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Progress of Theoretical Physics

Anderson Orthogonality Theorem: Overlap Integral between Two Ground States with Differnt Centers of Local Potential

T. Kitamura, Akihide Oguchi, Kei Yosida

Published Sep 1, 1987• 3 citations
Publisher

Abstract

The overlap integral between two ground states of the conduction electron system with a local potential at each different site is studied. The overlap integral is obtained for the system with 3 nonzero partial wave components of scattering, and this result is extended to the general case with n nonzero phase shifts. It is shown that the calculation of the exponent of the overlap integral can be reduced to solving an eigenvalue problem.

DOI
10.1143/ptp.78.583

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Physical Review Letters•Feb 22, 1971•22 citations
The overlap integral between two ground states of the conduction electron system with a local potential at each different site is studied. The overlap integral is obtained for the system with 3 nonzero partial wave components of scattering, and this result is extended to the general case with n nonzero phase shifts. It is shown that the calculation of the exponent of the overlap integral can be reduced to solving an eigenvalue problem.
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.
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 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 ${N}^{\ensuremath{-}\ensuremath{\alpha}}$, 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.