Defect Interactions Through Periodic Boundaries in Two-Dimensional $p$-atics

TL;DR

This paper derives an analytical model for defect interactions in two-dimensional $p$-atic liquid crystals with periodic boundaries, revealing anomalous defect interactions mediated by topological solitons, emphasizing the role of domain topology.

Contribution

It provides the first analytical expression for defect interactions in 2D $p$-atics with periodic boundaries, highlighting the influence of domain topology on defect behavior.

Findings

Defect interactions deviate from Coulomb law due to periodic boundaries.

Topological solitons mediate defect interactions in the system.

Periodic boundary conditions stabilize non-singular topological solitons.

Abstract

Periodic boundary conditions are a common theoretical and computational tool used to emulate effectively infinite domains. However, two-dimensional periodic domains are topologically distinct from the infinite plane, eliciting the question: How do periodic boundaries affect systems with topological properties themselves? In this work, I derive an analytical expression for the orientation fields of two-dimensional $p$-atic liquid crystals, systems with $p$-fold rotational symmetry, with topological defects in a flat domain subject to periodic boundary conditions. I show that this orientation field leads to an anomalous interaction between defects that deviates from the usual Coulomb interaction, which is confirmed through continuum simulations of nematic liquid crystals ($p = 2$). The interaction is understood as being mediated by non-singular topological solitons in the director field which are stabilized by the periodic boundary conditions. The results show the importance of considering domain topology, not only geometry, when analyzing interactions between topological defects.