Question

I am sending a couple of questions which seem a bit more specific than others on this site, partially to probe if there is a point in doing so. Not sure what is the range of expertise here, and no way to find out without trying. This one is also not terribly focused, but nonetheless here goes:

I am wondering if there are some well-known and well-studied examples of large N matrix models (in which the fields are adjoint rather than vectors) which are of use in describing some condensed matter phenomena.

There are lots of applications of matrix models in anything between nuclear physics to number theory, and there are well-known vector models which are useful in CM physics, but off the top of my head I cannot think about matrix models which are used to solve some condensed matter problems. Quite possibly I am missing something obvious...References or brief descriptions will be appreciated.

Answer 1

Susskind and Polychronakos - and, independently, Hellerman and Van Raamsdonk - have constructed a matrix model - based on string-theoretical D0-branes, which is the right fundamental way to think about any adjoint matrix model - to describe the fractional quantum Hall effect. A review is here:

http://iopscience.iop.org/1751-8121/42/30/304006

See also a Hellerman-Susskind realization of the quantum Hall effect in string theory

http://arxiv.org/abs/hep-th/0107200

Moreover, there has been a whole new industry started by the 1997 Maldacena's AdS/CFT correspondence. On the CFT side, one most typically has matrix-valued field theories, even though they're usually not just quantum-mechanical models but field theories in additional dimensions. Chern-Simons matrix models also have played a role.

AdS/something dualities have been successful to one extent or another in describing perfect fluids, Fermi liquids, non-Fermi liquids, superconductors, hydrodynamics, heavy ion physics, and other things. Phenomena are usually translated to the behavior of black holes in a higher-dimensional spacetime.