Symmetry Breaking and Restoration in the Ginzburg–Landau Model of Nematic Liquid Crystals

Abstract

Abstract. 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: > 0 which is small and represents the coherence scale of the system and a ≥ 0 which represents the intensity of the applied laser light. In particular we are interested in the phenomenon of symmetry breaking as a and 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 earlier in [7].

The connecting solution of the Painlevé phase transition model

“…where a ≥ 0 is a parameter and f = − 1 2 µ ′ , and in [13] we studied its analog for maps u : R 2 → R 2 . By proceeding as in [13], one can see that under the above assumptions there exists a global minimizer v…”

“…Theorem 2.1 gives account on how non smoothness of the limit of v ǫ is mediated near the circumference |x| = ρ, where µ changes sign, through the solution of (2.6). We should mention here that detailed description of the minimizers for yet more general setting of the energy can be found in [13,14].…”

Gradient theory of domain walls in thin, nematic liquid crystals films

“…Proceeding as in [5], one can see that under the above assumptions there exists a global minimizer v of E in H 1 (R 2 ), namely that E(v) = min H 1 (R 2 ) E. In addition, we show that v is a classical solution of (1.5), and v is even with respect to x 2 i.e. v(x 1 , x 2 ) = v(x 1 , −x 2 ).…”

“…In general we will assume that ǫ > 0 is small and a ≥ 0 is fixed. In our previous work [4] and [5], we examined respectively the cases of minimizers v : R → R, and v : R 2 → R 2 . In the present paper we follow the approach presented therein, and introduce several new ideas to address the specific issues occuring for minimizers v : R 2 → R. In particular, new variational arguments to determine the limit points of the zero level set of v, which is now a curve (cf.…”

“…Physically this means that as the intensity of the illumination, measured by a, is relatively small then no defect is visibly seen. For this reason and by analogy with [4,5] we call it the shadow domain wall. As a increases the shadow domain wall penetrates the bistable region becoming the standard domain wall, as described in (iii).…”