Hyunyong Lee, Stefan Kettemann
The quantum phase diagram of disordered electron systems with magnetic impurities is derived. The competition between the indirect exchange interaction of magnetic impurities and the Kondo effect is known to give rise to a rich quantum phase diagram, the Doniach Diagram. A Kondo screened phase is separated from a spin ordered phase at a quantum critical point (QCP), when the local exchange coupling $J$ and the concentration of magnetic moments $n$ are varied. We report here numerical results for disordered 2D electron systems which show that both the Kondo temperature and $|J_{\rm RKKY}|$ are widely distributed and the QCP is extended to a critical region. Calculating the distribution function of their ratio $|J_{\rm RKKY}|/T_K$ for various separations $R$, we find a sharp cutoff, and from that a critical density $n_c(J)$. For lower densities that ratio is smaller than one for all sites, which we can identify therefore as the Kondo phase. As the disorder amplitude is increased, a phase of coupled magnetic moments grows at the expense of the Kondo phase. In addition, a paramagnetic (PM) phase where magnetic moments remain free at all temperatures arises below a critical exchange coupling $J_c$. Besides a low temperature Kondo Fermi liquid phase we find a phase with non-Fermi liquid behaviour with anomalous power law temperature dependence in the magnetic susceptibility and the specific heat. We confirm that these anomalous powers are related to the multifractality parameter $\alpha_0$. The coupled moment phase shows a succession of 3 phases: a Griffiths phase with anomalous power laws determined by the distribution of $J_{\rm RKKY}$, a spin glass phase and a phase with long range magnetic order. We also report results on a honeycomb lattice, graphene, where we find that the magnetic phase is more stable against Kondo screening, but is more easily destroyed by disorder into a PM phase.
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http://arxiv.org/abs/1211.1734
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