Fig. 4: Dependence of the thermalization time on the range of interactions and thermal coupling to the environment.

a Thermalization time τ_σ as a function of T for different \({n}_{{{{{{{{\rm{s}}}}}}}}}^{{{{{{{{\rm{f}}}}}}}}}\), as shown. For the short-range case (solid symbols), the symbol shape indicates the sample (squares: 2 × 20, circles: 2 × 50, triangles: 1 × 90); \({n}_{{{{{{{{\rm{s}}}}}}}}}^{{{{{{{{\rm{i}}}}}}}}}(1{0}^{11}{{{{{{{{\rm{cm}}}}}}}}}^{-2})=(32.2\pm 0.3)\). Results for the long-range case (open symbols) are adapted from ref. ⁴⁸ in which a 2 × 50 sample was studied for different \({n}_{{{{{{{{\rm{s}}}}}}}}}^{{{{{{{{\rm{f}}}}}}}}}\). Dashed black lines guide the eye. In both cases, the samples were placed in vacuum. Electron-phonon coupling is weak or negligible for T < 2 K, and it becomes dominant for T > 2 K. b τ_σ vs. n_s for the short-range case with two different types of thermal coupling to the environment. The data from a, shown for T = (0.9 − 1.7) K, obtained with samples placed in vacuum and thus very weakly coupled to a thermal bath, demonstrate that τ_σ is anomalously long. When measured in ⁴He gas at T = 1.7 K, the samples and measurement leads are more strongly coupled to the environment. In that case, τ_σ is about an order of magnitude lower, i.e. the thermalization is much faster. Open symbols are data from another sample with the same dimensions; dashed lines guide the eye. The error bars were determined from the fits at long times, as illustrated in Supplementary Fig. 7. c Schematic of the effect of the Coulomb interaction range on quench dynamics in a disordered 2DES at low enough n_s. In the long-range, ~ 1/r case, the dynamics is glassy, but the system thermalizes, in principle at a finite τ_σ(T) at all T > 0 (τ_σ → ∞ as T → 0). In the short-range, ~ 1/r³ case, τ_σ is independent of T, but it is extremely long when coupling to the environment is very weak; when this coupling increases, τ_σ decreases. Therefore, in this case, the thermalization results from the residual coupling of the 2DES to the outside world, but on time scales short enough compared to τ_σ the system exhibits MBL-like properties. For a and b, source data are provided as a Source Data file.