Monday, October 22, 2012

Significance of the Kondo paradigm

This week we am part of new reading group which aims to work through Hewson's beautiful book The Kondo Problem to Heavy Fermions.
Why the choice of this topic?
Is it only of interest and relevance to people working on the Kondo physics and/or heavy fermion compounds?
I would say definitely no!

The Kondo problem represents a very important paradigm in quantum many-body physics. Perhaps the other main (well established and accepted) paradigms are
  • Fermi liquid theory,
  • BCS theory of superconductivity,
  • Mott insulators, 
  • spontaneous symmetry breaking
  • quantum antiferromagnets (Heisenberg models and Anderson superexchange),
  • Fractional Quantum Hall effect
1. Perhaps Mott insulators should not be on the list because there is no well-accepted theory beyond the "zeroth-order approximation".
2. Two other paradigms that I believe will eventually be accepted are Dynamical Mean-Field Theory and the RVB theory for superconductivity in proximity to a Mott insulator.
3. Is the shortness of the list a reflection of our success or failure? Is there a need for only a few paradigms or is it just we have solved so few problems?

Furthermore, the Kondo model/problem exhibits rich physics and key concepts in quantum many-body theory including
  • non-perturbative effects
  • asymptotic freedom: weak coupling scales to strong coupling as the temperature/energy decreases
  • scaling and universality [a single energy scale: the Kondo temperature]
  • the only (?) well-established realisation of a non-Fermi liquid fixed point [in multi-channel problems]
  • a simple physically transparent variational wave function [Yosida]

It is a benchmark for testing numerical and analytical methods since it can be solved exactly analytically using the Bethe ansatz and numerically using the Numerical Renormalisation Group. Its fixed points can be described by boundary conformal field theory.

Furthermore, Kondo physics is at the heart of the Dynamical Mean-Field Theory (DMFT) treatment of the Mott metal-insulator transition and electronic structure methods (e.g. DMFT+LDA).

Finally, Nozieres says the Kondo effect "is typical of what real theory should be, using tortuous roads towards simple final results".

I welcome additions and subtractions to my lists.


  1. I would keep Mott insulators on your list as a paradigm despite our only qualitative understanding.

    But I'd add Anderson insulators (demonstrating non-analyticities that result from disorder) and Luttinger liquids (the other well established paradigm for a non FL) as reasonably well understood paradigms.

  2. Thanks Peter.
    These are helpful suggestions.

  3. As a crucial paradigm, and not just for quantum many-body theory, definitely the renormalization group and all the ideas it is associated with: e.g. fixed points, scaling, classification of operators, conformal invariance and so on