From Knots to Crystals: Machine-Learned Potentials for Self-Assembling Topological Solitons in Liquid Crystals

TL;DR

The work addresses stabilizing knotted solitons in chiral liquid crystals and enabling their large-scale simulation by learning single-site coarse-grained potentials from fine-grained Frank-Oseen calculations. It introduces a symmetry-function-based expansion $\\Phi_{IJ}=\\sum_k w_k G_k$ to capture highly anisotropic, chiral heliknoton interactions in a cholesteric background and validates against FG results and experiments. The authors demonstrate self-assembly of heliknotons into rhombic and stretched kagome crystals, with voltage-controlled reconfiguration and interpolation of interaction potentials across voltages using cubic-spline weights $w_{ki}(U)$, achieving substantial speedups (CG runs orders of magnitude faster than FG). The framework generalizes to other knotted textures, enabling efficient exploration of topological metamatter in soft and hard condensed matter systems.

Abstract

Knotted fields in classical and quantum systems were long recognized for their non-trivial topologies and particle-like behavior, but practical applications have been limited by the difficulty of stabilizing them. Recently, stable knotted solitonic textures--heliknotons--have been discovered in chiral liquid crystals, forming adaptive crystal assemblies via elastic distortion-mediated interactions. We use machine learning to develop single-site coarse-grained potentials that accurately capture these chiral anisotropic interactions, enabling large-scale simulations beyond the reach of fine-grained methods. Our machine-learned potentials reproduce the experimentally observed assemblies and provide an efficient framework for modeling a wide range of topological textures.

Figures (3)

• Figure 1: Cross-sections through the horizontal mid-plane showing (a) the heliknoton's director field $\boldsymbol{n}(\boldsymbol{r})$ and (b) the helical field $\boldsymbol{\chi}(\boldsymbol{r})$. Non-polar $\boldsymbol{n}(\boldsymbol{r})$ admits a smooth vectorization while remaining nonsingular, as depicted using a color scheme based on two-sphere order parameter space of vectorized $\boldsymbol{n}(\boldsymbol{r})$. Linked loops in (a) are preimages of vertical orientations in smoothly vectorized $\boldsymbol{n}(\boldsymbol{r})$, and the light red tube in (b) shows the singular regions in nonpolar $\boldsymbol{\chi}(\boldsymbol{r})$ forming a trefoil knot; a colored sphere with diametrically opposite points identified decodes orientations of $\boldsymbol{\chi}(\boldsymbol{r})$. (c) Schematic illustrating the position and orientation coordinates used in the coarse-grained model of heliknoton-pair interactions. The gray isosurfaces surrounding the preimages and vortex knots highlight localized regions where $\boldsymbol{\chi}(\boldsymbol{r})$ exhibits significant deviations from its uniform far-field background.
• Figure 2: (a--c) Results for LC-1, $d=3p$, $U=2.2$V: (a) Pair interaction potential of heliknotons localized on the horizontal mid-plane of the LC cell, from fine-grained (FG) simulations, shown as a color map. (b) Radial profiles of the FG pair potential (symbols) compared with the coarse-grained (CG) model (lines), along different $\theta = \tan^{-1}(\Delta y/ \Delta x)$. (c) Heliknoton assemblies at successive stages of a Monte Carlo (MC) simulation using the CG model; the black symbols represent heliknoton cores and time is measured in MC sweeps (MCS). The bottom-right panel shows horizontal sections of $\boldsymbol{n}(\boldsymbol{r})$ at different heights in a unit cell from FG simulations; the color spheres depict the $\boldsymbol{n}(\boldsymbol{r})$ orientations. (d--e) Results for LC-2, $d=2p$: (d) Electrostriction of a heliknoton crystal obtained from MC (top) and FG (bottom) simulations, following a voltage change from $U = 3.3$V to $4.2$V. (e) Comparison of three-body energies versus the sum of pairwise energies from FG simulations. (f--i) Results for LC-1, $d=3p$, $U=2.08$V: (f) $z$-dependence of the potential experienced by an individual heliknoton. (g) Three-dimensional heliknoton pair potential shown for separation distances $1.75p$ and $2.25p$, color maps on spherical shells truncated by the sampled vertical range. (h) Horizontal sections of the pair potential at different vertical separations $\Delta z$. (i) Snapshots from MC simulations showing the stability of a stretched kagome crystal (top) and self-assembly into rhombic crystals (bottom); colors represent the $z$-position of the heliknotons. The insets show the two observed rhombic crystals (not unit cells) along with the corresponding energy per heliknoton. (j) The $S$-function pairs that contribute to the chirality of the CG model, visualized color maps on the unit sphere defined by $\hat{\boldsymbol{r}}_{IJ}$.
• Figure 3: Results for LC-2, $d=2p$: (a) Two-dimensional heliknoton pair potentials at voltages $U=4$V and $4.5$V shown as color maps. (b) Heliknoton assemblies from MC simulations using the CG models, as the voltage is cycled from $4.5$V to $4$V and back. (c) Interpolated heliknoton pair potentials (dotted dashed lines) compared with those from FG simulations (symbols) at different applied voltages. (d) Parity plots comparing the interpolated potentials ($\Phi$) versus the ground truth from the FG simulations ($\Delta F$) along with the RMS errors.