Friday, March 6, 2015

Is DMFT "the only game in town"?

Last week Peter Woit kindly recommended my blog to his readers. This immediately doubled the number of daily page views for several days thereafter!

Peter also drew attention to my recent paper with Nandan Pakhira that shows that the charge diffusion constant in a bad metal violates a conjectured lower bound. This bound was conjectured, partly on the basis of arguments from string theory techniques [holographic duality, AdS-CFT]. Our calculations were all based on a Dynamical Mean-Field Theory (DMFT) treatment of the Hubbard model.

One commenter "Bernd" wrote
The violation of the holographic duality bound is based on DMFT calculations, which is a bit like string theory for strongly correlated fermions in the sense that it is somtimes sold as “the only game in town”. Nobody knows how accurate these methods really are.
For background, "the only game in town" refers to a common argument of string theorists that string theory must be correct because there are no others options for a mathematically self-consistent theory of quantum gravity. Woit's blog and book has many counter arguments to this point of view, as for example here. Ironically, a  commenter Carl points out
The quote “the only game in town” is exceptionally apt. It’s most associated with con man “Canada Bill” Jones, master of the Three Card Monte, who ironically was himself addicted to gambling and lost his money to better professionals as fast as he took from marks. Supposedly, on being advised by a friend that the Faro game he was losing money at was rigged, he replied “I know, but it’s the only game in town!”.
I wish to make two points in response to "Bernd".
First, I think we have a pretty good idea of how accurate DMFT is.
Second, I think any analogy between DMFT and string theory is very weak.

We know that DMFT is exact in infinite dimensions. It is an approximation in lower dimensions, that can be compared to the results of other methods. Cluster versions of DMFT give a systematic way to look at how the neglect of spatial correlations in single site DMFT matter. It has also been benchmarked against a range of other numerical methods.

The "sociological" comparison between string theory and DMFT is debatable.
I have never heard anyone claim it is "the only game in town".  On the other hand, there are probably hundreds of talks that I have never gone to where someone might have been said this.
I am a big fan of DMFT. For example, I think combining DMFT and DFT is one of the most significant achievements of solid state theory from the past 20 years. Yet, this is a long way from claiming it is the "only game in town".

It is worth comparing the publication lists of string theorists and DMFT proponents.
String theorists virtually only publish string theory papers reflecting their belief that it is the "only game in town". However, if you look at the publication lists of DMFT originators and proponents, such as Georges, Kotliar, Vollhardt, Metzner, Jarrell, Millis, ... you will see that they also publish papers using techniques besides DMFT. They know its limitations.

If you look at the research profiles of physics departments the analogy also breaks down. Virtually only string theorists get hired to work on "beyond the Standard Model". In contrast, when it comes to correlated fermions, DMFT is a minority, and not even represented in many departments.

Most importantly, there are no known comparisons between experimental data and string theory. DMFT is completely different. For just one example, I show the figure below, taken from this paper, that compares DMFT calculations (left)  to  experiments (right) on a specific organic metal close to the Mott insulator transition.

This organic material is essentially two-dimensional, a long way from infinite dimensions!
Furthermore, the parameter regime considered here corresponds to the same parameter regime [bad metal] that DMFT gives violations of the conjectured bound in the diffusion constant.

DMFT is not "the only game in town". We have a pretty good idea of how and when it is reliable.


  1. 'Most importantly, there are no known comparisons between experimental data and string theory." String theory with the infinite nature hypothesis might predict the string landscape. However, let us assume 2 things: (1) String vibrations are confined to 3 copies of the Leech lattice.
    (2) Nature is finite and digital; the maximum physical wavelength equals the Planck length times the Fredkin-Wolfram constant.
    I claim that string theory with these two new hypotheses does indeed predict the Fernández-Rañada-Milgrom effect, the Space Roar Profile Prediction, and the 64 Particles Hypothesis. Google "peter woit fqxi essay contest 2015" and read my comment on Peter Woit's essay.

    1. I stand by my claim.

      I am looking for a paper in a Physical Review journal that compares actual experimental data with a specific prediction from string theory.

      In contrast, in the post I gave one example of a PRL paper that contained a comparison of experimental data with a DMFT calculation.

  2. For a better comparision of the success of DMFT to experiment, you could also show the non-DMFT calculations of the optical conductivity to illustrate how that presumably fails to match experiment.

    1. Ravindra,

      Thanks for making this important point.

      The reason I have no such comparison is that I am unaware of a non-DMFT calculation [besides a brute force "black box" numerical approach] that can even qualitatively reproduce the data. Specifically, one needs to produce a Drude peak at zero frequency AND the broad feature associated with the Hubbard bands, AND have the Drude peak collapse as the temperature is increased. If someone knows such an approximation scheme, please point it out.

      However, I should stress that my main point is NOT that DMFT is great. Rather, it is that DMFT is not like string theory and we have a pretty good idea of situations in which it works well.