# Matrix models and condensed matter physics

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.

• I assume you mean apart from Random Matrix Theory, which is a kind of 0 dimensional matrix model. Aug 28 '11 at 3:00

Matrix models have been used to study RNA folding. For example, see: G. Vernizzi, H. Orland, and A. Zee, "Enumeration of RNA Structures by Matrix Models", Phys. Rev. Lett. 94, 168103 (2005), arXiv:q-bio/0411004.

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:

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

• S. Hellerman and L. Susskind, Realizing the Quantum Hall System in String Theory, arXiv: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.

• Hey Lubos, nice to see you here, I can see you are determined to answer all questions on the site :-). Yes, I know about these, my question was a bit vague, I was mainly thinking about planar limit of some matrix model for small 'tHooft coupling being related to specific phenomena. In other words, I was probing to see if a different community has done something I should be aware of, and if members of such community frequent this site.
– user566
Jan 14 '11 at 0:00