Wednesday, January 20, 2016

The Sommerfeld model is a Pauling point

A basic question that comes up in introductory solid state physics is:
Why does the Sommerfeld model for metals work so well?
It assumes that electrons are non-interacting fermions. Yet if you calculate the first order correction (in e^2 where e is the electronic charge) in the Coulomb energy you find it is comparable to the kinetic energy associated with the ground state.

Aside: the success of Sommerfeld is such a puzzle that Wigner mentioned it (for the wrong reasons in my view) at the end of his famous 1962 essay, The Unreasonable Effectiveness of Mathematics in the Physical Sciences.

The standard answer we give students is screening  plus Landau's Fermi liquid theory.
However, an interesting question is what happens if you try to actually do some sort of systematic many-body expansion with respect to the Coulomb interaction. Can you get the calculation to converge to experiment and see why Sommerfeld is good?

In Telluride last (northern) summer I heard a nice talk by Timothy Berkelbach that is relevant to this issue. The message I took away was that the Sommerfeld model is a Pauling point, i.e. by accident it gets the right answer for the wrong reasons.

The relevant paper has now appeared on the arXiv.

Spectral Functions of the Uniform Electron Gas via Coupled-Cluster Theory and Comparison to the GW and Related Approximations 
James McClain, Johannes Lischner, Thomas Watson, Devin A. Matthews, Enrico Ronca, Steven G. Louie, Timothy C. Berkelbach, Garnet Kin-Lic Chan

A key graph is below. It shows the quasi-particle dispersion relation for a uniform electron gas with r_s=4, the value relevant to sodium.

For comparison the "binding energy" [i.e. the k=0 energy] deduced from experiment is -2.6 eV, and the values for Sommerfeld and LDA are about -3.1 eV. Hartree-Fock gives -7.3 eV!

As you increase the "level of theory" [i.e. the sophistication of treatment] of electron correlations you go from Sommerfeld to HF to HF+GW to CCSD [Coupled Cluster Singles and Doubles].
Then one sees the answer at first gets worse and then improves and you almost get back to where you started!

Aside: This also illustrates how LDA is a Pauling point too!

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