Exponential Orthogonality Catastrophe at the Anderson Metal-Insulator Transition

We consider the orthogonality catastrophe at the Anderson metal-insulator transition (AMIT). The typical overlap $F$ between the ground state of a Fermi liquid and the one of the same system with an added potential impurity is found to decay at the AMIT exponentially with system size $L$ as $F\sim \exp (-c{L}^{\eta })$, where $\eta$ is the power of multifractal intensity correlations. Thus, strong disorder typically increases the sensitivity of a system to an added impurity exponentially. We recover, on the metallic side of the transition, Anderson’s result that the fidelity $F$ decays with a power law $F\sim L^{-q(E_F)}$ with system size $L$. Its power increases as the Fermi energy $E_F$ approaches the mobility edge $E_M$ as $q(E_F)\sim [(E_F-E_M)/E_M{]}^{-\nu \eta }$, where $\nu$ is the critical exponent of the correlation length ${\xi }_c$. On the insulating side of the transition, $F$ is constant for system sizes exceeding the localization length $\xi$. While these results are obtained for the typical fidelity $F$, we find that $\log F$ is widely, log normally, distributed with a width diverging at the AMIT. As a consequence, the mean value of the fidelity $F$ converges to one at the AMIT, in strong contrast to its typical value which converges to zero exponentially fast with system size $L$. This counterintuitive behavior is explained as a manifestation of multifractality at the AMIT.

Figure 1

Correlation function of intensities $C$ as function of their energy difference $|E-E_M|$ in a disordered system with $d=3$, $\eta \approx 2$. Inset: Density of states (DOS) as function of energy $E$ with a transition between localized and extended states at the mobility edge $E_M$. Intensities are correlated within an energy window $2E_c$ around $E_M$.

Figure 2

The average Anderson integral $⟨I_A⟩$ as function of the Fermi energy $E_F$ in units of mobility edge $E_M$ for a 3D disordered system, Eqs. (6), (7), for various system sizes $L$ (in units of microscopic length $a_c$): $L=10$ (blue), $L=20$ (orange), and $L=50$ (red).

Figure 3

Average Anderson integral $⟨I_A⟩$ as a function of disorder parameter $g=E_F\tau$ for a 2D disordered system, substituting ${\xi }_{2D}=g\exp (\pi g)$ and ${\gamma }_{2D}=1/(\pi g)$ into Eq. (6), with $\mathrm{\Delta }\to \max ({\mathrm{\Delta }}_{\xi }=D/{\xi }^2,\mathrm{\Delta }=D/L^2)$ for various system sizes $L$.

Figure 4

Average fidelity $⟨F⟩$ as a function of the logarithm of system size $L$ for $\lambda =1$: (a) the 3D AMIT (blue line) with $\eta =2.096$ [18], (b) the 2D integer quantum Hall transition with $\eta =0.524$ [19], and (c) the 2D AMIT with spin-orbit interaction, with $\eta =0.344$ [19].