Anomalous Coupling Between Topological Defects and Curvature

We investigate a counterintuitive geometric interaction between defects and curvature in thin layers of superfluids, superconductors, and liquid crystals deposited on curved surfaces. Each defect feels a geometric potential whose functional form is determined only by the shape of the surface, but whose sign and strength depend on the transformation properties of the order parameter. For superfluids and superconductors, the strength of this interaction is proportional to the square of the charge and causes all defects to be repelled (attracted) by regions of positive (negative) Gaussian curvature. For liquid crystals in the one elastic constant approximation, charges between 0 and $4\pi$ are attracted by regions of positive curvature while all other charges are repelled.

Figure 1

A corrugated substrate and its downward projection on a flat plane. The shaded strip surrounding $P$ is more stretched than the one surrounding $Q$ despite their projections onto the plane having the same area. The energy stored in the field will be lower if the core of the defect is located at $Q$ rather than $P$.

Figure 2

Disclinations of charge $2\pi$ located on top of bumps with different aspect ratios. The amount of splay in the liquid crystal director on the taller bump is reduced and hence the energy density is lower.

Figure 3

(a) An isolated disclination on a deformed plane feels a force that depends only on the enclosed Gaussian curvature. (b) The deformed plane is compactified to the sphere by placing a shadow defect at the south pole.

Figure 4

Conformal mapping of the target surface $T$ onto the reference space $R$. The continuous disks on both surfaces represent the “physical” cores of constant radius $a$. The dashed lines represent the position-dependent images on $R$ of the defect cores on $T$ with variable radii $a_i$. Note that the energy stored in the annuli comprised by the dashed and continuous lines in $R$ must be added or subtracted to $E_R$ to obtain $E_T$.