Umbilic Lines in Orientational Order

Three-dimensional orientational order in systems whose ground states possess nonzero gradients typically exhibits linelike structures or defects: $\lambda$ lines in cholesterics or Skyrmion tubes in ferromagnets, for example. Here, we show that such lines can be identified as a set of natural geometric singularities in a unit vector field, the generalization of the umbilic points of a surface. We characterize these lines in terms of the natural vector bundles that the order defines and show that they give a way to localize and identify Skyrmion distortions in chiral materials—in particular, that they supply a natural representative of the Poincaré dual of the cocycle describing the topology. Their global structure leads to the definition of a self-linking number and helicity integral which relates the linking of umbilic lines to the Hopf invariant of the texture.

Popular Summary

Ordered materials—including magnets, superfluids, Bose condensates, and liquid crystals—display complex three-dimensional structures that are central to modern developments in novel metamaterials, fluid photonics, and ultralow-current spintronics. A common theme in these structures is their geometrical and topological properties, an understanding of which is key to unlocking potential applications. Here, we provide a thorough geometric analysis of orientational order and show how it identifies the topology in complex three-dimensional fields.

Materials described by orientational order contain linelike geometrical features such as cores of vortices, lambda lines in cholesterics, and Skyrmions in chiral magnets. We show that such lines can be identified as a set of natural geometric singularities in a unit vector field, the generalization of the umbilic points of a surface, and that they encode the topology of the order. We illustrate the relationship between geometry and topology for a series of examples from modern experiments on Skyrmions and solitons in liquid crystals and chiral ferromagnets. In three dimensions, umbilic lines may become twisted and entangled, and these geometric singularities can form loops whose linking conveys more subtle topological information about the order, which we relate to helicity and Chern-Simons theory. Our work provides a precise geometric and topological characterization of orientational order, clarifies the structures seen in recent experiments, and presents a set of tools to develop and understand ordered media.

We expect that our theoretical framework will motivate new experiments exploring the linking and twisting of umbilic loops.

Figure 1

Umbilic lines. (a) ${\lambda }^{+}$ ${\lambda }^{-}$ dislocation in a cholesteric. It contains two generic umbilics. This combination of two generic umbilics can be identified as half a Skyrmion, or a meron. (b) Eigenvector field of $\mathrm{\Pi }$ for the dislocation. The two umbilics have opposite windings of the eigenvectors, but are counted with the same sign as $\mathbf{n}$ rotates by $\pi$ when passing from one to the other. (c) Umbilic lines can be oriented in three dimensions. (d),(e) Typical centers of vortices, based on curvature and torsion distortions, respectively, that correspond to winding number two nongeneric umbilics.

Figure 2

Skyrmion lattice. Each unit cell possesses a single central $s=2$ umbilic, with the six outer $s=1$ umbilics each shared between three unit cells giving a total count of $2+6\times 1/3=4$ umbilics $=1$ Skyrmion per unit cell. Top left: Unit vector field $\mathbf{n}$ in a Skyrmion lattice. Top right: Contour plot of $|\mathrm{\Delta }|$ showing the umbilic lines forming a lattice of tubes. Bottom right: Eigenvector field of $\mathrm{\Pi }$, illustrating the $+1$ winding around the central umbilics and $-1/2$ winding around the outer umbilics. Note that both are counted with the same sign as the orientation of $\mathbf{n}$ changes from one to another. Bottom left: Umbilics form lines in three dimensions.

Figure 3

Transient monopoles and umbilic lines observed in a simulated quench of the free energy [Eq. (1)]. Left: Contours of constant value of $|\mathrm{\Delta }|$. The blue lines are umbilics, the beige spheres are monopoles (point defects). Each umbilic has winding $n=1$. They are all oriented similarly, so that the sum of the umbilics entering (leaving) the negative (positive) point defect is 4, in accordance with Eq. (20) and corresponding to the Skyrmion created between the monopoles. Top right: Cross section showing $\mathbf{n}$ on the shaded plane. As a map from the plane to $S^2$, it has degree 1, indicating the presence of a Skyrmion. Bottom right: Contour plot showing the magnitude of $\log (|\mathrm{\Delta }|)$ on the shaded plane; the umbilics are clearly picked out.

Figure 4

Umbilic lines in a toron. (a) Simulation results showing point defects and characteristic umbilic loop in a toron. The umbilic loop is transverse and has zero self-linking number. The point defects have opposite charge; due to lattice effects the simulation does not resolve the umbilic lines connecting them, though they are clear in the topology of the eigenvector field. (b) Diagram illustrating the umbilic structure of the toron in $S^3$ when the boundary of the cell is collapsed to a point. Note that the umbilic loops connecting the point defects each have strength $s=2$, for a total of 4, as required by Eq. (20). (c) Simulation results showing director field for the toron. (d) Contour plot showing the value of $|\mathrm{\Delta }|$. (e) Eigenvector field of $\mathrm{\Pi }$. Note that the umbilic loop is generic and has a monstar profile.

Figure 5

Umbilic lines in the blue phases. The umbilics are indicated by blue isosurfaces, while the network of disclination lines is shown by milk chocolate isosurfaces. The textures BPI and BPII are the cubic structures observed experimentally. $\mathrm{O}{8}^{+}$ is a cubic structure with the same space group as BPI but an interchange between disclination lines and umbilics. BPX is a tetragonal texture observed in an applied electric field. The arrangement of disclinations and umbilics is the same as in ${\mathrm{O}}^{8+}$, although the symmetry is different.

Figure 6

The splitting $T{\mathbb{R}}^3\cong L_{\mathbf{n}}\oplus \xi$ induced by orientational order. Left: A simple helical director profile, corresponding to the ground state of a cholesteric liquid crystal or helimagnet. Right: An idealized, axisymmetric Skyrmion profile, the central portion of which is a standard idealized model of double twist cylinders in blue phases.

Figure 7

Anatomy of an umbilic line. Top: The vector $\mathbf{\nabla }\times \mathit{B}$ is tangent to generic umbilic lines, giving them a natural orientation. This orientation is compatible with that induced by $A$. Middle: The vector field $\mathbf{n}$ and associated orthogonal plane $\xi$. Note that, in general, the tangent vector of the umbilic does not align with $\mathbf{n}$. $\xi$ is oriented by $\mathbf{n}$ and so gives a natural cycle along which to measure the winding. For a generic umbilic, if $\mathbf{n}·\mathbf{\nabla }\times \mathit{B}>0$, then the eigenvectors of $\mathrm{\Delta }$ have a $+1/2$ profile, and if $\mathbf{n}·\mathbf{\nabla }\times \mathit{B}<0$, the eigenvectors have a $-1/2$ profile. Bottom: A measuring cycle $\gamma$ bounding a surface $\mathrm{\Sigma }$. Whether $\gamma$ measures the winding of the umbilic as positive or negative depends on whether the umbilic—oriented by $\mathbf{\nabla }\times \mathit{B}$—intersects $\mathrm{\Sigma }$ positively or negatively.

Figure 8

The self-linking number for transverse umbilic loops. The red curve $U^{\prime }$ is formed by pushing $U$ along an eigenvector field ${\mathbf{p}}_{\mathrm{\Delta }}$ of $\mathrm{\Delta }$. The self-linking number is then computed as the linking number of $U$ with $U^{\prime }$. The umbilic must be transverse to ensure that $U^{\prime }$ does not cross $U$. Because the eigenvectors are degenerate on the umbilic, one evaluates ${\mathbf{p}}_{\mathrm{\Delta }}$ on a line close to $U$, $\widehat{U}$, that has zero self-linking number with $U$. Such a line can be found by drawing an orientable surface whose boundary is $U$ and pushing $U$ into that surface. Note that, because eigenvectors are line fields, the self-linking number can be a half-integer, corresponding to $\mathbf{p}$ being nonorientable along the longitude of $U$.

Figure 9

Umbilic lines in the Hopf fibration. Top: Umbilic lines given as isosurfaces of $|\mathrm{\Delta }|$ for a simulated Hopf fibration. The two lines in the midplane of the cell are both generic, with the same orientation. For symmetry reasons the central umbilic is not generic and has $s=2$. On the boundary of the cell $\mathbf{n}$ is uniform, which means that the Hopf invariant is well defined. Note that the system is totally umbilic on the boundary. If you compactify the domain by collapsing the boundary to a point, then one can interpret the central vertical umbilic line as passing through this point. Bottom left: Director field for the Hopf texture. Bottom right: Schematic of the relationship between umbilic lines in $S^3$. Two generic $s=1$ umbilic lines each link with one $s=2$ umbilic line that passes through the point at infinity.