Non-fermi-liquid behavior in a disordered Kondo-alloy model

We study a mean-field model of a Kondo alloy using numerical techniques and analytic approximations. In this model, randomly distributed magnetic impurities interact with a band of conduction electrons and have a residual Ruderman-Kittel-Kasuya-Yosida coupling of strength J. This system has a quantum-critical point at ${J=J}_c\sim T_K^0,$ the Kondo scale of the problem. The T dependence of the spin susceptibility near the quantum critical point is singular with $\chi \mathrm{}\left(0\right)\mathrm{}-\chi \mathrm{}\left(T\right)\mathrm{}\propto T^{\gamma }$ and noninteger $\gamma .$ At $J_c,\hspace{0.5em}\gamma =3/4.\mathrm{}$ For $J\lesssim J_c$ there are two crossovers with decreasing T, first to $\gamma =3/2\mathrm{}$ and then to $\gamma =2,$ the Fermi-liquid value. The dissipative part of the time-dependent susceptibility ${\chi }^{″}\left(\omega \right)\propto \omega$ as $\overrightarrow{\omega }0$ except at the quantum-critical point where we find ${\chi }^{″}\left(\omega \right)\propto \sqrt{\omega }.$ The characteristic spin-fluctuation energy vanishes at the quantum-critical point with ${\omega }_{\mathrm{sf}}\sim \left(1\mathrm{}-{J/J}_c\right)$ for $J\lesssim J_c,$ and ${\omega }_{\mathrm{sf}}\propto T^{3/2}$ at the critical coupling.