Low Energy Properties of $M$-State Tunneling Systems in Metals: New Candidates for Non-Fermi-Liquid Systems

We construct a generalized multiplicative renormalization group transformation to study the low energy dynamics of a heavy particle tunneling among $M$ different positions and interacting with $N_f$ independent conduction electron channels. Using a $1/N_f$ expansion we show that this M-level system scales towards a fixed point equivalent to the $N_f$ channel $\mathrm{SU}\left(M\right)\times \mathrm{SU}\left(N_f\right)$ Coqblin-Schrieffer model. Solving numerically the scaling equations we find that a realistic M-level system scales close to this fixed point and its Kondo temperature is in the experimentally observable range $1–10\mathrm{K}$.