Using the Renormalized Energy from Ginzburg-Landau Theory to Compute Solution Landscapes in the Reduced Landau-de Gennes Theory
Abstract
We study nematic liquid crystal phases confined to bounded, two-dimensional (2D) domains, in the reduced Landau-de Gennes (rLdG) theoretical framework. The rLdG framework is equivalent to the celebrated Ginzburg-Landau (GL) theoretical setting, under certain assumptions. In the asymptotic limit of large domains (or low temperatures), we recall the concept of the finite-dimensional renormalized energy in the GL theory [Bethuel et al., GL Vortices, Prog. Nonlinear Differ. Equ. Appl. 13, Birkhäuser, 1994.] and generalize it to polygons with sharp corners. The renormalized energy is fully defined by the triplet, (a,c, d), of topological degrees of the interior defects (d) and the winding number contribution of polygon vertices (c) and interior defect locations (a), subject to sum rules dictated by the imposed Dirichlet boundary condition. We use the critical points of the renormalized energy to construct initial conditions for the rLdG solver, recover known energy-minimizing and non-energy minimizing saddle points of the rLdG energy on a disc, regular hexagon and rectangle along with the discovery of previously unreported rLdG solution branches. The renormalized energy approach is computationally substantially more efficient than a direct PDE-solver in the rLdG setting but misses solution branches with line defects and finite-size effects.
Related
- Global minimality of the degree-one Ginzburg-Landau vortex solution in the unit ball in dimensions n ≥ 2
- A Solution of Problem 2.5 by Brezis on the Planar Ginzburg--Landau Equation
- Comparison between renormalized volume and area in the Weil--Petersson topology
- Dynamical classical-field limit of Bosonic Gibbs states: Renormalized Hartree NLS correlations in 2D and 3D
- From wall structures to closed mirror symmetry. The case of Kℙ2: Renormalized periods over the positive real locus, closed Gromov-Witten invariants from wall functions, and tropical enumeration
BibTeX
@misc{han2026using,
title = {{Using the Renormalized Energy from Ginzburg-Landau Theory to Compute Solution Landscapes in the Reduced Landau-de Gennes Theory}},
author = {Yucen Han and Apala Majumdar},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16083},
eprint = {2610.16083},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16083},
}
RIS
TY - UNPB
TI - Using the Renormalized Energy from Ginzburg-Landau Theory to Compute Solution Landscapes in the Reduced Landau-de Gennes Theory
AU - Yucen Han
AU - Apala Majumdar
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16083
UR - https://arxiv.org/abs/2610.16083
AB - We study nematic liquid crystal phases confined to bounded, two-dimensional (2D) domains, in the reduced Landau-de Gennes (rLdG) theoretical framework. The rLdG framework is equivalent to the celebrated Ginzburg-Landau (GL) theoretical setting, under certain assumptions. In the asymptotic limit of large domains (or low temperatures), we recall the concept of the finite-dimensional renormalized energy in the GL theory [Bethuel et al., GL Vortices, Prog. Nonlinear Differ. Equ. Appl. 13, Birkhäuser, 1994.] and generalize it to polygons with sharp corners. The renormalized energy is fully defined by the triplet, (a,c, d), of topological degrees of the interior defects (d) and the winding number contribution of polygon vertices (c) and interior defect locations (a), subject to sum rules dictated by the imposed Dirichlet boundary condition. We use the critical points of the renormalized energy to construct initial conditions for the rLdG solver, recover known energy-minimizing and non-energy minimizing saddle points of the rLdG energy on a disc, regular hexagon and rectangle along with the discovery of previously unreported rLdG solution branches. The renormalized energy approach is computationally substantially more efficient than a direct PDE-solver in the rLdG setting but misses solution branches with line defects and finite-size effects.
ER -