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Progress of Theoretical Physics

Orthogonality Catastrophe Due to Local Electron Interaction

K. Yamada, Kei Yosida

Published Apr 1, 1978• 21 citations
Publisher

Abstract

The overlap integral between two ground state wave functions of the conduction electron system with different local situations is discussed for a general case in which local electron interaction exists in addition to a local impurity potential. The Anderson theorem which states that the overlap integral is expressed by exp[-δ/π×logN] is proved in this general case.

DOI
10.1143/ptp.59.1061

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The Anderson theorem concerning the overlap integral between two ground-state wave-functions of the electron system with different local potentials is discussed from a general point of view. It is shown that this theorem generally holds in the case where the initial Hamiltonian also includes electron interactions.

Progress of Theoretical Physics•Aug 1, 1978•14 citations

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New Formalism for an Overlap Integral of the Anderson Orthogonality Theorem

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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.

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Orthogonality Catastrophe Revisited

Nic Shannon

We prove a simple theorem on the overlap of the wavefunctions of a manybody system with and without a single impurity and show how, and under which conditions, this leads to the ``Orthogonality Catastrophe'' (OC) described by Anderson. A compact new derivation of the known result for the OC exponent in a free electron gas, $α=2 \sum_l(2l+1) (δ_l(ε_f)/π)^2 $ %$α= 2\sum_l [2l+1] (δ_l(ε_f)/π)^2$, is given. We also introduce a simple way of calculating core level photoemission (XPS) lineshapes from the finite size scaling of groundstate energies, reproducing and extending results previously derived using boundary conformal field theory. Different conditions under which the OC fails are discussed, and simple physical predictions for XPS lineshapes made in each case.

arXiv:cond-mat/9910121•Oct 7, 1999•0 citations
The overlap integral between two ground state wave functions of the conduction electron system with different local situations is discussed for a general case in which local electron interaction exists in addition to a local impurity potential. The Anderson theorem which states that the overlap integral is expressed by exp[-δ/π×logN] is proved in this general case.
It is shown that the overlap integral between two ground-state wave functions for the Hamiltonian of the interacting electron system with different local potentials can be written generally in terms of the S-matrix describing potential scattering. Here, the local potential is considered to be a general inseparable one and the Green functions and self-energies are treated as operaors with off-diagonal elements. This orthogonality theorem expressed in a general form is applied to some special cases.
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.
We prove a simple theorem on the overlap of the wavefunctions of a manybody system with and without a single impurity and show how, and under which conditions, this leads to the ``Orthogonality Catastrophe'' (OC) described by Anderson. A compact new derivation of the known result for the OC exponent in a free electron gas, $α=2 \sum_l(2l+1) (δ_l(ε_f)/π)^2 $ %α=2∑l[2l+1](δl(εf)/π)2α= 2\sum_l [2l+1] (δ_l(ε_f)/π)^2α=2∑l​[2l+1](δl​(εf​)/π)2, is given. We also introduce a simple way of calculating core level photoemission (XPS) lineshapes from the finite size scaling of groundstate energies, reproducing and extending results previously derived using boundary conformal field theory. Different conditions under which the OC fails are discussed, and simple physical predictions for XPS lineshapes made in each case.