Wednesday, September 14, 2016

Relating the Hall coefficient to thermodynamic quantities

Previously, I have posted about how in certain contexts one can relate non-equilibrium transport quantities to equilibrium thermodynamic quantities. This is particularly nice because for theorists it is usually a lot easier to calculate the latter than the former.
But, it should be stressed that all of these results are an approximation or only hold in certain limits.

Here are some examples.

The thermoelectric power can be related to the temperature derivative of the chemical potential through the Kelvin formula (illuminated by Peterson and Shastry).

A paper argues that the Weidemann-Franz ratio in a non-Fermi liquid can be related to the ratio of two different susceptibilities.

Work of Shastry showing that the high frequency limit of the Hall coefficient, Lorenz ratio and thermopower can be related to equilibrium correlation functions.

It has been suggested that the transverse thermoelectric conductivity (Nernst signal) due to superconducting fluctuations is closely related to the magnetisation.

Haerter, Pederson, and Shastry conjectured that for a doped Mott insulator on a triangular lattice that the Hall coefficient can be related to the temperature dependence of diamagnetic susceptibility.

Here I want to discuss some interesting results for the Hall coefficient R_H.
First,  remember that in a simple Fermi liquid (or a classical Drude model) with only one species of charge carrier of charge q and density n that

R_H=1/q n

Clearly this is a case where a rather complex transport quantity (which is actually a correlation function involving three currents) reduces to a simple thermodynamic quantity.

But, what about in strongly correlated systems?

There is a rarely cited paper from 1993

Sign of equilibrium Hall conductivity in strongly correlated systems 
 A. G. Rojo, Gabriel Kotliar, and G. S. Canright

It gives an argument of just a few lines that relates the Hall coefficient to the orbital magnetic susceptibility (Landau diamagnetism)

BTW. I think there is a typo in the very last equation. It should also contain a factor of the charge compressibility.

I am a bit puzzled by the derivation, because it appears to be completely rigorous and general. The derivation of the Kelvin formula for thermopower, also has this deceptive general validity. It turns out that "devil in the details" turns out to be that the two limits of sending frequency and wave vector to zero do not commute.

A related paper shows that with a certain limiting procedure the Hall response (at zero temperature) is related to the derivative of the Drude weight with respect to the density.

Reactive Hall Response
X. Zotos, F. Naef, M. Long, and P. Prelovšek

This is valuable because it gives a simple explanation of why in a doped Mott insulator the Hall coefficient can change sign as the doping changes.

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