Topology of Smectic Order on Compact Substrates

TL;DR

This paper develops a differential forms framework to analyze smectic order on curved surfaces like spheres and tori, revealing multiple topologically distinct low-energy states characterized by integer charges, with implications for defect dynamics.

Contribution

It introduces a differential forms formalism to classify smectic orders on curved substrates and uncovers the topological structure of low-energy states and defect behavior.

Findings

Multiple topologically distinct low-energy states characterized by two integer charges.

Number of low-energy states scales with the square root of substrate area.

Disclination dynamics on a sphere have topological significance.

Abstract

Smectic orders on curved substrates can be described by differential forms of rank one (1-forms), whose geometric meaning is the differential of the local phase field of density modulation. The exterior derivative of 1-form is the local dislocation density. Elastic deformations are described by superposition of exact differential forms. Applying this formalism to study smectic order on torus as well as on sphere, we find that both systems exhibit many topologically distinct low energy states, that can be characterized by two integer topological charges. The total number of low energy states scales as the square root of the substrate area. For smectic on a sphere, we also explore the motion of disclinations as possible low energy excitations, as well as its topological implications.