Relating non-Fermi liquid transport properties to thermodynamics

On tuesday I had nice discussion with Raghu Mahajan, Maissam Barkeshli, and Sean Hartnoll about their recent preprint Non-Fermi liquids and the Wiedemann-Franz law.

Aside: I generally find that discussing a paper with the authors before/after I have read it greatly increases my understanding. Here are a few things that became clearer to me.

In this paper "almost conserved quantities" means quantities for which the relaxation time is very long. Thus in a Fermi liquid the quasi-particles have very long lifetimes and so one can think of the quasi-particle number for every wave-vector near the Fermi surface as being "almost conserved". This means there are many conserved quantities.

However, they consider a system in which there is a Drude peak in the frequency dependent conductivity but fermionic quasi-particles are poorly defined due to large scattering. Optimally doped cuprates might be an example of a real material with this property. I thought that one dimensional models that exhibit this are Luttinger liquids. They have a Drude peak due to a collective bosonic mode but no fermionic quasi-particles. However, they are close to integrability which corresponds to having an infinite number of conserved quantities.

Note, this is different from most of the bad metals I discuss on this blog: they have no Drude peak and no quasi-particles. Although Aristomenis Donos and Sean recently considered a model (based on the holographic correspondence) that does have this property.

A Drude peak but no quasi-particles means there is one dominant relaxation timescale, that for momentum relaxation. This is what they mean by only one almost conserved quantity. This is a bit like hydrodynamics.

Central to the paper is a "memory matrix formalism" for transport properties. Some justification (and an intuitive understanding) for that is given in this paper. Central to that is the real part of static correlation functions [thermodynamic quantities] between the total momentum P and the electrical current J and heat current Q.

A Wiedemann-Franz type ratio can given in terms of these thermodynamic functions. The actual Lorenz ratio is much less than one.

This is because of a cancellation of the two terms in 

where the first term obeys the modified ratio

This is the central result of the paper. The ratio of two transport quantities is determined by the ratio of two thermodynamic quantities.

It will be nice to see extensions of this approach to give the thermopower (alpha/sigma=Seebeck coefficient) and the Hall coefficient. Both these quantities are fairly independent of scattering time in a Fermi liquid.

I think that in the absence of thermal conductivity due to phonons (unrealistic) the thermoelectric figure of merit could be larger than one.

In some sense this work is similar in spirit to that of Shastry on the Hall coefficient and thermopower. He considered the high frequency limits of these quantities for strongly correlated electron models and showed they could be related to equal time expectation values of operators (thermodynamic quantities).
He also considered Kelvin's formula for the thermopower.

I have one minor quibble. They say that CeCoIn5 violates Wiedemann-Franz (WF) at low temperatures. However, in a PRL Michael Smith and I showed that the relevant experimental paper in Science involves a spurious extrapolation to low temperatures. At sufficiently low temperatures we claim WF will hold. I think this alternative point of view should be stated in the paper.
It does seem awfully hard to find violations of Wiedemann-Franz.