Exponential Orthogonality Catastrophe at the Anderson Metal-Insulator Transition

Abstract

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 ∼ exp(− IA /2) = exp(−cL η ), where IA is the so called Anderson integral, η is the power of multifractal intensity correlations and ... denotes the ensemble average. Thus, strong disorder typically increases the sensitivity of a system to an additional impurity exponentially. We recover on the metallic side of the transition Anderson's result that fidelity F decays with a power law F ∼ L −q(E F ) with system size L. This power increases as Fermi energy EF approaches mobility edge EM as q(EF ) ∼ (where ν is the critical exponent of correlation length ξc. On the insulating side of the transition F is constant for system sizes exceeding localization length ξ. While these results are obtained from the mean value of IA, giving the typical fidelity F , we find that IA is widely, log normally, distributed with a width diverging at the AMIT. As a consequence, the mean value of 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.PACS numbers: 72.10. Fk,72.15.Rn,72.20.Ee,74.40.Kb,75.20.Hr, Anderson showed in Ref.1 that the addition of a static potential impurity to a system of N fermions changes its groundstate such that the overlap between the original ψ | and the new ground state ψ | has an upper bound,where the Anderson integral I A is for noninteracting electrons given in terms of the single particle eigenstates of the original system |n and the new system |m byIf the added impurity is short ranged and of strength λ, Anderson found for a clean metal I A = 2π 2 λ 2 ln N, diverging with the number of fermions N , so that F , also called fidelity, decays with a power law with N, leading to the so called orthogonality catastrophe (AOC). This implies that the local perturbation connects the system to a macroscpic number of excited states which has important consequences like the singularities in the X-Ray absorption and emission of metals [2]. Furthermore, the zero bias anomaly in disordered metals [3] and anomalies in the tunneling density of states in quantum Hall systems [4] are related to the AOC The concept of fidelity can be generalised to any parametric perturbation of a system and be used to characterise quantum phase transitions [5]. The AOC has been explored in mesoscopic systems [6,7]. With the advent of engineered many-body systems in ensembles of ultracold atoms it is possible to study nonequilibrium quantum dynamics of such systems in a controlled way so that conseuqences of parameter changes become measurable directly [8].An intriguing question is, if the system becomes less or more sensitive to the addition of another impurity if it alread...

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Mesoscopic fluctuations of the single-particle Green's function at Anderson transitions with Coulomb interaction

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