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.