# A universal thin film model for Ginzburg-Landau energy with dipolar interaction

## TL;DR

This paper derives a reduced two-dimensional Ginzburg-Landau model for thin films with dipolar interactions, analyzes its asymptotic behavior, and identifies a phase transition threshold between homogeneous and modulated states.

## Contribution

It introduces a universal thin film model incorporating dipolar interactions and analyzes phase transitions using $ ext{Gamma}$-convergence, extending Ginzburg-Landau theory to thin film geometries.

## Key findings

- Reduced 2D model is asymptotically equivalent to 3D model for small thickness.
- Identifies a sharp threshold for transition from homogeneous to modulated states.
- Recover the local 2D Ginzburg-Landau model in the vanishing thickness limit.

## Abstract

We present an analytical treatment of a three-dimensional variational model of a system that exhibits a second-order phase transition in the presence of dipolar interactions. Within the framework of Ginzburg-Landau theory, we concentrate on the case in which the domain occupied by the sample has the shape of a flat thin film and obtain a reduced two-dimensional, non-local variational model that describes the energetics of the system in terms of the order parameter averages across the film thickness. Namely, we show that the reduced two-dimensional model is in a certain sense asymptotically equivalent to the original three-dimensional model for small film thicknesses. Using this asymptotic equivalence, we analyze two different thin film limits for the full three-dimensional model via the methods of $\Gamma$-convergence applied to the reduced two-dimensional model. In the first regime, in which the film thickness vanishes while all other parameters remain fixed, we recover the local two-dimensional Ginzburg-Landau model. On the other hand, when the film thickness vanishes while the sample's lateral dimensions diverge at the right rate, we show that the system exhibits a transition from homogeneous to spatially modulated global energy minimizers. We identify a sharp threshold for this transition.