Finite-Size Scaling of the Level Compressibility at the Anderson Transition

We compute the number level variance $\Sigma_{2}$ and the level compressibility $\chi$ from high precision data for the Anderson model of localization and show that they can be used in order to estimate the critical properties at the metal-insulator transition by means of finite-size scaling. With $N$, $W$, and $L$ denoting, respectively, system size, disorder strength, and the average number of levels in units of the mean level spacing, we find that both $\chi(N,W)$ and the integrated $\Sigma_{2}$ obey finite-size scaling. The high precision data was obtained for an anisotropic three-dimensional Anderson model with disorder given by a box distribution of width $W/2$. We compute the critical exponent as $\nu \approx 1.45 \pm 0.12$ and the critical disorder as $W_{\rm c} \approx 8.59 \pm 0.05$ in agreement with previous transfer-matrix studies in the anisotropic model. Furthermore, we find $\chi\approx 0.28 \pm 0.06$ at the metal-insulator transition in very close agreement with previous results.