi\):  
\[I(E)\propto\sinh(NeEa/T)\exp[-(2 NeE_{th}a/\pi T)] \tag{1}\]  
\(E_{th}=V_{th}/L\) (\(L\) is the length between the voltage contacts) is the threshold electric field. At the threshold, \(eE_{th}a\) is \(\sim\) 0.01 \(\mu\)eV (or 0.1 mK)\(\ll k_{B}T\), confirming a collective nature. As shown in Figure 3a, fitting the 32 mK curve to equation (1) produces \(N\sim\) 200 or \(\xi\sim\) 14\(a\sim\) 1.4 \(\mu\)m at \(R^{2}=0.9995\). Nevertheless, this fitting makes an assumption that \(N\) is only affected by \(T\) and is independent of \(V\). This, however, is incorrect if a melting instead of depinning occurs. Note that these results associated with soft pinning are qualitatively consistent with the previous results [28]. Figure 3b shows the DC-IV obtained for the same \(r_{s}\) at 29 mK where a threshold behavior appears.