In september 2008 I received my Masters degree in Theoretical Physics cum laude, with the research I did under supervision of Koenraad Schalm at the Leiden University. My Masters thesis was titled “Phase Transitions in Matrix Models” and can be downloaded here (pdf, 716 kB). The summary of the thesis is:

Matrix models are toy models applicable in various ļ¬elds of physics. The overall propertiesĀ of such a matrix model are deļ¬ned by its partition function, which is an integral over $latex N \times N$ Hermitian matrices *M* with energy/action *S[M]* invariant under similarity transformations.

Upon integrating over the rotational degrees of freedom, the action can be described in termsĀ of the eigenvalues of *M*.Ā If the action has one unique absolute minimum, then the free energy $latex F = – \log Z_N$ canĀ be approximated via a perturbation series around that minimum. Generically, however, theĀ matrix model action will have multiple extrema, e.g. in the $latex gM^4$ model. Using the eigenvalueĀ representation, we show that the $latex gM^4$ model exhibits a phase transition for a speciļ¬c range ofĀ coupling constants. For high $latex \mu_C = m^2/4g$ (the depth of the potential well) the ground stateĀ consists of a superposition of multiple solitons. For low $latex \mu_C$ there exists one single minimumĀ of the action, which allows a perturbation expansion of the free energy.

We ļ¬nd that the phase transition of the $latex gM^4$ model is analytic in the macroscopic parameters, but is non-analytic when the action is coupled to external sources for eigenvalues.Ā This can be veriļ¬ed by computing the correlation function in both phases. Physically theĀ source term preselects one speciļ¬c set of microscopic variables. The non-analyticity in thisĀ microscopic parameter while analytic in all macroscopic parameters suggests that we areĀ dealing with a Kosterlitz-Thouless phase transition.

Finally we construct a renormalization group ļ¬ow of the theory with respect to changesĀ in the matrix dimension *N* and show that the lines of constant *Z* (aka the renormalizationĀ group ļ¬ows) do cross the line of critical Āµin the phase diagram.