K. Vladár

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.

A general Hamiltonian for the interaction between conduction electrons and the two-level system is considered. Renormalization-group equations of second order are constructed with the use of the multiplicative renormalization-group technique. The mass renormalization is treated in detail to determine the effect of screening by conduction electrons on the energy splitting $E$. The crossover temperature ${T}_{K}=D{({v}^{x}{v}^{z})}^{\frac{1}{2}}{(\frac{{v}^{x}}{4{v}^{z}})}^{\frac{1}{4{v}^{z}}}$ between the weak and strong coupling regions is determined, and it is reduced by 2 orders of magnitude compared to the expression obtained in first-order scaling. The scaled values of the couplings are calculated analytically. In the crossover region the off-diagonal couplings are ${v}^{x}\ensuremath{\sim}{v}^{y}\ensuremath{\sim}\frac{1}{8}$. The crossover temperature can be found in the region of physical interest (${T}_{K}>1$ K) if the initial diagonal coupling ${v}^{z}>0.2$. In this case, the energy splitting calculated is reduced by more than 2 orders of magnitude. That reduction results in a large enhancement in the distribution of the energy splitting at the low-energy side. The position of the lower end of the scaling region is discussed where scaling in terms of temperature is hindered by the energy splitting.

In the two immediately preceding papers a theory of interaction between electrons and two-level systems (TLS) has been developed. According to that theory, the electron scattering on the TLS is resonant scattering below a characteristic crossover temperature ${T}_{k}$. This resonant scattering contributes to the lifetime ${T}_{1}$ of the TLS due to a Korringa-type mechanism. The relaxation time ${T}_{1}$ has been calculated by using the renormalized enhanced-coupling constants. Making use of the different ultrasound experimental data, we have given the effective coupling strengths for different alloys. The average coupling is small in PdCuSi and NiP alloys to form a resonant state. For PdZr and NbZr alloys, however, the averaged coupling strengths are too small---only by less than a factor of 2---to have a resonant state. Thus in these alloys a portion of the TLS may have sufficiently large coupling to have resonant scattering if a distribution for the coupling is assumed. The electrical resistivity, as a function of temperature, is calculated in detail. At the crossover temperature a logarithmic temperature dependence with a negative coefficient is found for one decade of the temperature. At lower temperature a crossover between the logarithmic and Fermi-liquid-type behaviors is suggested; therefore the resistivity near $T=0$ must behave as $\ensuremath{\Delta}R\ensuremath{\sim}(1\ensuremath{-}a{T}^{2})$. This overall behavior is in agreement with the available experimental data. The amplitude of the resistivity maximum arount $T=0$ is calculated, and the experimentally observed values can be explained by assuming reasonable densities for the TLS. It has been suggested that an enhanced density of state may be due to the renormalization (reduction) of the energy splitting of the TLS. That enhancement is necessary to explain the amplitude of the resistivity maximum at $T=0$ if only a small portion of the TLS has sufficiently large coupling to form the resonant state. Finally, the inelastic inverse scattering lifetime ${\ensuremath{\tau}}_{\mathrm{in}}$ for conduction electrons is calculated. Assuming a strong coupling case, the amplitude of the ${\ensuremath{\tau}}_{\mathrm{in}}$ obtained is of the same order of magnitude supported by experimental data. The assumption of dealing with the strong coupling case offers a possibility of resolving the discrepancy concerning the shortness of ${\ensuremath{\tau}}_{\mathrm{in}}$ relevant in localization theory. Finally, it is emphasized that a systematic study combining the ultrasound, resistivity, and electron inelastic scattering lifetime measurements may justify the applicability of the present theory to real metallic glasses.

A two-level system is treated which interacts with a degenerate fermionic heat bath. Arbitrarily strong screening by fermions is taken into account. The hopping of the two-level system may be spontaneous or assisted by the fermionic bath. By derivation of scaling equations it is shown for the spin-(1/2) case that because of the assisted hops the two-level system cannot be localized in one of the states.

The spinless resonant level model is studied when it is coupled by hopping to one of the arbitrary numbers of conduction-electron channels. The Coulomb interaction acts between the electron on the impurity and in the different channels. In the case of a repulsive or attractive interaction the conduction electrons are pushed away or attracted to ease or hinder the hopping by creating unoccupied or occupied states, respectively. In the screening of the hopping orthogonality catastrophe plays an important role. At equilibrium in the weak- and strong-coupling limits the renormalizations are treated by perturbative, numerical, and Anderson-Yuval Coulomb gas methods. In the case of two leads the current due to applied voltage is treated in the weak-coupling limit. The presented detailed study should help to test other methods suggested for nonequilibrium transport.