Good Fibrations: Packing Rules for Diabolic Domains

TL;DR

This paper develops a geometric theory for packing hyperboloid diabolic domains in liquid crystal textures, revealing how Lorentz transformations and phase structures can lower elastic energy and relate to complex topologies.

Contribution

It introduces a novel framework for understanding bend-free liquid crystal textures using hyperboloid domains and Lorentz transformations, connecting geometry, topology, and energy minimization.

Findings

Domains can be related by Lorentz transformations to reduce elastic energy

The textures include phases analogous to blue phases in liquid crystals

The geometry relates to Milnor fibrations and four-dimensional unification

Abstract

We describe a theory of packing hyperboloid 'diabolic' domains in bend-free textures of liquid crystals. The domains sew together continuously, providing a menagerie of bend-free textures akin to the packing of focal conic domains in smectic liquid crystals. We show how distinct domains may be related to each other by Lorentz transformations, and that this process may lower the elastic energy of the system. We discuss a number of phases that may be formed as a result, including splay-twist analogues of blue phases. We also discuss how these diabolic domains may be subject to "superluminal boosts", yielding defects analogous to shocks waves. We explore the geometry of these textures, demonstrating their relation to Milnor fibrations of the Hopf link. Finally, we show how the theory of these domains is unified in four-dimensional space.