For an experimentalist one of the "easiest" quantities to measure for a metal is the electrical resistivity. Yet, for a many-body theorist working on models for strongly correlated electron systems this is one of the most difficult quantities to calculate, without making strong and debatable assumptions. One of the key questions is whether vertex corrections do matter. Ten years ago I summarised some of the issues.

This issue is nicely addressed in this nice paper from 2019.

Conductivity in the Square Lattice Hubbard Model at High Temperatures: Importance of Vertex Corrections

J. Vučičević, J. Kokalj, R. Žitko, N. Wentzell, D. Tanasković, and J. Mravlje

Besides the general issue of understanding the importance of vertex corrections, the paper is partly motivated by recent experiments on ultracold atoms that were compared to the results of calculations for a Hubbard model, using the finite-temperature Lanczos method (which essentially gives exact results on small finite lattices (e.g. 4 x 4)) and cluster Dynamical Mean-Field Theory (DMFT) (which does not include vertex corrections and has some momentum dependence in the self energy).

Before looking at the results I should point out the parameter values for the calculations. They are done for a Hubbard model on a square lattice. The half-bandwidth D=4t where t is the hopping parameter and U=10t. For the graphs below the doping p=0.1 (comparable to optimal doping in the cuprates).

Most importantly, the lowest temperature for which reliable calculations can be performed is T=0.2D=0.8t. In the cuprates, t is about 0.3 eV and so this lowest temperature corresponds to about 3000 K!, i.e. well above the superconducting Tc and the range of resistivity measurements on real materials. Most solids melt at these high temperatures.

Nevertheless, the results are important for two reasons.

First, the experiments on ultracold atoms are in this temperature regime. [Aside: again this shows how fermion cold atom experiments are a long long way from simulating cuprates, contrary to some hype a decade ago].

Second, we are desperate for reliable results, and so it is worth knowing something about the possible importance of vertex corrections, even at very high temperatures. [Aside: my first guess would have been that they are not very important since I would have thought that correlations would be short-range and hand waving from Ward's identity would suggest that it follows the vertex corrections are small. This is wrong.]

In the figure above the top panel is the charge compressibility versus temperature. This is a thermodynamic quantity and the results show that most of the methods give similar results suggesting that the corresponding vertex corrections are small, at least above 0.1D.

The lower panel shows the temperature dependence of the resistivity and suggests that vertex corrections do lead to quantitative, but not qualitative differences. I guess the resistivity is in units of the quantum of resistance. Each rectangle has a vertical dimension of 5 units and so the resistivity is in excess of the Mott-Ioffe-Regel limit, i.e. the system is a bad metal.

The figure above shows the frequency dependence of the optical conductivity for T=0.5D. There is a Drude peak at zero frequency and the broad peak near omega=2.5D=U corresponds to transitions between the lower and upper Hubbard band. DMFT is qualitatively correct but does differ from FTLM, showing the importance of vertex corrections.