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

Cody D. Schimming

TL;DR

This work tackles how periodic boundary conditions modify topological defect interactions in two-dimensional $p$-atics. It develops a periodic orientation field using Weierstrass functions, showing that the naive Coulomb-like interaction from the infinite plane does not survive under PBC due to topological constraints, and introduces topological solitons as mediators of an asymmetric defect force. The analysis introduces global windings $n_1$ and $n_2$ and demonstrates, with continuum simulations at $p=2$, that defects preferentially move along paths dictated by soliton distribution rather than by symmetric Coulomb forces. The results highlight the essential role of domain topology in defect dynamics and offer guidance for interpreting simulations and experiments in annular or toroidal geometries where curvature and windings interplay with defect interactions.

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.

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

TL;DR

This work tackles how periodic boundary conditions modify topological defect interactions in two-dimensional -atics. It develops a periodic orientation field using Weierstrass functions, showing that the naive Coulomb-like interaction from the infinite plane does not survive under PBC due to topological constraints, and introduces topological solitons as mediators of an asymmetric defect force. The analysis introduces global windings and and demonstrates, with continuum simulations at , that defects preferentially move along paths dictated by soliton distribution rather than by symmetric Coulomb forces. The results highlight the essential role of domain topology in defect dynamics and offer guidance for interpreting simulations and experiments in annular or toroidal geometries where curvature and windings interplay with defect interactions.

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 -atic liquid crystals, systems with -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 (). 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.

Paper Structure

This paper contains 13 sections, 41 equations, 6 figures.

Table of Contents

  1. Introduction
  2. $p$-atic topological defects in the plane
  3. Periodic boundary conditions
  4. Failure of the electrostatic analogy
  5. Periodic configuration and interaction between defects
  6. Topological Solitons
  7. Global Solitons
  8. Physically Realizable Domains
  9. Conclusion
  10. Halperin-Mazenko formalism for $p$-atic defect velocities
  11. The Coulomb interaction and Weierstrass functions
  12. Real-valued orientation configuration and defect velocity
  13. Computational details

Figures (6)

  • Figure 1: Schematic of the periodic system. Positive defects are located at $z^+$, negative defects at $z^-$, and a test defect at $z$. $\omega_1$ and $\omega_2$ give the principal directions of the periodic lattice.
  • Figure 2: Orientation fields compatible with defects interacting as Coulomb charges, Eq. \ref{['eqn:CoulombOrderParam']}, for a system with $\omega_1 = 0.5$, $\omega_2 = 0.5i$, $z^+ = 0.25+0.25i$, $z^- = -0.25 - 0.25i$ and (a) $p=1$ or (b) $p=2$. In this case, periodic boundary conditions are not satisfied.
  • Figure 3: Orientation fields corresponding to Eq. \ref{['eqn:ThetaFinalAnswer']} for the same system of defects as in Figs. \ref{['fig:NonPeriodicConfiguration']}(a,b)
  • Figure 4: (a,b) Complex color plots of the velocity of the positive $p$-atic defect, Eq. \ref{['eqn:DefectVel']} and force on the positive Coulomb charge, Eq. \ref{['eqn:CoulombVel']} for a configuration with two defects or charges in a square domain with $\omega_1 = 0.5$ and $\omega_2 = 0.5 i$ as a function of defect separation $\Delta z = \Delta x + i\Delta y$. The color represents the argument of the function while the brightness indicates the magnitude. The black line indicates the subset of the plot that is shown in (e). (c,d) Corresponding vector plots of Eqs. \ref{['eqn:DefectVel']} and \ref{['eqn:CoulombVel']} where the arrows indicate the argument of the function while the color indicates the magnitude. (e) Velocity of the positive defect for the case $\Delta y = 0$ (green solid line). The purple dashed line shows the force on the positive Coulomb charge in the same configuration. The black dot indicates the initial separation for the simulation shown in (f). (f) Time snapshots of a simulation of two annihilating defects in a nematic liquid crystal ($p = 2$) with initial separation $\Delta x = 0.81$. The color is the scalar order parameter $S$ and the white lines are the nematic director $\mathbf{\hat{n}}$
  • Figure 5: (a) Defects in the $XY$-model subject to periodic boundary conditions. The curve between the defects measures a charge [Eq. \ref{['eqn:DefectCharge']}] $k=-1$, indicating a topological excitation. If the curve is moved past either defect, it measures $k=0$, as shown by the curve to the right of the positive defect. (b) Time snapshots of a nematic liquid crystal simulation with defects initially separated by $\Delta x =0.2$ and $n_2 = 1$. Color is the scalar order parameter $S$ while the white lines are the orientation field. (c) Time snapshots of a simulation similar to (b) but with initial separation $\Delta x = 0.1$.
  • ...and 1 more figures