Symmetry breaking and restoration in the Ginzburg-Landau model of nematic liquid crystals

In this paper we study qualitative properties of global minimizers of the Ginzburg-Landau energy which describes light-matter interaction in the theory of nematic liquid crystals near the Friedrichs transition. This model is depends on two parameters: $\epsilon>0$ which is small and represents the coherence scale of the system and $a\geq 0$ which represents the intensity of the applied laser light. In particular we are interested in the phenomenon of symmetry breaking as $a$ and $\epsilon$ vary. We show that when $a=0$ the global minimizer is radially symmetric and unique and that its symmetry is instantly broken as $a>0$ and then restored for sufficiently large values of $a$. Symmetry breaking is associated with the presence of a new type of topological defect which we named the shadow vortex. The symmetry breaking scenario is a rigorous confirmation of experimental and numerical results obtained in our earlier work.