Theory of the interaction between electrons and the two-level system in amorphous metals. I. Noncommutative model Hamiltonian and scaling of first order

The general form of the interaction between tunneling two-level systems (TLS) and conduction electrons is discussed for metallic glasses. The particular form of the Hamiltonian is given in the case where only a single atom tunnels between two positions. There are two couplings corresponding to the two basic scattering processes: In the first one, the tunneling atom does not change position; the second process is the conduction-electron-assisted tunneling process. The two coupling parameters are estimated. The difference in the angular dependence of these couplings on the directions of the incoming and of the outgoing electrons is responsible for the appearance of logarithmic corrections in the scattering amplitude. Scaling equations are derived for the couplings in terms of changing the bandwidth cutoff. It is shown that the scaling equations lead to especially strong coupling in two conduction-electron scattering channels which are linear combinations of the $s$-, $p$-, and $d$- like spherical wave functions. The Hamiltonian scales to a spin $S=\frac{1}{2}$ antiferromagnetic Kondo Hamiltonian, which indicates the formation of a "bound state," where the motions of the tunneling atom and of the conduction-electron screening cloud around the TLS are strongly correlated; thus the Friedel oscillations follow the tunneling atom. The crossover temperature, below which the correlation becomes especially strong, is determined in the leading logarithmic approximation.