Exponential Orthogonality Catastrophe at the Anderson Metal-Insulator Transition

Area of Science:

• Condensed Matter Physics
• Disordered Systems
• Quantum Mechanics

Background:

• The orthogonality catastrophe describes the loss of overlap between quantum states due to perturbations.
• Understanding this phenomenon is crucial for systems near the Anderson metal-insulator transition (AMIT).

Purpose of the Study:

• To investigate the behavior of fidelity at the Anderson metal-insulator transition.
• To analyze the impact of disorder and system size on quantum state overlap.

Main Methods:

• Calculation of the typical overlap (fidelity) F between ground states.
• Analysis of fidelity decay as a function of system size L.
• Investigation of the distribution of logF.

Main Results:

• Fidelity F decays exponentially with system size L (F∼exp(-cL^{η})) at the AMIT, indicating enhanced sensitivity to impurities.
• On the metallic side, fidelity follows a power law (F∼L^{-q(E_{F})}), with the exponent increasing as Fermi energy approaches the mobility edge.
• On the insulating side, fidelity becomes constant for sizes larger than the localization length.
• LogF is log-normally distributed with a diverging width at the AMIT, leading to a mean fidelity converging to one, contrasting with the typical value's exponential decay.

Conclusions:

• Strong disorder significantly increases system sensitivity to impurities near the AMIT.
• Multifractality at the AMIT explains the counterintuitive behavior of mean fidelity converging to one.
• The study provides insights into quantum phenomena at phase transitions in disordered materials.