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Physical Review Letters

Exact Calculation of the Orthogonality Catastrophe in Metals

N. Rivier, Eugen Šimánek

Published Feb 22, 1971• 22 citations
Publisher

Abstract

We give exact calculation, using the linked-cluster theorem, of the overlap between the ground-state wave functions of a system of $N$ fermions in the absence and in the presence of a localized potential. It is found that the overlap vanishes as ${N}^{\ensuremath{-}\ensuremath{\alpha}}$, where $\ensuremath{\alpha}(\ensuremath{\delta})$ contains both a nonperiodic and a $\ensuremath{\pi}$-periodic term in the phase shift $\ensuremath{\delta}({\ensuremath{\epsilon}}_{\mathrm{F}})$ due to the potential. The present result reduces to that of Anderson and of Nozi\`eres and de Dominicis for small phase shifts.

DOI
10.1103/physrevlett.26.435

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We give exact calculation, using the linked-cluster theorem, of the overlap between the ground-state wave functions of a system of NNN fermions in the absence and in the presence of a localized potential. It is found that the overlap vanishes as N\ensuremath−\ensuremathα{N}^{\ensuremath{-}\ensuremath{\alpha}}N\ensuremath−\ensuremathα, where \ensuremathα(\ensuremathδ)\ensuremath{\alpha}(\ensuremath{\delta})\ensuremathα(\ensuremathδ) contains both a nonperiodic and a \ensuremathπ\ensuremath{\pi}\ensuremathπ-periodic term in the phase shift \ensuremathδ(\ensuremathϵF)\ensuremath{\delta}({\ensuremath{\epsilon}}_{\mathrm{F}})\ensuremathδ(\ensuremathϵF​) due to the potential. The present result reduces to that of Anderson and of Nozi`eres and de Dominicis for small phase shifts.
We propose a simple calculation method for an overlap integral between two ground states of the system with a local potential at each different site. Our formalism is applied to the overlap integral limited to l-wave scattering and the exact result is obtained. We also apply our formalism to the overlap integral for the system with l- and l’-partial wave components of scattering. The overlap integral in this case is exactly obtained in a closed form as a function of the phase shifts of l- and l’-waves. Our result covers that of Yamada, Sakurai, Miyazima and Hwang.