Thursday, November 1, 2012

The Nozieres exhaustion problem

The trivial problem is that I find reading the papers of Nozieres difficult and exhausting. They are big on hand waving and physical insight. An earlier post considered his classic paper on a Fermi liquid treatment of the Kondo problem.

Now the real problem.
Today I giving a cake meeting talk about a short 2005 paper Kondo lattices and the Mott Metal-Insulator which Nozieres published in a JPSJ volume "Kondo effect- 40 years after the discovery". It also has several other nice (short) review articles. This paper reviews the resolution of the "exhaustion problem" for Kondo lattices, first raised by Nozieres in a 1998 paper.

In the Kondo effect a single magnetic impurity forms a spin singlet with a collection of metallic electrons within an energy range T_K (the Kondo temperature) of the Fermi energy, E_F. Typically, T_K is much smaller than E_F.

Now consider the Kondo lattice where there are magnetic "impurities" (i.e. localised spins) placed on a lattice of N_L sites. The metal contains of order N_L electrons. How many electrons can form Kondo spin singlets with (i.e. screen) the localised electrons?

N_eff ~ DOS(E_F)*T_K ~ N_L/E_F * T_K  is much less than N_L

Hence, just a few itinerant electrons must screen (form singlets with) many localised spins.
Why is this a problem?
Because in heavy fermion materials (which are described by a Kondo lattice model) one does observe a Fermi liquid ground state, which is a spin singlet with all the localised spins "screened", below some coherence temperature T_coh
Nozieres has a simple argument [which I don't follow and it turns out to be wrong] that
T_coh ~ N_eff/N_L * T_K ~ T_K^2/E_F is much less than T_K

This prediction that T_coh can be much less than T_K and that their ratio has a non-perturbative dependence on the Kondo coupling turns out to be incorrect. This was shown in 2000 with a slave boson theory by Burdin, Georges, and Grempel and confirmed with DMFT by Costi and Manini. There is a nice review of the problem by Burdin.
Nozieres says there is an "obscure puzzle" here: "a phase transition that generates an energy scale much bigger than its transition temperature". He refers to the fact that in the Kondo lattice problem the hybridisation energy Delta [which mixes the localised spin states with the conduction band] is much larger than T_K [its value in the single impurity problem].

Another puzzle for me is how all this relates to a 2008 Nature paper
Scaling the Kondo lattice
by Yang, Fisk, Lee, Thompson, and Pines
which gives an exhaustive (!) analysis of a wide range of experimental data on heavy fermions and only invokes a single temperature scale T* which is much larger than T_K.

What does all this have to do with the Mott transition and the DMFT treatment of it?
The latter can be viewed as a "Kondo lattice problem". At half filling each site contains a "localised" electron which hybridizes and (in the metallic state) forms Kondo singlets with all the remaining electrons.
Nozieres then makes an obscure argument that leads him to the "physical insight" that "the narrow resonance behaviour [present in the metallic phase] is a direct consequence of the residual entropy [of the Mott insulator]"


  1. Thank you Ross! Nice talk and nice post. Things are much more clear in such a "resonant" mode of presentation

  2. Hi Ross,

    concerning the work of Fism et al. you mention, showing T* >> T_K, there is reason to believe that this can be a result of inter-site RKKY interactions. This mechanism was discussed using cluster DMFT in the following paper we recently published:

    However, there are still some unresolved issues left open by this study, associated by limitations of the cluster approach. I think this issues deserves more careful further study using cluster DMFT.