Level-number variance and spectral compressibility in a critical two-dimensional random-matrix model

We study level-number variance in a two-dimensional random matrix model characterized by a power-law decay of the matrix elements. The amplitude of the decay is controlled by the parameter $b$. We find analytically that at small values of $b$ the level number variance behaves linearly, with the compressibility $\chi$ between 0 and 1, which is typical for critical systems. For large values of $b$, we derive that $\chi =0$, as one would normally expect in the metallic phase. Using numerical simulations we determine the critical value of $b$ at which the transition between these two phases occurs.

Figure 1

Level number variance as a function of the average level number, for different values of $b<1$ and $L=64$.

Figure 2

Spectral compressibility as a function of $b$, obtained from the slopes of the lines in Fig. 1. The data is presented for three different system sizes. The error bars are smaller than the symbol sizes.

Figure 3

Level number variance as a function of the average level number, for different values of $b>1$ and $L=64$. Dots are numerical data. The solid line at $b=5$ is Eq. (16). The other solid lines are the results of fitting the data with Eq. (17).

Figure 4

Spectral compressibility as a function of $b$, extracted from the data in Fig. 3 using Eq. (17). The error bars are smaller than the symbol size.