Theory of the interaction between electrons and the two-level system in amorphous metals. II. Second-order scaling equations

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 $T_K=D{\left(v^x{v}^z\right)}^{\frac{1}{2}}{\left(\frac{v^x}{4v^z}\right)}^{\frac{1}{4v^z}}$ 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 $v^x\sim v^y\sim \frac{1}{8}$. The crossover temperature can be found in the region of physical interest ($T_K>1\mathrm{}$ K) if the initial diagonal coupling $v^z>0.2\mathrm{}$. 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.