Factorization of the -Electron Wave Function in the Kondo Ground State

The multielectron wave function of an interacting electron system depends on the size of the system, that is, the number of electrons. Here the question investigated is how the wave function changes for a symmetric Friedel-Anderson impurity when the volume is doubled. It turns out that for sufficiently large volume (when the level spacing is smaller than the resonance width) the change in the wave function can be expressed in terms of a universal single-electron state centered at the Fermi level. This electron state is independent of the number of electrons and independent of the parameters of the Friedel-Anderson impurity. It is even the same universal state for a Kondo impurity and a symmetric Friedel impurity independent of any parameter. The only requirement is that the impurity has a resonance exactly at the Fermi level and that the level spacing is smaller than the resonance width. This result clarifies recent fidelity calculations. 1. Introduction In the late 1960s Anderson [1] showed that the potential of a weak impurity in a metal host changes the total -electron wave function of the conduction electrons dramatically. Actually with increasing number of electron states (which is achieved by increasing the volume) the scalar product between the wave functions of the -electron host without and with the impurity approaches zero. This phenomenon is generally called the Anderson orthogonality catastrophe (AOC). In recent years this phenomenon has been somewhat generalized and decorated with the romantic name fidelity. The generalization is that one applies the AOC to an arbitrary system, which depends on one or several parameters . If the system consists of electrons then it is described by its Hamiltonian. The Hamiltonian may contain a term, which is proportional to a parameter . Suppose that one can calculate the ground state of the system for and for finite . Then the scalar product of the two wave functions is defined as the fidelity of the system. Here is the number of conduction electrons states, which is proportional to the volume. The fidelity depends on the size of the system and of particular interest is the limit for increasing towards infinity. If approaches zero in this limit (the thermodynamic limit), then one faces an AOC. Our group studied recently the fidelity of the Friedel-Anderson impurity. This is an electron system with a -atom as impurity. The energy of the -electron lies at below the Fermi level. If one removes a -electron, that is, creates a -hole, then the conduction electrons can hop into the empty -state with a hopping