## Vici project “Opposites attract” starts

Today I start officially as PhD-student in Theoretical Physics under the supervision of Hans Hilgenkamp, Jan Zaanen and Jeroen van den Brink. The latter two are professors in theoretical physics at the Lorentz Institute, Leiden University. Professor Hans Hilgenkamp, from Twente University, is the main supervisor since he got a Vici-grant for our research.

The Vici is granted to Hilgenkamps’ proposal which is titled “Opposites attract; Electron-hole dances in coupled p- and n-type Mott-conductors“. The goal of the research is to realize and investigate new states of matter by coupling p-type and n-type Mott materials.

## Masters thesis: Phase Transitions in Matrix Models

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 ﬁelds of physics. The overall properties of such a matrix model are deﬁned 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 terms of the eigenvalues of M. If the action has one unique absolute minimum, then the free energy $latex F = – \log Z_N$ can be approximated via a perturbation series around that minimum. Generically, however, the matrix model action will have multiple extrema, e.g. in the $latex gM^4$ model. Using the eigenvalue representation, we show that the $latex gM^4$ model exhibits a phase transition for a speciﬁc range of coupling constants. For high $latex \mu_C = m^2/4g$ (the depth of the potential well) the ground state consists of a superposition of multiple solitons. For low $latex \mu_C$ there exists one single minimum of the action, which allows a perturbation expansion of the free energy.

We ﬁnd 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. This can be veriﬁed by computing the correlation function in both phases. Physically the source term preselects one speciﬁc set of microscopic variables. The non-analyticity in this microscopic parameter while analytic in all macroscopic parameters suggests that we are dealing with a Kosterlitz-Thouless phase transition.

Finally we construct a renormalization group ﬂow of the theory with respect to changes in the matrix dimension N and show that the lines of constant Z (aka the renormalization group ﬂows) do cross the line of critical µin the phase diagram.