Two-dimensional skyrmions and other solitonic structures in confinement-frustrated chiral nematics

Particle-based model of liquid crystal skyrmion dynamics

Motivated by recent experimental results that reveal rich collective dynamics of thousands-to-millions of active liquid crystal skyrmions, we have developed a coarse-grained, particle-based model of the dynamics of skyrmions in a dilute regime. The basic physical mechanism of skyrmion motion is related to squirming undulations of domains with high director twist within the skyrmion cores when the electric field is turned on and off. The motion is not related to mass flow and is caused only by the reorientation dynamics of the director field. Based on the results of the "fine-grained" Frank-Oseen continuum model, we have mapped these squirming director distortions onto an effective force that acts asymmetrically upon switching the electrical field on or off. The resulting model correctly reproduces the skyrmion dynamics, including velocity reversal as a function of the frequency of a pulse width modulated driving voltage. We have also obtained approximate analytical expressions for the phenomenological model parameters encoding their dependence upon the cholesteric pitch and the strength of the electric field. This has been achieved by fitting coarse-grained skyrmion trajectories to those determined in the framework of the Frank-Oseen model.

Temperature reconfigurable skyrmionic solitons in cholesteric liquid crystals

In this work, a reversible transformation between torons and cholesteric fingers is realized by continuously changing the pitch through temperature variation of the chiral nematic liquid crystal twist inversion system. By decreasing the pitch, the torons act as seeds from which cholesteric fingers gradually grow. By increasing the pitch, the cholesteric fingers gradually shorten and transform back to the initial state. We find that although the morphology of the torons is severely deformed and cannot be distinguished during the heating-cooling loops, the torons are very well topologically protected and can hardly be destroyed.

Seeding and Emergence of Composite Skyrmions in a van der Waals Magnet

Topological charge plays a significant role in a range of physical systems. In particular, observations of real-space topological objects in magnetic materials have been largely limited to skyrmions - states with a unitary topological charge. Recently, more exotic states with varying topology, such as antiskyrmions, merons, or bimerons and 3D states such as skyrmion strings, chiral bobbers, and hopfions, have been experimentally reported. Along these lines, the realization of states with higher-order topology has the potential to open new avenues of research in topological magnetism and its spintronic applications. Here, real-space imaging of such spin textures, including skyrmion, skyrmionium, skyrmion bag, and skyrmion sack states, observed in exfoliated flakes of the van der Waals magnet Fe_3-x GeTe₂ (FGT) is reported. These composite skyrmions may emerge from seeded, loop-like states condensed into the stripe domain structure, demonstrating the possibility to realize spin textures with arbitrary integer topological charge within exfoliated flakes of 2D magnets. The general nature of the formation mechanism motivates the search for composite skyrmion states in both well-known and new magnetic materials, which may yet reveal an even richer spectrum of higher-order topological objects.

Topological steering of light by nematic vortices and analogy to cosmic strings

Liquid crystals are widely known for their technological uses in displays, electro-optics, photonics and nonlinear optics, but these applications typically rely on defining and switching non-topological spatial patterns of the optical axis. Here, we demonstrate how a liquid crystal's optical axis patterns with singular vortex lines can robustly steer beams of light. External stimuli, including an electric field and light itself, allow us to reconfigure these unusual light-matter interactions. Periodic arrays of vortices obtained by photo-patterning enable the vortex-mediated fission of optical solitons, yielding their lightning-like propagation patterns. Predesigned patterns and spatial trajectories of vortex lines in high-birefringence liquid crystals can steer light into closed loops or even knots. Our vortex lattices might find technological uses in beam steering, telecommunications, virtual reality implementations and anticounterfeiting, as well as possibly offering a model system for probing the interaction of light with defects, including the theoretically predicted, imagination-capturing light-steering action of cosmic strings, elusive defects in cosmology.

Optical Imaging and Analytical Design of Localized Topological Structures in Chiral Liquid Crystals

We combine numerical modeling and analytical design techniques to study several of the most common localized topological structures in frustrated chiral nematic liquid crystal cells. An energy minimization procedure is applied to the lattice model to simulate the director field distributions. These distributions are also approximated using the suitably designed analytical ansatz. We present both simulated and approximated results for optical polarizing microscopy textures and different visualizations of director field structure such as distributions of the azimuthal director angle and isolines for the normal component of the director in coordinate planes. The ansatz correctly mimicked the geometry and optical properties of the solitonic structures under consideration.

Continuous generation of topological defects in a passively driven nematic liquid crystal

Synthetic active matter is emerging as the prime route for the realisation of biological mechanisms such as locomotion, active mixing, and self-organisation in soft materials. In particular, passive nematic complex fluids are known to form out-of-equilibrium states with topological defects, but their locomotion, activation and experimental realization has been developed and understood to only a limited extent. Here, we report that the concentration-driven flow of small molecules triggers turbulent flow in the thin film of a nematic liquid crystal that continuously generates pairs of topological defects with an integer topological charge. The diffusion results in the formation of counter-rotating vortex rolls in the liquid crystal, which above a velocity threshold transform into a turbulent flow with continuous generation and annihilation of the defect pairs. The pairs of defects are created by the self-amplifying splay instability between the vortices, until a pair of oppositely charged defects is formed.

It has been known that spontaneous defect formation and annihilation can be triggered by turbulent flows in active nematic liquid crystals. Here, Mur et al. show a complementary mechanism induced by the flow of foreign organic molecules into the liquid crystal following the concentration gradient.

Fabrication of Arrays of Topological Solitons in Patterned Chiral Liquid Crystals for Real-Time Observation of Morphogenesis

Topological solitons have knotted continuous field configurations embedded in a uniform background, and occur in cosmology, biology, and electromagnetism. However, real-time observation of their morphogenesis and dynamics is still challenging because their size and timescale are enormously large or tiny. Liquid crystal (LC) structures are promising candidates for a model-system to study the morphogenesis of topological solitons, enabling direct visualization due to the proper size and timescale. Here, a new way is found to rationalize the real-time observation of the generation and transformation of topological solitons using cholesteric LCs confined in patterned substrates. The experimental demonstration shows the topologically protected structures arise via the transformation of topological defects. Numerical modeling based on minimization of free energy closely reconstructs the experimental findings. The fundamental insights obtained from the direct observations pose new theoretical challenges in understanding the morphogenesis of different types of topological solitons within a broad range of scales.

Light-Induced Structures and Microparticle Transportation in a Free-Surface Frustrated Chiral Nematic Film

Local illumination with a light beam leads to thermo-orientational processes in a frustrated chiral nematic film with a free surface. Light-induced hydrodynamic flow and orientational structure create an adaptive platform for the collection, translation and rotation of suspended spherical microparticles. The demonstrated approach has potential applications in soft robotics, micro-object delivery systems, and other micro- and nanotechnologies.

Programming Solitons in Liquid Crystals Using Surface Chemistry

AC electric fields cause three-dimensional orientational fluctuations (solitons) to form and rapidly propagate in confined films of liquid crystals (LCs), offering the basis of a new class of active soft matter (e.g., for accelerating mixing and transport processes in microscale chemical systems). How surface chemistry impacts the formation and trajectories of solitons, however, is not understood. Here, we show that self-assembled monolayers (SAMs) formed from alkanethiols on gold, which permit precise control over surface chemistry, are electrochemically stable over voltage and frequency windows (<100 V; 1 kHz) that lead to soliton formation in achiral nematic films of 4'-butyl-4-heptyl-bicyclohexyl-4-carbonitrile (CCN-47). By comparing soliton formation in LC films confined by SAMs formed from hexadecanethiol (C₁₆SH) or pentadecanethiol (C₁₅SH), we reveal that the electric field required for soliton formation increases with the LC anchoring energy: surfaces patterned with regions of C₁₆SH and C₁₅SH SAMs thus permit spatially controlled creation and annihilation of solitons necessary to generate a net flux of solitons. We also show that solitons propagate in orthogonal directions when confined by obliquely deposited gold films decorated with SAMs formed from C₁₆SH or C₁₅SH and that the azimuthal direction of propagation of solitons within achiral LC films possessing surface-induced twists is not unique but reflects variation in the spatial location of the solitons across the thickness of the twisted LC film. Finally, discontinuous changes in LC orientation induced by patterned surface anchoring lead to a range of new soliton behaviors including refraction, reflection, and splitting of solitons at the domain boundaries. Overall, our results provide new approaches for the controlled generation and programming of solitons with complex and precise trajectories, principles that inform new designs of chemical soft matter.

Topological structures in chiral media: Effects of confined geometry

We theoretically study orientational structures in chiral magnetics and cholesteric liquid crystal (CLC) nanosystems confined in the slab geometry. Our analysis is based on the model that, in addition to the exchange and the Dzyaloshinskii-Moriya interactions, takes into account the bulk and surface anisotropies. In CLC films, these anisotropies describe the energy of interaction with external magnetic/electric field and the anchoring energy assuming that magnetic/electric anisotropy is negative and the boundary conditions are homeotropic. We have computed the phase diagram and found that the ground state of the film is represented by various delocalized structures depending on the bulk and surface anisotropy parameters, κ^{b} and κ^{s}. These include the z helix and the z cone states, the oblique, and the x helicoids. The minimum energy paths connecting the ground state and metastable helicoids and the energy barriers separating these states are evaluated. We have shown that there is a variety of localized topological structures such as the skyrmion tube, the toron, and the bobber that can be embedded in different ground states including the z cone (conical phase) and tilted fingerprint states. We have also found the structure called the leech that can be viewed as an intermediate state between the toron and the skyrmion tube.

Skyrmion tubes in achiral nematic liquid crystals

We analyze the interaction with uniform external fields of nematic liquid crystals within a recent generalized free energy posited by Virga and falling in the class of quartic functionals in the spatial gradients of the nematic director. We review some known interesting solutions, i.e., uniform heliconical structures, which correspond to the so-called twist-bend nematic phase and we also study the transition between this phase and the standard uniform nematic one. The twist-bend phase is further reproduced by three-dimensional simulations. Moreover, we find liquid crystal configurations, which closely resemble some novel, experimentally detected, structures called Skyrmion tubes. Skyrmion tubes are characterized by a localized cylindrically symmetric pattern surrounded by either twist-bend or uniform nematic phase. We study the equilibrium differential equations and find numerical solutions and analytical approximations.

Recent Progresses on Experimental Investigations of Topological and Dissipative Solitons in Liquid Crystals

Solitons in liquid crystals have received increasing attention due to their importance in fundamental physical science and potential applications in various fields. The study of solitons in liquid crystals has been carried out for over five decades with various kinds of solitons being reported. Recently, a number of new types of solitons have been observed, among which, many of them exhibit intriguing dynamic behaviors. In this paper, we briefly review the recent progresses on experimental investigations of solitons in liquid crystals.

Coarse-grained theory for motion of solitons and skyrmions in liquid crystals

Recent experiments have found that applied electric fields can induce motion of skyrmions in chiral nematic liquid crystals. To understand the magnitude and direction of the induced motion, we develop a coarse-grained approach to describe dynamics of skyrmions, similar to our group's previous work on the dynamics of disclinations. In this approach, we represent a localized excitation in terms of a few macroscopic degrees of freedom, including the position of the excitation and the orientation of the background director. We then derive the Rayleigh dissipation function, and hence the equations of motion, in terms of these macroscopic variables. We demonstrate this theoretical approach for 1D motion of a sine-Gordon soliton, and then extend it to 2D motion of a skyrmion. Our results show that skyrmions move in a direction perpendicular to the induced tilt of the background director. When the applied field is removed, skyrmions move in the opposite direction but not with equal magnitude, and hence the overall motion may be rectified.

Dynamics of flowing 2D skyrmions

We investigate, numerically, the effects of externally imposed material flows on the structure and temporal evolution of liquid crystal (LC) skyrmions. The dynamics of a 2D system of skyrmions is modeled using the Ericksen-Leslie theory, which is based on two coupled equations, one for material flow and the other for the director field. As the time scales of the velocity and director fields differ by several orders of magnitude for realistic values of the system parameters, we have simplified the calculations by assuming that the velocity relaxes instantaneously when compared to the relaxation of the director field. Thus, we have used a finite-differences method known as artificial compressibility with adaptive time step to solve the velocity field and a fourth-order Runge-Kutta method for the director field. We characterized the skyrmion shape or configuration as a function of the time and the average velocity of the flow field. We found that for velocities above a certain threshold, the skyrmions stretch in the direction perpendicular to the flow, by contrast to the regime of weak flows where the skyrmions stretch along the streamlines of the flow field. These two regimes are separated by an abrupt (first-order) dynamical transition, which is robust with respect to e.g., the LC elastic anisotropy. Additionally, we have found how the presence of a second skyrmion affects the evolution of the shape of the skyrmions, by comparing the evolution of pairs of skyrmions to the evolution of a single-skyrmion.

Layering transitions and metastable structures of cholesteric liquid crystals in cylindrical confinement

Our study of cholesteric lyotropic chromonic liquid crystals in cylindrical confinement reveals the topological aspects of cholesteric liquid crystals. The double-twist configurations we observe exhibit discontinuous layering transitions, domain formation, metastability, and chiral point defects as the concentration of chiral dopant is varied. We demonstrate that these distinct layer states can be distinguished by chiral topological invariants. We show that changes in the layer structure give rise to a chiral soliton similar to a toron, comprising a metastable pair of chiral point defects. Through the applicability of the invariants we describe to general systems, our work has broad relevance to the study of chiral materials.

Flow-assisted self-healing of the helical structure in a cholesteric liquid crystal

We employ nonequilibrium molecular dynamics simulations to investigate the structure and dynamics of a cholesteric liquid crystal confined between atomically corrugated solid walls. By choosing walls normal to the helical axis, we can study systems with an arbitrary cholesteric pitch without exposing the cholesteric helix to a spurious stress. We investigate the effects of local heating and flow and their joint effects. A steady-state laminar Poiseuille flow is initiated by means of an external body force. Flow alone (i.e., without local heating) in a direction normal to the helical axis does not affect the cholesteric pitch. If the liquid crystal is heated in a small region, the cholesteric helix becomes unstable and melts locally. However, if local heating and flow are combined, a nontrivial synergistic effect is observed in that the helical structure recuperates the better, the higher the speed of the flow is.

Skyrmion Spin Ice in Liquid Crystals

We propose the first skyrmion spin ice, realized via confined, interacting liquid crystal skyrmions. Skyrmions in a chiral nematic liquid crystal behave as quasiparticles that can be dynamically confined, bound, and created or annihilated individually with ease and precision. We show that these quasiparticles can be employed to realize binary variables that interact to form ice-rule states. Because of their unique versatility, liquid crystal skyrmions can open entirely novel avenues in the field of frustrated systems. More broadly, our findings also demonstrate the viability of liquid crystal skyrmions as elementary degrees of freedom in the design of collective complex behaviors.

Nonuniform localized distortions in generalized elasticity for liquid crystals

We analyze a recent generalized free-energy for liquid crystals posited by Virga and falling in the class of quartic functionals in the spatial gradients of the nematic director. We review some known interesting solutions, i.e., uniform heliconical structures, and we find new liquid crystal configurations, which closely resemble some novel, experimentally detected, structures called Skyrmion tubes. These new configurations are characterized by a localized pattern given by the variation of the conical angle. We study the equilibrium differential equations and find numerical solutions and analytical approximations.

Review: knots and other new topological effects in liquid crystals and colloids

Humankind has been obsessed with knots in religion, culture and daily life for millennia, while physicists like Gauss, Kelvin and Maxwell already involved them in models centuries ago. Nowadays, colloidal particles can be fabricated to have shapes of knots and links with arbitrary complexity. In liquid crystals, closed loops of singular vortex lines can be knotted by using colloidal particles and laser tweezers, as well as by confining nematic fluids into micrometer-sized droplets with complex topology. Knotted and linked colloidal particles induce knots and links of singular defects, which can be interlinked (or not) with colloidal particle knots, revealing the diversity of interactions between topologies of knotted fields and topologically nontrivial surfaces of colloidal objects. Even more diverse knotted structures emerge in nonsingular molecular alignment and magnetization fields in liquid crystals and colloidal ferromagnets. The topological solitons include hopfions, skyrmions, heliknotons, torons and other spatially localized continuous structures, which are classified based on homotopy theory, characterized by integer-valued topological invariants and often contain knotted or linked preimages, nonsingular regions of space corresponding to single points of the order parameter space. A zoo of topological solitons in liquid crystals, colloids and ferromagnets promises new breeds of information displays and a plethora of data storage, electro-optic and photonic applications. Their particle-like collective dynamics echoes coherent motions in active matter, ranging from crowds of people to schools of fish. This review discusses the state of the art in the field, as well as highlights recent developments and open questions in physics of knotted soft matter. We systematically overview knotted field configurations, the allowed transformations between them, their physical stability and how one can use one form of knotted fields to model, create and imprint other forms. The large variety of symmetries accessible to liquid crystals and colloids offer insights into stability, transformation and emergent dynamics of fully nonsingular and singular knotted fields of fundamental and applied importance. The common thread of this review is the ability to experimentally visualize these knots in real space. The review concludes with a discussion of how the studies of knots in liquid crystals and colloids can offer insights into topologically related structures in other branches of physics, with answers to many open questions, as well as how these experimentally observable knots hold a strong potential for providing new inspirations to the mathematical knot theory.

Chiral Liquid Crystal Lenses Confined in Microchannels

It is known that the liquid crystalline smectic-A phase has geometric defects, called focal conic domains, which can be used as gradient-index microlenses. Cholesteric (chiral nematic) phases also have topological defects with a central symmetry and a singularity at their center. We explore a weakly chiral system in which both types of defects can be present in the same material at different temperatures, and with this strategy we create lenses whose focal length is tunable with temperature. We measure the focal length of the tunable lenses, and we investigate the behavior of the defects near the phase transition. We identify the experimental conditions that make the simultaneous presence of the smectic focal conic domains and the circular cholesteric domains possible, such as the concentration of chiral dopant and the rate of heating and cooling. The transformation of focal conic domains into circular cholesteric domains is a new example of memory at the phase transition between smectic-A and nematic liquid crystals.

Hopf Solitons in Helical and Conical Backgrounds of Chiral Magnetic Solids

Three-dimensional topological solitons attract a great deal of interest in fields ranging from particle physics to cosmology, but remain experimentally elusive in solid-state magnets. Here we numerically predict magnetic heliknotons, an embodiment of such nonzero-Hopf-index solitons localized in all spatial dimensions while embedded in a helical or conical background of chiral magnets. We describe conditions under which heliknotons emerge as metastable or ground-state localized nonsingular structures with fascinating knots of magnetization field in widely studied materials. We demonstrate magnetic control of three-dimensional spatial positions of such solitons, as well as show how they interact to form moleculelike clusters and possibly even crystalline phases comprising three-dimensional lattices of such solitons with both orientational and positional order. Finally, we discuss both fundamental importance and potential technological utility of magnetic heliknotons.

Geometry of Bend: Singular Lines and Defects in Twist-Bend Nematics

We describe the geometry of bend distortions in liquid crystals and their fundamental degeneracies, which we call β lines; these represent a new class of linelike topological defect in twist-bend nematics. We present constructions for smecticlike textures containing screw and edge dislocations and also for vortexlike structures of double twist and Skyrmions. We analyze their local geometry and global structure, showing that their intersection with any surface is twice the Skyrmion number. Finally, we demonstrate how arbitrary knots and links can be created and describe them in terms of merons, giving a geometric perspective on the fractionalization of Skyrmions.

Dynamic dissipative solitons in nematics with positive anisotropies

Electric field induced instabilities of nematic molecules are of importance for both fundamental science and practical applications. Complex electro-hydrodynamic (EHD) effects such as electro-convection, fingerprint textures, spatiotemporal chaos, and solitons in nematics have been broadly investigated and generated much attention. In this work, dissipative solitons as a novel EHD phenomenon are realized in nematics with positive anisotropies, presumably for the first time. Unlike the ones reported recently in nematics with negative anisotropies whose formation and dynamics are mainly attributed to the flexoelectric and electro-convection effects, the solitons discussed here arise from the nonlinear coupling between the director field and the isotropic flow induced by ion motion. The structure and dynamics of the solitons are demonstrated and the influences of chirality, azimuthal anchoring and ion concentration are also investigated. Finally, we show that the propagation trajectory of solitons can be manipulated by patterned photoalignment and micro-particles can be trapped by them as vehicles for micro-cargo transport.

Surface anchoring as a control parameter for stabilizing torons, skyrmions, twisted walls, fingers, and their hybrids in chiral nematics

Chiral condensed matter systems, such as liquid crystals and magnets, exhibit a host of spatially localized topological structures that emerge from the medium's tendency to twist and its competition with confinement and field coupling effects. We show that the strength of perpendicular surface boundary conditions can be used to control the structure and topology of solitonic and other localized field configurations. By combining numerical modeling and three-dimensional imaging of the director field, we reveal structural stability diagrams and intertransformation of twisted walls and fingers, torons, and skyrmions and their crystalline organizations upon changing boundary conditions. Our findings provide a recipe for controllably realizing skyrmions, torons, and hybrid solitonic structures possessing features of both of them, which will aid in fundamental explorations and technological uses of such topological solitons. Moreover, we discuss how other material parameters can be used to determine soliton stability and how similar principles can be systematically applied to other liquid crystal solitons and solitons in other material systems.

Commensurate states and pattern switching via liquid crystal skyrmions trapped in a square lattice

Using continuum based simulations we show that a rich variety of skyrmion liquid crystal states can be realized in the presence of a periodic obstacle array. As a function of the number of skyrmions per obstacle we find hexagonal, square, dimer, trimer and quadrimer ordering, where the n-mer structures are a realization of a molecular crystal state of skyrmions. As a function of external field and obstacle radius we show that there are transitions between the different crystalline states as well as mixed and disordered structures. We discuss how these states are related to commensurate effects seen in other systems, such as vortices in type-II superconductors and colloids interacting with two dimensional substrates.

Self-assembled toron-like structures in inverse nematic gels

A novel form of nematic gel (N-gel) wherein bright flower-like domains (BFDs) rich in gelator fibres are embedded in a matrix of liquid crystal (LC) molecules has been reported. These gels which we denote as inverse N-gels are unlike typical N-gels in which the LC is encapsulated within an aggregated network of gelator molecules. The self-organization of the helical gelator fibres within the BFDs leads to the creation of localized toron-like structures that are topologically protected due to their skyrmion director profile. Optical and confocal microscopy have been used to deduce the LC director configuration, in order to understand possible intermolecular interactions that can lead to the formation of the twisted structures and the inverse N-gels.

Electrically powered motions of toron crystallites in chiral liquid crystals

Malleability of metals is an example of how the dynamics of defects like dislocations induced by external stresses alters material properties and enables technological applications. However, these defects move merely to comply with the mechanical forces applied on macroscopic scales, whereas the molecular and atomic building blocks behave like rigid particles. Here, we demonstrate how motions of crystallites and the defects between them can arise within the soft matter medium in an oscillating electric field applied to a chiral liquid crystal with polycrystalline quasi-hexagonal arrangements of self-assembled topological solitons called "torons." Periodic oscillations of electric field applied perpendicular to the plane of hexagonal lattices prompt repetitive shear-like deformations of the solitons, which synchronize the electrically powered self-shearing directions. The temporal evolution of deformations upon turning voltage on and off is not invariant upon reversal of time, prompting lateral translations of the crystallites of torons within quasi-hexagonal periodically deformed lattices. We probe how these motions depend on voltage and frequency of oscillating field applied in an experimental geometry resembling that of liquid crystal displays. We study the interrelations between synchronized deformations of the soft solitonic particles and their arrays, and the ensuing dynamics and giant number fluctuations mediated by motions of crystallites, five-seven defects pairs, and grain boundaries in the orderly organizations of solitons. We discuss how our findings may lead to technological and fundamental science applications of dynamic self-assemblies of topologically protected but highly deformable particle-like solitons.

Topological solitons, cholesteric fingers and singular defect lines in Janus liquid crystal shells

Topological solitons are non-singular but topologically nontrivial structures in fields, which have fundamental significance across various areas of physics, similar to singular defects. Production and observation of singular and solitonic topological structures remain a complex undertaking in most branches of science - but in soft matter physics, they can be realized within the director field of a liquid crystal. Additionally, it has been shown that confining liquid crystals to spherical shells using microfluidics resulted in a versatile experimental platform for the dynamical study of topological transformations between director configurations. In this work, we demonstrate the triggered formation of topological solitons, cholesteric fingers, singular defect lines and related structures in liquid crystal shells. We show that to accommodate these objects, shells must possess a Janus nature, featuring both twisted and untwisted domains. We report the formation of linear and axisymmetric objects, which we identify as cholesteric fingers and skyrmions or elementary torons, respectively. We then take advantage of the sensitivity of shells to numerous external stimuli to induce dynamical transitions between various types of structures, allowing for a richer phenomenology than traditional liquid crystal cells with solid flat walls. Using gradually more refined experimental techniques, we induce the targeted transformation of cholesteric twist walls and fingers into skyrmions and elementary torons. We capture the different stages of these director transformations using numerical simulations. Finally, we uncover an experimental mechanism to nucleate arrays of axisymmetric structures on shells, thereby creating a system of potential interest for tackling crystallography studies on curved spaces.

Optically enriched and guided dynamics of active skyrmions

Light provides a powerful means of controlling physical behavior of materials but is rarely used to power and guide active matter systems. We demonstrate optical control of liquid crystalline topological solitons dubbed "skyrmions", which recently emerged as highly reconfigurable inanimate active particles capable of exhibiting emergent collective behaviors like schooling. Because of a chiral nematic liquid crystal's natural tendency to twist and its facile response to electric fields and light, it serves as a testbed for dynamic control of skyrmions and other active particles. Using ambient-intensity unstructured light, we demonstrate large-scale multifaceted reconfigurations and unjamming of collective skyrmion motions powered by oscillating electric fields and guided by optically-induced obstacles and patterned illumination.

Helicoids in chiral liquid crystals under external fields

Cholesteric liquid crystals, subject to externally applied magnetic fields and confined between two parallel planar surfaces with strong homeotropic anchoring conditions, are found to undergo transitions to different types of helicoidal configurations with disclinations. Analytical and numerical studies are performed in order to characterize their properties. In particular, we produce a phase diagram for the transitions from the nematic state to the helicoidal phases in terms of the molecular chirality and the strength of the applied magnetic field.

Thermo-Optical Generation of Particle-Like Structures in Frustrated Chiral Nematic Film

The creation of metastable particle-like structures in frustrated (unwound) chiral nematic film containing light-absorbing additive is studied. It is shown that such localized structures can be generated by the thermo-optical action of a focused laser beam or arise spontaneously at a phase transition from an isotropic to a liquid crystal state. Observed axisymmetric patterns resemble cholesteric spherulites with toroidal double-twisted director-field configuration.

Schools of skyrmions with electrically tunable elastic interactions

Coexistence of order and fluidity in soft matter often mimics that in biology, allowing for complex dynamics and applications-like displays. In active soft matter, emergent order can arise because of such dynamics. Powered by local energy conversion, this behavior resembles motions in living systems, like schooling of fish. Similar dynamics at cellular levels drive biological processes and generate macroscopic work. Inanimate particles capable of such emergent behavior could power nanomachines, but most active systems have biological origins. Here we show that thousands-to-millions of topological solitons, dubbed “skyrmions”, while each converting macroscopically-supplied electric energy, exhibit collective motions along spontaneously-chosen directions uncorrelated with the direction of electric field. Within these “schools” of skyrmions, we uncover polar ordering, reconfigurable multi-skyrmion clustering and large-scale cohesion mediated by out-of-equilibrium elastic interactions. Remarkably, this behavior arises under conditions similar to those in liquid crystal displays and may enable dynamic materials with strong emergent electro-optic responses.

While flocking and schooling are more often associated with birds and fish, these types of behaviour can also be observed in inanimate systems. Here the authors demonstrate schooling of topological solitons in a liquid crystal system powered by oscillating electric fields.

Light-controlled skyrmions and torons as reconfigurable particles

Topological solitons, such as skyrmions, arise in field theories of systems ranging from Bose-Einstein condensates to optics, particle physics, and cosmology, but they are rarely accessible experimentally. Chiral nematic liquid crystals provide a platform to study skyrmions because of their natural tendency to form twisted structures arising from the lack of mirror symmetry at the molecular level. However, large-scale dynamic control and technological utility of skyrmions remain limited. Combining experiments and numerical modeling of chiral liquid crystals with optically controlled helical pitch, we demonstrate that low-intensity, unstructured light can control stability, dimensions, interactions, spatial patterning, self-assembly, and dynamics of these topological solitons.

Orientational structures in cholesteric droplets with homeotropic surface anchoring

The dependency of orientational structures in cholesteric droplets with homeotropic surface anchoring on the helicity parameter has been studied by experiment and simulations. We have observed a sequence of structures, in which the director configurations and topological defects were identified by comparison of polarized microscopy pictures with simulated textures. A toron-like and low-symmetry intermediate layer-like structures have been revealed and studied in detail. The ranges of stability of the observed structures have been summarized in a general diagram and explained by the helicity parameter dependence of the free energy terms.

Phase Transition in Frustrated Magnetic Thin Film-Physics at Phase Boundaries

In this review, we outline some principal theoretical knowledge of the properties of frustrated spin systems and magnetic thin films. The two points we would like to emphasize: (i) the physics in low dimensions where exact solutions can be obtained; (ii) the physics at phase boundaries where interesting phenomena can occur due to competing interactions of the two phases around the boundary. This competition causes a frustration. We will concentrate our attention on magnetic thin films and phenomena occurring near the boundary of two phases of different symmetries. Two-dimensional (2D) systems are in fact the limiting case of thin films with a monolayer. Naturally, we will treat this case at the beginning. We begin by defining the frustration and giving examples of frustrated 2D Ising systems that we can exactly solve by transforming them into vertex models. We will show that these simple systems already contain most of the striking features of frustrated systems such as the high degeneracy of the ground state (GS), many phases in the GS phase diagram in the space of interaction parameters, the reentrance occurring near the boundaries of these phases, the disorder lines in the paramagnetic phase, and the partial disorder coexisting with the order at equilibrium. Thin films are then presented with different aspects: surface elementary excitations (surface spin waves), surface phase transition, and criticality. Several examples are shown and discussed. New results on skyrmions in thin films and superlattices are also displayed. By the examples presented in this review we show that the frustration when combined with the surface effect in low dimensions gives rise to striking phenomena observed in particular near the phase boundaries.

Liquid-crystalline half-Skyrmion lattice spotted by Kossel diagrams

Skyrmions are swirl-like topological entities that have been shown to emerge in various condensed matter systems. Their identification has been carried out in different ways including scattering techniques and real-space observations. Here we show that Kossel diagrams can identify the formation of a hexagonal lattice of half-Skyrmions in a thin film of a chiral liquid crystal, in which case Kossel lines appear as hexagonally arranged circular arcs. Our experimental observations on a hexagonal lattice of half-Skyrmions and other defect structures resembling that of a bulk cholesteric blue phase are perfectly accounted for by numerical calculations and a theoretical argument attributing strong reflections yielding Kossel lines to guided mode resonances in the thin liquid crystal film. Our study demonstrates that a liquid crystal is a model system allowing the investigation of topological entities by various optical means, and also that Kossel techniques are applicable to the investigation of thin systems with non-trivial photonic band structures including topologically protected optical surface states.

Super-resolution stimulated emission depletion microscopy of director structures in liquid crystals

Extending the optical imaging of director structures in liquid crystals (LCs) beyond the diffraction limit is poised to provide insights into previously elusive LC physics at the nanoscale. Here, we develop and characterize super-resolution stimulated emission depletion microscopy with molecular orientation sensitivity and apply it to reveal spatially localized director structures in chiral nematic and smectic LCs. As examples, we demonstrate director imaging and reconstruction of nanoscale LC configurations, including solitonic Bloch walls and two-dimensional skyrmions, both observed in sub-micrometer-thick strongly confined LC films, and focal conic domains in smectic LCs. The ≲100  nm resolution of our orientation-resolved STED imaging is three times better than that of fluorescence confocal polarizing microscopy and polarized nonlinear imaging techniques, but can be potentially improved even further.

Comparing skyrmions and merons in chiral liquid crystals and magnets

When chiral liquid crystals or magnets are subjected to applied fields or other anisotropic environments, the competition between favored twist and anisotropy leads to the formation of complex defect structures. In some cases, the defects are skyrmions, which have 180^{∘} double twist going outward from the center, and hence can pack together without singularities in the orientational order. In other cases, the defects are merons, which have 90^{∘} double twist going outward from the center; packing such merons requires singularities in the orientational order. In the liquid crystal context, a lattice of merons is equivalent to a blue phase. Here we perform theoretical and computational studies of skyrmions and merons in chiral liquid crystals and magnets. Through these studies, we calculate the phase diagrams for liquid crystals and magnets in terms of dimensionless ratios of energetic parameters. We also predict the range of metastability for liquid crystal skyrmions and show that these skyrmions can move and interact as effective particles. The results show how the properties of skyrmions and merons depend on the vector or tensor nature of the order parameter.

Dynamics of topological solitons, knotted streamlines, and transport of cargo in liquid crystals

Active colloids and liquid crystals are capable of locally converting the macroscopically supplied energy into directional motion and promise a host of new applications, ranging from drug delivery to cargo transport at the mesoscale. Here we uncover how topological solitons in liquid crystals can locally transform electric energy to translational motion and allow for the transport of cargo along directions dependent on frequency of the applied electric field. By combining polarized optical video microscopy and numerical modeling that reproduces both the equilibrium structures of solitons and their temporal evolution in applied fields, we uncover the physical underpinnings behind this reconfigurable motion and study how it depends on the structure and topology of solitons. We show that, unexpectedly, the directional motion of solitons with and without the cargo arises mainly from the asymmetry in rotational dynamics of molecular ordering in liquid crystal rather than from the asymmetry of fluid flows, as in conventional active soft matter systems.

Revolving supramolecular chiral structures powered by light in nanomotor-doped liquid crystals

Molecular machines operated by light have been recently shown to be able to produce oriented motion at the molecular scale^1,2 as well as do macroscopic work when embedded in supramolecular structures^3-5. However, any supramolecular movement irremediably ceases as soon as the concentration of the interconverting molecular motors or switches reaches a photo-stationary state^6,7. To circumvent this limitation, researchers have typically relied on establishing oscillating illumination conditions-either by modulating the source intensity^8,9 or by using bespoke illumination arrangements^10-13. In contrast, here we report a supramolecular system in which the emergence of oscillating patterns is encoded at the molecular level. Our system comprises chiral liquid crystal structures that revolve continuously when illuminated, under the action of embedded light-driven molecular motors. The rotation at the supramolecular level is sustained by the diffusion of the motors away from a localized illumination area. Above a critical irradiation power, we observe a spontaneous symmetry breaking that dictates the directionality of the supramolecular rotation. The interplay between the twist of the supramolecular structure and the diffusion ¹⁴ of the chiral molecular motors creates continuous, regular and unidirectional rotation of the liquid crystal structure under non-equilibrium conditions.

Reflection spectra and near-field images of a liquid crystalline half-Skyrmion lattice

We investigate numerically the optical properties of a hexagonal half-Skyrmion lattice exhibited by a highly chiral liquid crystal confined between two parallel plates. Our study focuses on the near and far-field reflection for normally incident light with different polarizations. We show that, when the wavelength of the incident light is longer than a threshold value, the reflectivity is almost insensitive to the polarization of the incident light, although the intensity profiles of the reflected light, in particular in the near-field regime, depend significantly on the polarization. The former property is attributable to the quasi two-dimensional nature of the half-Skyrmion lattice, that is, almost uniform orientational order along the direction normal to the confining plates. Our results for the intensity of reflected light generated by evanescent as well as propagating contributions suggest that direct evidence of the formation and structure of half-Skyrmions could be provided by near-field optics with resolutions higher than that of conventional optical microscopy.

Filamentous phages as building blocks for reconfigurable and hierarchical self-assembly

Filamentous bacteriophages such as fd-like viruses are monodisperse rod-like colloids that have well defined properties of diameter, length, rigidity, charge and chirality. Engineering these viruses leads to a library of colloidal rods, which can be used as building blocks for reconfigurable and hierarchical self-assembly. Their condensation in an aqueous solution with additive polymers, which act as depletants to induce attraction between the rods, leads to a myriad of fluid-like micronic structures ranging from isotropic/nematic droplets, colloid membranes, achiral membrane seeds, twisted ribbons, π-wall, pores, colloidal skyrmions, Möbius anchors, scallop membranes to membrane rafts. These structures, and the way that they shape-shift, not only shed light on the role of entropy, chiral frustration and topology in soft matter, but also mimic many structures encountered in different fields of science. On the one hand, filamentous phages being an experimental realization of colloidal hard rods, their condensation mediated by depletion interactions constitutes a blueprint for the self-assembly of rod-like particles and provides a fundamental foundation for bio- or material-oriented applications. On the other hand, the chiral properties of the viruses restrict the generalities of some results but vastly broaden the self-assembly possibilities.

Topology-commanded optical properties of bistable electric-field-induced torons in cholesteric bubble domains

Nowadays, complicated topological defects enable many experimental manipulations and configurational simulations of active soft matter for optical and photonic applications. Investigation of topological defects in soft anisotropic materials enables one to better understand three-dimensional orientation fields in cholesteric liquid crystals. Here, we describe optical properties of bistable bubble domain (BD) texture torons in a thin layer of cholesteric liquid crystal (CLC), frustrated by homeotropic anchoring conditions, and reliably switchable by a random process. The control of macroscopic optical density and diffraction efficiency of the BD texture is demonstrated by a selection of a confinement ratio of the CLC. Experimentally reconstructed CLC director profile reveals the topology of BD torons allowing consideration of naturally occurring BD texture for applications in optical and photonic devices, which are bistably switchable between active and transparent optical states.

Bipolar configuration with twisted loop defect in chiral nematic droplets under homeotropic surface anchoring

Optical textures and appropriate orientational structures have been studied within droplets of chiral nematic dispersed in polymer assigning the homeotropic anchoring. The helix axis of the chiral structure inside droplets forms the bipolar configuration. The optical droplet textures were analysed in the unpolarised light, analyser switching-off scheme and in crossed polarisers. The twisted loop defect reveals itself convincingly in all schemes. Its appearance at the optical patterns of the chiral nematic droplets has been examined depending on their size and the aspect direction. The existence of the defect has been verified by the structural and optical calculations. The effect of an electric field on both the defect line shape and the orientational structure of chiral nematic has been studied.

Squirming motion of baby skyrmions in nematic fluids

Skyrmions are topologically protected continuous field configurations that cannot be smoothly transformed to a uniform state. They behave like particles and give origins to the field of skyrmionics that promises racetrack memory and other technological applications. Unraveling the non-equilibrium behavior of such topological solitons is a challenge. We realize skyrmions in a chiral liquid crystal and, using numerical modeling and polarized video microscopy, demonstrate electrically driven squirming motion. We reveal the intricate details of non-equilibrium topology-preserving textural changes driving this behavior. Direction of the skyrmion’s motion is robustly controlled in a plane orthogonal to the applied field and can be reversed by varying frequency. Our findings may spur a paradigm of soliton dynamics in soft matter, with a rich interplay between topology, chirality, and orientational viscoelasticity.

A skyrmion is a topological object originally introduced to model elementary particles and a baby skyrmion is its two-dimensional counterpart which can be realized as a defect in liquid crystals. Here the authors show that an electric field can drive uniform motion of baby skyrmions in liquid crystals.

Nonlinear Ultrafast Spin Scattering in the Skyrmion Phase of Cu_{2}OSeO_{3}

Ultrafast x-ray scattering studies of the topological Skyrmion phase in Cu_{2}OSeO_{3} show the dynamics to be strongly dependent on the excitation energy and fluence. At high photon energies, where the electron-spin scattering cross section is relatively high, the excitation of the topological Skyrmion phase shows a nonlinear dependence on the excitation fluence, in contrast to the excitation of the conical phase which is linearly dependent on the excitation fluence. The excitation of the Skyrmion order parameter is nonlinear in the magnetic excitation resulting from scattering during electron-hole recombination, indicating different dominant scattering processes in the conical and Skyrmion phases.

Theory of helicoids and skyrmions in confined cholesteric liquid crystals

Cholesteric liquid crystals experience geometric frustration when they are confined between surfaces with anchoring conditions that are incompatible with the cholesteric twist. Because of this frustration, they develop complex topological defect structures, which may be helicoids or skyrmions. We develop a theory for these structures, which extends previous theoretical research by deriving exact solutions for helicoids with the assumption of constant azimuth, calculating numerical solutions for helicoids and skyrmions with varying azimuth, and interpreting the results in terms of competition between terms in the free energy.