Geometric transformation and three-dimensional hopping of Hopf solitons

Abstract

Arising in many branches of physics, Hopf solitons are three-dimensional particle-like field distortions with nontrivial topology described by the Hopf map. Despite their recent discovery in colloids and liquid crystals, the requirement of applied fields or confinement for stability impedes their utility in technological applications. Here we demonstrate stable Hopf solitons in a liquid crystal material without these requirements as a result of enhanced stability by tuning anisotropy of parameters that describe energetic costs of different gradient components in the molecular alignment field. Nevertheless, electric fields allow for inter-transformation of Hopf solitons between different geometric embodiments, as well as for their three-dimensional hopping-like dynamics in response to electric pulses. Numerical modelling reproduces both the equilibrium structure and topology-preserving out-of-equilibrium evolution of the soliton during switching and motions. Our findings may enable myriads of solitonic condensed matter phases and active matter systems, as well as their technological applications.

Field-controlled dynamics of skyrmions and monopoles

“…While the analogy of our skyrmions and monopoles to their high-energy physics counterparts such as Dirac monopoles ( 15 ) and normalπ3false(S3false) Skyrme solitons ( 2 ), which in particle-physics models describe subatomic particles with different baryon numbers, is distant, the observed findings closely mimic what were found in chiral magnets ( 5 ) where even its Hamiltonian can have a form similar to that of a chiral LC, so that LCs could potentially serve as model systems and provide useful insights. The diversity of observed geometrically distinct but topologically equivalent torons or monopole-terminating skyrmions points to possible topology-preserving switching between them, in analogy to the recently studied intertransformation between geometrically distinct states of hopfions ( 16 ). Last, we discuss the fundamental implications of monopole control in LC systems and their potential technological applications.…”

“…In a homeotropic cell, when the confined chiral LC has an elastic anisotropy such that K 33 / K 22 ≤ 1, i.e., the energetic penalty of bend deformation is smaller than that of twist deformation, the background n ( r ) can be switched between the helical state and the uniform state by an electric field. For a material with positive dielectric anisotropy, the background is helical at no field and can be unwound to the uniform state when an electric field is applied ( 16 , 24 ). Vertical skyrmions stabilized in a helical background with no field display attractive pairwise interaction and form a close-packed hexagonal lattice when several of them are in close proximity (Fig.…”

“…Numerical modeling of configurations and energetics of skyrmions and monopoles was based on energy minimization of Frank-Oseen free energy that describes energetic penalties of different elastic deformations modes in n(r) supplemented by dielectric and magnetic (for LC ferromagnets) coupling terms (11,16,21). For a chiral LC, the elastic and dielectric terms read as:…”

“…Minimization of free energy can be performed iteratively using standard numerical methods of finite difference discretization in space and forward Euler method in time and has been implemented in commercial software and custom energy-minimization routines (16,17,21,45). Previously relaxed configurations or those obtained from experimental 3PEF-PM images were used as initial conditions for n(r) or m(r).…”

Emergent Magnetic Field and Nonzero Gyrovector of the Toroidal Magnetic Hopfion

“…In his pioneering work, [ 1 ] Faddeev emphasized that the study of hopfions could become relevant in several theoretical and experimental situations. In fact, the field of study of hopfions turned out to be even wider than Faddeev assumed, covering condensed matter, [ 5 ] liquid crystals, [ 6 ] photonics, [ 7 ] optics, [ 8 ] electromagnetism, and gravitation. [ 9 ] Recently, stability of the hopfions was studied numerically in chiral ferromagnets [ 10,11 ] and in confined ferroelectric nanoparticles.…”