Quantum Critical Properties of the Bose-Fermi Kondo Model in a Large-$N$ Limit

Studies of non-Fermi-liquid properties in heavy fermions have led to the current interest in the Bose-Fermi Kondo model. Here we use a dynamical large-$N$ approach to analyze an $\mathrm{S}\mathrm{U}(N)\times \mathrm{S}\mathrm{U}(\kappa N)$ generalization of the model. We establish the existence in this limit of an unstable fixed point when the bosonic bath has a sub-Ohmic spectrum ($|\omega {|}^{1-ϵ}\mathrm{s}\mathrm{g}\mathrm{n}\omega$, with $0<ϵ<1$). At the quantum-critical point, the Kondo scale vanishes and the local spin susceptibility (which is finite on the Kondo side for $\kappa <1$) diverges. We also find an $\omega /T$ scaling for an extended range (15 decades) of $\omega /T$. This scaling violates (for $ϵ\ge 1/2$) the expectation of a naive mapping to certain classical models in an extra dimension; it reflects the inherent quantum nature of the critical point.

Figure 1

The large-$N$ Feynman diagrams.

Figure 2

Static and dynamical local susceptibilities for $ϵ=0.1$. The solid line corresponds to the quantum critical point $g=g_c$.

Figure 3

Static and dynamical susceptibilities for $ϵ=0.9$.

Figure 4

$\omega /T$ scaling of the dynamical spin susceptibility at $g=g_c$, for $ϵ=0.9$ and $\kappa =1/2$. The scaling exponent is 0.1.