Critical Fidelity at the Metal-Insulator Transition

Using a Wigner Lorentzian random matrix ensemble, we study the fidelity, $F(t)$, of systems at the Anderson metal-insulator transition, subject to small perturbations that preserve the criticality. We find that there are three decay regimes as perturbation strength increases: the first two are associated with a Gaussian and an exponential decay, respectively, and can be described using linear response theory. For stronger perturbations $F(t)$ decays algebraically as $F(t)\sim t^{-D_2^{\mu }}$, where $D_2^{\mu }$ is the correlation dimension of the local density of states.

Figure 1

Fidelity of an ES, for (a) $x=0.01$ (the standard perturbative regime), (b) $x=0.8$ (FGR regime), and (c) $x=20$ (nonperturbative regime). The solid lines are the LRT results from Eq. (7) while the crosses are the outcomes of the numerical simulations with model (3, 4). In these simulations $L=1000$ and $b=10$. The mean level spacing of the unperturbed system is set to $\Delta \approx 1$. In this case, $x_c\approx 0.59$ and $x_{\mathrm{prt}}\approx 1.88$. The dotted line in (c) is plotted to guide the eye on the power-law behavior.

Figure 2

$F_I$ for $b=0.32$, 1.00, and 3.16. The initial state was chosen to be an ES. The mean level spacing of the unperturbed Hamiltonian is set to be $\Delta \approx 1$, while the size of the matrices is $L=5000$. In the cases reported here we have choose $x=5$. In the inset, we also present the fidelity for RS for $b=0.32$ and $b=1.00$ using the same parameters except $L=1000$. The straight lines are plotted to guide the eye.

Figure 3

The fitting parameter $\gamma$ for $b=0.03$, 0.10, 0.32, 1.00, 3.16, and 10. We are using $\Delta =1$, $x=5$. The analytical (solid lines) Eq. (5) and numerical (crosses) [46] results for $D_2$ are also shown for comparison.