Topology of smectic order on compact substrates

Area of Science:

• Condensed matter physics
• Materials science
• Geometric mechanics

Background:

• Smectic liquid crystals exhibit ordered phases.
• Curved substrates introduce geometric complexity to these ordered phases.
• Understanding topological defects is crucial for characterizing material behavior.

Purpose of the Study:

• To develop a differential geometric framework for describing smectic order on curved substrates.
• To investigate the topological properties and low-energy states of smectic order on spherical and toroidal surfaces.
• To explore the dynamics and topological implications of disclinations in smectic systems on spheres.

Main Methods:

• Utilizing differential forms (1-forms) to represent the local phase field of density modulation.
• Employing the exterior derivative to define local dislocation density.
• Describing elastic deformations as superpositions of exact differential forms.
• Analyzing smectic order on torus and sphere geometries.

Main Results:

• A formalism based on differential forms effectively describes smectic order on curved substrates.
• Both toroidal and spherical systems exhibit numerous topologically distinct low-energy states.
• The number of low-energy states scales with the square root of the substrate area.
• Motion of disclinations on a sphere was explored as low-energy excitations.

Conclusions:

• The differential form approach provides a powerful tool for understanding topological defects and low-energy states in curved smectic systems.
• Topological charges play a key role in characterizing the distinct states.
• The scaling law suggests a fundamental relationship between system size and topological complexity.
• Disclination dynamics offer insights into the excitations and topological implications within these systems.