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.
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.
Can you explain what vertex corrections mean physically in the context of condensed matter more generally? I am only used to the very formal viewpoint of QED where Ward identities save a lot of work. I also ready your previous blog posts on the topic but am still confused.
ReplyDeleteBest regards,
Anonymous Masters Student
There is no any intuition for vertex corrections. They are just a calculation tool of Feynman perturbation theory.
DeleteThe only physical interpretation of the vertex function I can offer is within the quasiparticle picture. The vertex corrections captures the angle of scattering, where forward scattering does not decrease the current or conductivity, while the more backward scattering decreases it the most. In pure bubble calculation any scattering gives the same decrease of conductivity. This is e.g. discussed to some extent in the book "Bruus, Flensberg: Many-body quantum theory in condensed matter physics" in chapter "Impurity scattering and conductivity".
ReplyDeleteI would like to mention two other indications of the importance of vertex corrections.
i) The difference of calculated charge susceptibility dn/d\mu within DMFT by direct differentiation of density via chemical potential or by bubble calculation (as dn/d\mu= /T). These give quite different results as the second one does not include the vertex corrections. The difference is shown in first graph (blue area).
ii) Comparison of spin and charge conductivity. Without vertex correction one obtains the same result for both. This is clearly wrong, e.g., in a Mott insulator being charge insulator and spin conductor. The vertex corrections are expected to turn spin insulating result to a spin conducting result. This is briefly mentioned in arXiv:2011.05750.
Jure, Thanks for the helpful and insightful comment.
ReplyDelete