Quantum ergodicity for electrons in 2d

The results obtained [102,105] show that the Coulomb interaction between two electrons in excited states leads to their delocalization for $1< r_s < r_L^{4/3}$ while for $r_s > r_L^{4/3}$ they remain localized. Here the parameted $r_s$ is the ration of the Coulomb interaction to the Fermi energy and $r_L$ is the $r_s$-value taken at such a density when one electron is located in a box of one-electron localization length size. The transition between these phases is similar to the Anderson transition in an effective dimension $d_{eff}=3$. The numerical studies of the spectral statistics for many polarized electrons in the 2D Anderson model [104,106] show that for $3 < r_s < 9$ and $5 \leq W/V \leq 15$ the ground state is nonergodic (localized) and is characterized by the Poisson statistics for the total energy $E< E_c$ (here $W, V$ are the disorder and hopping strengths, respectively). However, the transition to quantum ergodicity and the Wigner-Dyson statistics takes place at a fixed total energy $E_c$ independent of system size (for $r_s \approx 3.2$, $5 \leq W/V \leq 10$ and fixed filling $\nu \approx 1/32$). This implies a delocalization at zero temperature $T$. At the critical point $E_c$ the critical statistics approaches to the Poisson limit with the increase of disorder strength. In a certain sence the situation is similar to the Anderson transition in high dimensions $d > 3$ where a similar tendency had been observed. In analogy with this result and the case of two electrons in 2D we make a conjecture that the transition at $E_c$ is similar to a transition in some effective dimension $3 \leq d_{eff}$. This $d_{eff}$ is growing with the disorder strength $W$.

The interaction induced ergodicity at $T=0$ is in favor of the metal-insulator transition observed experimentally by Kravchenko et al.. However, the data are not yet sufficient to determine the behavior of resistivity on temperature in the ergodic phase at $E>E_c$. Therefore, it is not excluded that at strong disorder this ergodic phase will show a resistivity growth at low $T$. In this case one can suppose that the ergodicity induced by the Coulomb interaction is responsible for the phononless VRH conductivity as it was argued by Shklovskii and indicated by the experiments of Shlimak, Pepper et al.. More detailed investigations are required to understand the properties of the Coulomb ergodic phase at $E>E_c$.