Effective model of the electronic Griffiths phase

The recent discovery of a number of heavy fermion materials with non-Fermi liquid (NFL) thermodynamic and transport properties has been followed by a significant theoretical effort to understand the origin of the NFL behavior.1 In the cleaner systems, the proximity to a quantum critical point appears to dominate many of the observed properties.2–10 In other heavy fermion systems disorder seems to play a more essential role and appears to be crucial for understanding the NFL behavior.11–17 Many experiments can be explained by the disordered Kondo model,11 which has recently been put on a much stronger microscopical foundation.18–20 The emergence of electronic Griffiths phases in models of correlated electrons has been established18–21 as a universal phenomenon, within a class of extended (“statistical”) Dynamical mean field theory (DMFT) approaches.21 This statDMFT method provides an exact (numerical) treatment of localization in the absence of interactions, and reduces to the standard DMFT equations22 in the absence of disorder. When both interactions and localization are present, nonFermi liquid behavior emerges universally,18 as a precusor of a disorder-driven metal-insulator transition, due to a very broad distribution PsTKd of local Kondo temperatures. This distribution has a low-TK tail of the form PsTKd,TK , independent of the microscopic details or the specific form of disorder. The exponent a=asWd is found to be a smooth, monotonically decreasing function of the disorder strength W, and the NFL behavior emerges for W greater than the critical value WNFL corresponding to aø1, when PsTKd becomes singular at small TK. As in other Griffiths phases, the thermodynamic and transport properties in this NFL region are dominated by rare events, which in this model correspond to sites with the lowest Kondo temperatures. In this paper, we show that the same behavior is found in a simpler, standard DMFT version of the model with a judicious choice of bare disorder. We should emphasize that localization is not present in this effective model, but the Griffiths phase emerges in qualitatively the same fashion as in the above more realistic calculations. We discuss how the specific disorder distribution which is hand-picked in the effective model is dynamically generated by fluctuation effects within the statDMFT formulation, elucidating the origin of the universality of the Griffiths phase behavior. In addition, the simplicity of this DMFT effective model makes it possible to describe all the qualitative features of the solution using simple analytical arguments, thus eliminating the need for large scale numerical computations in the description of the electronic Griffiths phase. This may be crucial in order to address more complicated issues, such as the role of additional Ruderman-Kittel-Kasuya-Yosida (RKKY) interactions in disordered Kondo alloys.23 This paper is organized as follows. Section II introduces the effective model for the electronic Griffiths phase as a DMFT model with a Gaussian distribution of random site energies. This model is solved analytically in the Kondo limit in Sec. III, and numerically in Sec. IV. The arguments explaining the universal aspects of the form of the renormalized disorder are presented in Sec. V. Section VI establishes a connection with the Griffiths phase in a single band Hubbard model, and Sec. VII contains our conclusions.