Tuesday, October 18, 2011

Deconstructing Transport properties of strongly correlated electron metals

I have been working through a really nice article Electrothermal transport coefficients at finite frequencies by Sriram Shastry. The key idea is that there are certain transport coefficients [the Hall coefficient (Hall resistivity), Lorenz ratio, and Thermopower] for which have a weak frequency dependence and so one can obtain a reliable estimate of the dc value from the high frequency value. This greatly simplifies the computation because the latter is determined by the expectation value of a specific operator in the ground state (or thermal ensemble). Unlike the dc transport coefficient this expectation value is not particularly sensitive to finite size effects and so can be evaluated from Lanczos (exact diagonalization) on a small lattice. Alternatively it can be evaluated from a high temperature series expansion.

Is this high frequency approximation justified? It can motivated in a heuristic manner from the fact that in the Drude model the relevant transport coefficients [the Hall coefficient, Lorenz ratio, and Thermopower] are all independent of the relaxation time.
For the t-J model on the triangular lattice Shastry also compares explicit evaluations of the Hall coefficient at zero, non-zero, and infinite frequency and finds there is little variation between them.
Here are a few highlights of the paper.

1. Strong correlations cause qualitative differences. Consider the Hubbard model as a function of doping. There are three changes in the sign of the Hall and Seebeck coefficients, in contrast to the one change in sign (at half filling) that occurs for the uncorrelated (U=0) band. In particular one can have a "hole-like" band structure and Fermi surface but an "electron-like" Hall coefficient.
[I think the solid line in the above graph is the Heikes formula which holds in the infinite temperature limit and is related to the entropy of the charge carriers in the Hubbard model in the atomic limit U >> |t|].

2. On the triangular lattice changing the sign of the hopping t can lead to significant changes in the magnitude and temperature dependence of the thermopower. [Although I wonder if some of this difference is related to the relative proximity to van Hove singularities and the associated differences in the non-interacting density of states near the Fermi energy as discussed here].

3. On the triangular lattice at high temperatures there are contributions to the thermopower and Hall resistance which are first order in t/T. In contrast on the square lattice the leading terms are of order (t/T)^2. This arises because on the triangular lattice one can perform closed loop hops involving only 3 lattice sites.

4. A connection is made [with some interesting history] to the expression of Thomson [Lord Kelvin] for the thermopower in terms of entropy. This is relevant to this post.

I thank Subroto Mukerjee for helping me gain a better understanding of Shastry's work.

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