Wednesday, November 21, 2012

The Kondo effect is non-perturbative

Last week we read "Beyond perturbation theory", chapter 3 of Hewson's The Kondo Problem to Heavy Fermions.

First he gives, without derivation, perturbation expressions for the impurity spin susceptibility and specific heat. The results exhibit logarithmic divergences at temperatures of the order of the Kondo temperature.

Hewson discusses some of the herculean efforts in the 1960s of people such as Abrikosov, Suhl, and Hamann, to come up with new diagrammatic techniques and summations to get rid of, or at least reduce, the divergences.
The results still have logarithmic temperature dependences. None give the Fermi liquid like dependences at low temperatures that experiments hinted at.

The Kondo effect is non-perturbative. n.b. the Kondo temperature has a non-analytic dependence on J.

What does one do?
An important insight was variational wave function proposed by Yosida in 1966.
One finds that the ground state is a spin singlet between the impurity spin and a superposition of the electrons above the Fermi sea. The binding energy has a similar non-analytic dependence on J as the Kondo temperature. Indeed if the wave function is generalised to include an infinite number of particle-hole pair excitations one finds that the binding energy is the Kondo temperature. Furthermore, the spin susceptibility is finite and inversely proportional to the Kondo temperature.

How can one describe the crossover with temperature to formation of these Kondo singlets and the emergence of the Kondo energy scale?
Anderson's poor mans scaling does that. Since it is such a profound and monumental achievement it deserves a separate post!

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