Power spectrum characterization of the Anderson transition

We examine the power spectrum of the energy level fluctuations of a family of critical power-law random banded matrices whose spectral properties are similar to those of a disordered conductor at the Anderson transition. It is shown analytically and numerically that at the Anderson transition the power spectrum presents $1∕f^2$ noise for small frequencies but $1∕f$ noise for larger frequencies. The analysis of the region between these two power-law limits gives an accurate estimation of the Thouless energy of the system. Finally we discuss in what circumstances these findings may be relevant in the case of nonrandom Hamiltonians.

Figure 1

(Color online) Power spectrum $S\left(k\right)$ as a function of $k$. Symbols represent numerical results (the matrix size is 3000 and the number of eigenvalues considered is $N=1024$ around the center of the band) for the critical random banded model Eq. (3) for different bandwidths $b$, the power spectrum was evaluated from Eq. (7). Lines represent the analytical prediction of critical statistics Eq. (11) with the TLCF given by Eq. (6). For all bandwidths $b$ the agreement between analytical and numerical results is excellent. For $b⪢1$ we observe two different regions: $N∕k⪢2\pi b$ corresponding with $S\left(k\right)\sim 1∕k^2$ and $N∕k⪡2\pi b$ with $S\left(k\right)\sim 1∕k$ similar to the prediction of WD (GUE). However for $b⪡1$, $S\left(k\right)\sim 1∕k^2$ for almost all $k$.

Figure 2

(Color online) Power spectrum $S\left(k\right)$ versus $N∕k$ in the limit $b⩽1$. Crosses and circles represent the numerical results (the matrix size is 3000 and the number of eigenvalues considered is $N=1024$ around the center of the band) for the critical random banded model Eq. (3) with $b=0.25,1$, respectively. The power spectrum $S\left(k\right)$ was obtained from Eq. (7). The dashed line corresponds to the best fit $S\left(k\right)\propto 1∕k^{\alpha }$ (the statistical error of $\alpha$ is about $\Delta \alpha =\pm 0.02$) in the limit $N∕k⪢2\pi b$ and the solid line is the prediction of WD statistics obtained from the GUE. The intersection between the linear fit and the WD statistics prediction corresponds in principle with the dimensionless conductance $g$ (the Thouless energy in units of the mean level spacing) of the system. However we observe that, for $b⩽1$, $S\left(k\right)$ is always different from the WD result and consequently no Thouless energy can be defined.

Figure 3

(Color online) Power spectrum $S\left(k\right)$ versus $N∕k$ in the limit $b⪢1$. Crosses and circles represent the numerical results (the matrix size is 3000 and the number of eigenvalues considered is $N=1024$ around the center of the band) for the critical random banded model Eq. (3) with $b=4,16$, respectively. The power spectrum $S\left(k\right)$ was evaluated from Eq. (7). The dashed line corresponds to the best fit $S\left(k\right)\propto 1∕k^{\alpha }$ (the statistical error of $\alpha$ is about $\Delta \alpha =\pm 0.02$) in the limit $N∕k⪢2\pi b$ and the solid line is the prediction of WD statistics obtained from the GUE. The intersection between the linear fit and the WD prediction corresponds with the Thouless energy, $E_c$, of the system. Units have been chosen such that the intersection point yields the dimensionless conductance of the system $g=E_c∕\Delta$. Thus $g\sim 25$ for $b=4$ and $g\sim 100$ for $b=16$.

Figure 4

(Color online) (Left) Power spectrum $S\left(k\right)$ obtained from 25 sets of 256 eigenvalues from the numerical diagonalization of the evolution matrix associated to Eq. (13) with $V\left(q\right)=0.2\ln \mid q\mid$ (cross). The dashed line is the prediction of SP statistics, the dotted-dashed line is the best fit $S\left(k\right)\sim 1∕k^{\alpha }$ and the dotted line is the prediction of WD statistics as is obtained from the Gaussian orthogonal ensemble (GOE) of random matrices. (Right) Poincare section from a single initial condition $p_0=0.2$ and $q_0=0.6$ after 300 000 iterations.