Sunday, January 2, 2011

Glueing together a theory

In elemental superconductors Cooper pairs form due to an attractive retarded electron-electron interaction that is mediated by exchange of phonons. A perennial and controversial question concerning superconductivity in strongly correlated electron materials (particularly the cuprates and organic charge transfer salts) is whether there is a similar "pairing glue". In a 2007 Science paper Phil Anderson claimed there is not. 

A nice PRL by Maier, Poilblanc, and Scalapino addresses the question in a precise manner. The abstract has a nice summary of the issue.
The question of whether one should speak of a “pairing glue” in the Hubbard and t-J models is basically a question about the dynamics of the pairing interaction. If the dynamics of the pairing interaction arises from virtual states, whose energies correspond to the Mott gap, and give rise to the exchange coupling J, the interaction is instantaneous on the relative time scales of interest. In this case, while one might speak of an “instantaneous glue”, this interaction differs from the traditional picture of a retarded pairing interaction. However, as we will show, the dominant contribution to the pairing interaction for both of these models arises from energies reflecting the spectrum seen in the dynamic spin susceptibility. In this case, the basic interaction is retarded, and one speaks of a spin-fluctuation glue which mediates the d-wave pairing.
One thing that I found interesting and impressive about the paper is that it shows how one can define a dynamical pairing function in a non-perturbative manner. The relevant three equations are below. The Gorkov pairing (anomalous) Greens function F is defined by
which then leads to the pairing function 
 where G is the normal Greens function. This function has a real and imaginary part which can be related by Kramers-Kronig (or Cauchy) relations. The static part gives the pairing which can be written
The paper concludes:

The question regarding whether there is a “pairing glue” is then a question of whether the dominant contribution to φ1(kA,ω = 0) comes from the integral of φ2(kA, ω)/ω. From the results presented here we conclude that both the t-J and Hubbard models have spin-fluctuation “pairing glue”. However, they also exhibit a smaller, non-retarded contribution. For the cuprate materials, the relative weight of the retarded and non-retarded contributions to the pairing interaction remains an open question. Thus the continuing experimental search for a pairing glue in the cuprates is important and will play an essential role in determining the origin of the high Tc pairing interaction.

1 comment:

  1. This point about the glue can be reinforced by looking at the coincidence of the structures in the spin spectral function and in the anomalous self-energy, in a way analogous to what was done for phonons in Eliashberg theory: "Pairing dynamics in strongly correlated superconductivity" Phys. Rev. B 80, 205109 (2009), in particular Fig. 1