Theory of hybrid defects, with coupled orientational order parameters, on flat and curved surfaces

Abstract: Many physical systems involve two types of orientational order, which are coupled together. For example, ferroelectric nematic liquid crystals have coupled polar and nematic order, and tilted hexatic phases have coupled polar and hexatic order. In these systems, defect structures can be quite complex. Here, we investigate phases with two types of two-dimensional orientational order, $m$-atic and $n$-atic, where $m$ and $n$ are two distinct integers. We simulate these phases in a flat disk with strong radial anchoring, and on a spherical surface, because both of these geometries require the presence of defects. If the coupling between the two types of order is weak, then the defects are connected by a network of diffuse walls, and the system forms a stable domain structure. As the coupling increases, the domain walls become sharper and shorter. For very strong coupling, the higher-order defects merge into the lower-order defects, forming stretched defect cores.

• The paper develops a general classification of hybrid defects in coupled m-atic and n-atic systems, showing that divisible orders form clusters of n/m fractional defects connected by domain walls.
• Simulations on disks and spheres recover polar–nematic strings, nematic–tetratic strings, and polar–hexatic stars, while an analytic model predicts shorter walls as coupling increases and defect cores merge.
• For non-divisible orders such as tetratic–hexatic coupling, the greatest-common-divisor rule predicts fractional clusters, but simulations often produce extended networks whose ground-state status remains unresolved.

Overview

This paper develops a general theory and numerical study of "hybrid defects"—topological structures that combine point defects of one orientational order parameter with domain walls of another—on two geometries that force defect formation: a flat disk with strong radial anchoring (requiring total topological charge +1) and a spherical surface (requiring total charge +2). The authors, Paik and Selinger, generalize the classic string defects of Lee and Grinstein [Lee1985] and star defects of Dierker, Pindak, and Meyer [Dierker1986] to arbitrary pairs of $m$-atic and $n$-atic order parameters, motivated by ferroelectric nematic liquid crystals (coupled polar and nematic order), tilted hexatic films (polar tilt plus hexatic bond order), colloidal crystals on spheres, active matter, and epithelial tissue packing.

The central result is a classification rule: when $n$ is a multiple of $m$, hybrid defects consist of $n/m$ point defects of charge $+1/n$ bound by domain walls into clusters of total charge $+1/m$; as coupling strengthens, walls shorten and eventually the higher-order defects merge into stretched cores of the lower-order defect. When $n$ is not a multiple of $m$, the expected cluster charge is $+1/l$ where $n$0, but simulations show persistent global networks rather than isolated clusters.

Model and methods

For a single $n$1-atic order on the disk, the authors use an XY-type lattice Hamiltonian

$n$2

on a triangular finite-element mesh generated with pygmsh/Gmsh ($n$3, mesh size $n$4), with radial anchoring enforced at edge sites. Simulated annealing via Metropolis Monte Carlo, with trial moves including large rotations of $n$5, reliably finds ground states containing exactly $n$6 point defects of charge $n$7.

On the sphere, the authors avoid coordinate singularities at the poles by working entirely in 3D Cartesian vectors $n$8, constrained to the local tangent plane through a penalty term $n$9. The $n$0-atic interaction is written as polynomials in $n$1 for $n$2. Meshes are generated with trimesh's icosphere subdivision. Visualization uses a Mercator projection (sphere → cylinder → plane), which reveals all defects at once at the cost of distortion near the poles—a caveat the authors acknowledge.

Baseline results confirm expectations: the disk hosts $n$3 defects of charge $n$4; the sphere hosts $n$5 such defects totaling $n$6.

Coupled polar–nematic order

With both polar ($n$7) and nematic ($n$8) couplings, the disk and sphere reproduce string defects: pairs of $n$9 nematic point defects connected by a domain wall across which polar order reverses while nematic order is unchanged. Each string carries total charge $m$0; the sphere holds two strings.

The paper goes beyond qualitative description with an analytic estimate of the equilibrium string length. Balancing elastic repulsion between the two half-charges against domain-wall line tension gives

$m$1

with effective line tension $m$2 and stiffness $m$3. This predicts divergence of $m$4 as $m$5 (free nematic defects) and monotonic shrinkage with increasing $m$6, in agreement with the simulations for $m$7 from 0.1 to 0.75. At the strongest coupling simulated, each pair merges into a single polar defect with an extended core.

Nematic–tetratic and polar–hexatic cases

The nematic–tetratic case is mathematically equivalent to polar–nematic under $m$8, so string defects of total charge $m$9 appear: pairs of $n/m$0 tetratic defects joined by walls where nematic order reverses (visualized via black/blue arrow decoration). On the sphere, four such strings account for the required charge $n/m$1. The authors draw a connection to recent hard-particle packing simulations of rounded tetrahedra on spheres [Jones2025], speculating that the observed woven motif and extended defect networks correspond to strong tetratic order with weak nematic order induced by curvature—an interpretation offered explicitly as speculation, not a derivation.

The polar–hexatic case recovers the experimentally known star defects of tilted hexatic films: a central $n/m$2 hexatic defect that is simultaneously a $n/m$3 polar defect, with five arms (domain walls across which polar order rotates by $n/m$4) terminating in five peripheral $n/m$5 hexatic defects. One star fills the disk; two fill the sphere. The simulated stars are less symmetric than the experimental ones, with kinked arms attributed to mesh artifacts—a limitation the authors state plainly.

Tetratic–hexatic order: the non-divisible case

The tetratic–hexatic combination ($n/m$6, $n/m$7) is qualitatively different because $n/m$8 is not a multiple of $n/m$9. Since $+1/n$0, the authors predict hybrid defects of charge $+1/n$1, each combining two $+1/n$2 tetratic defects with three $+1/n$3 hexatic defects. The interaction potential itself changes character with the ratio $+1/n$4: minima shift continuously between four-fold and six-fold patterns, with only the minima at relative angles 0 and $+1/n$5 fixed.

Simulations partially confirm this picture. In strongly hexatic or strongly tetratic regimes on the disk, the system does organize into two hybrid defects of charge $+1/n$6 each. However, on the sphere—and at comparable coupling strengths on both geometries—the defects and domain walls form networks spanning the entire system rather than isolated clusters. The authors concede that isolated hybrid defects may still be the true ground state but that their annealing algorithm cannot reach it, hypothesizing a rugged energy landscape with many near-degenerate configurations. This is the paper's most significant unresolved point: the clean classification by greatest common divisor is validated only in limited regimes.

Limitations and open questions

Several limitations bear directly on the generality of the conclusions. First, the model contains no electrostatics, elasticity anisotropy, or substrate effects, so quantitative comparison with ferroelectric nematic experiments remains indirect. Second, the interpretation of the tetrahedral-particle colloidal simulations as weak-nematic/strong-tetratic order is speculative and untested quantitatively. Third, the tetratic–hexatic network states may be metastable artifacts of annealing; whether isolated $+1/n$7 hybrid defects are genuine ground states, and what controls the crossover, is left open. Fourth, mesh-induced kinks in domain walls suggest that continuum-level predictions of wall shapes require finer meshes or adaptive methods. Finally, the analytic string-length formula assumes isotropic elasticity and a simple line tension; its accuracy beyond the parameter ranges tested here is not established.

Conclusion

This work unifies string and star defects as instances of a general class of hybrid defects in systems with coupled $+1/n$8-atic and $+1/n$9-atic order parameters, and provides a predictive structural rule based on the ratio $+1/m$0 and, in the non-divisible case, the greatest common divisor. Simulations on disks and spheres confirm the rule when $+1/m$1 is a multiple of $+1/m$2, including the predicted shrinkage and eventual core-stretching of domain walls with increasing coupling. The non-divisible case exposes the limits of both the classification and the simulation methodology, leaving open whether global defect networks are thermodynamic ground states or kinetic traps.