Topological solitons in chiral liquid crystals

ABSTRACT Topological solitons are quasi-particles of field deformations that are topologically nontrivial. Conceptualised by celebrated mathematicians and scientists including Carl Friedrich Gauss, Lord Kelvin, and Tony Skyrme, the ideas of knots, topology, and solitons have found a wealth of experimental embodiments in chiral liquid crystals in recent years. This is largely due to the exceptional capacity of the material for stabilising, characterising, and controlling topological solitons. In this review, I report the recent progress in realising and characterising topological solitons in chiral liquid crystals. Following the homotopy classification, I begin with the 1D twist walls and subsequently introduce 2D skyrmions and skyrmion bags, followed by 3D Hopf solitons including both hopfions and heliknotons. Physical properties of topological solitons are discussed, which include stability, soliton–soliton interaction, field-controlled transformation, field-activated dynamics, and crystalline self-assembly. Finally, the review ends by illustrating how liquid-crystal solitons can provide insights into topological solitons in other physical systems, and perspectives of future studies on the subject. The purpose of this review is to provide a fundamental overview of topological solitons in chiral liquid crystals.


Introduction
Topology is the study of properties that are preserved under continuous deformations [1].Topological properties include genus, knot invariant, linking number, etc. (Figure 1a).Fields, such as continuous vector fields, can also be topological.This can be illustrated by the fact that certain field configurations cannot be deformed continuously into one another without introducing singularities that tear the continuum, making these fields topologically distinct (Figure 1bd).Topological solitons are topologically nontrivial particle-like field configurations embedded in a farfield background [5,6].
The concepts of solitons and knots have their roots in the early nineteenth century when Carl Friedrich Gauss envisaged that knots in fields could behave like particles and Lord Kelvin proposed a theory of chemical elements as different knots in Aether [7].Tony Skyrme, in the second half of the twentieth century, developed a model that describes nucleons as topological solitons in a nonlinear field theory [8].The model was later demonstrated to be a low-energy effective model of quantum chromodynamics [9].Today, topological solitons have been discovered or realised in many branches of physics, including nuclear physics, condensed matters, optics, and cosmology [10][11][12][13][14][15][16][17][18][19].Not only are topological solitons of great fundamental interest, but they also promise technological applications because of their unique particle-like properties and the capability of encoding information in topology [20].
Liquid crystals (LCs) are ordered fluids with longrange orientational order and partial or no positional order [21,22].The long-range orientational order allows the director field n r ð Þ to be defined in nematic LCs, which is a non-polar line field of the average molecular orientation (Figure 1e).Beyond the multi-billion-dollar industry of electro-optic devices including displays [23], LCs have served as a platform for studying topological structures and defects [18,24,25].LCs are excellent for studying topological solitons due to their unparalleled accessibility in generating and manipulating topological solitons, as well as structurally analysing topological solitons they host [26].Moreover, solitons in chiral LCs have been shown to gain additional stability.In chiral LCs, n r ð Þ has an intrinsic tendency to twist due to molecular chirality; the distance over which n r ð Þ twists by 2π at equilibrium is the helical pitch p (Figure 1f).The extra chirality term in the energy functional allows the system to evade the prediction of the Derrick-Hobart theorem, which shows solitons cannot be stabilised in dimensions of three and beyond within a linear field theory [27][28][29].
CONTACT Jung-Shen Benny Tai jungshen.tai@colorado.edu This review is organised as follows.Homotopy groups and LC free-energy models are first introduced for classifying the topology and modelling the field configurations of topological solitons.Following the classification of homotopy groups, I provide a brief and non-comprehensive review of the progress in topological solitons in chiral LCs.I begin with the simplest one-dimensional (1D) twist walls, and progress to 2D skyrmions.The highlights include the stability of skyrmions, different geometric configurations, field- ð Þ can also be physically achieved by the dispersion of ferromagnetic particles with magnetization vectors m r ð Þ aligning with n r ð Þ.The orientation of vectorized n r ð Þ is visualized by a 3D arrow.Reproduced with permission from ref [3].(f) Schematic of the helical pitch p in the helical state of n r ð Þ shown by ellipsoids.(g) The target space (order-parameter space) of a field is S 1 for a 2D unit vector (left) and S 2 for a 3D unit vector (right).(h) Homotopy groups used for classification of topological solitons.The green, yellow, and blue highlight topologically nontrivial configurations discussed in this review, while the π 3 S 3 À � solitons (red) arise in Skyrme's nucleon model.(i) A 2D unit vector field in R 1 with a 2π rotation forming a π 1 S 1 À � soliton with a topological degree of 1.The uniform far-field in R 1 (represented by the red arrow pointing upwards) allows the configuration space R 1 to be compactified into S 1 .(j) The target space of a 3D director is S 2 =Z 2 , namely, S 2 with diametrically opposite points identified.Reproduced from [4] with permission.
induced dynamics, and novel 2D field configurations with arbitrary skyrmion number -skyrmion bags.Next, I introduce 3D Hopf solitons, also known as hopfions, with the focus on their field-induced topological transformation.This is followed by heliknotons, a new class of 3D topological solitons discovered in the helical background of chiral LCs.I detail the fascinating properties of heliknotons, including 3D localisation, stable knotted vortex lines, self-assembled crystals, electrostriction, etc.I also delve into the geometric connection between hopfions and heliknotons and how they can reversibly transform into one another.Additionally, the 3D hopping dynamics of Hopf solitons during repeated intertransformation between the two topologically identical, but geometrically distinct configurations is discussed.Finally, this review concludes by discussing how insights gained from studying LC topological solitons can be applied to studying solitons in other systems.

Homotopy classification of topological solitons
How do we describe or classify the topology of a field?A field n r ð Þ is a map from the physical configuration space X to the target space Y, n : X7 !Y, where r 2 X and n 2 Y: While the configuration space can be intuitively understood as the space the field lives in, the target space (or the order-parameter space in physical systems) depends on the nature of the field.For example, when the field is a unit vector confined to 2D orientations, the target space is the space composed of all possible orientations of the 2D unit vector, which is a unit circle S 1 (Figure 1g).In topology, if a field n 1 r ð Þ can be continuously deformed into n 2 r ð Þ, n 1 r ð Þ and n 2 r ð Þ are homotopic and belong to the same homotopy class.Moreover, the set of homotopy classes can form a homotopy group.Homotopy groups are often defined in terms of spheres; the n th homotopy group of S m , or π n S m ð Þ, summarises topologically distinct maps from the n-dimensional sphere S n to the m-dimensional sphere S m and can be used to classify topological solitons (Figure 1h) [1,5].As an example, a 2D unit vector field in the 1D Euclidean space R 1 is a map R 1 7 !S 1 .In this case, R 1 can be compactified into S 1 by identifying the positive and negative infinity in R 1 as one point without introducing discontinuity if n À 1 ð Þ ¼ n 1 ð Þ, thus allowing the implementation of homotopy groups for topological classification (Figure 1i).The integer number of 2π rotations that n r ð Þ goes through from À 1 to 1 is invariant under smooth geometric perturbations and is referred to as the topological degree or charge of the field.Thus, π 1 S 1 À � identifies with Z, the group of integers under addition.In nematic LCs, the nonpolar nature of LC director leads to the order-parameter space being S 2 =Z 2 , a sphere with diametrically opposite points identified (Figure 1j).A major consequence is the existence of LC defect lines in 3D, since is a trivial group.However, as most homotopy groups of S 2 are homeomoric to homotopy groups of S 2 =Z 2 , and a continuous director field in a simply connected manifold can always be 'vectorised' into a vector field [30], a director field can be replaced with a vector field for simplicity in many cases.

Free energy of chiral LCs and chiral LC ferromagnets
An important tool for investigating the structure and stability of topological solitons is numerical modelling of field configurations by free-energy minimisation.In chiral LCs, the free energy associated with deformations in the director field n r ð Þ (treated as a unit vector field here) can be described by the Frank-Oseen free energy functional [3,21] where p is the helical pitch and K 11 , K 22 , K 33 and K 24 are the Frank elastic constants describing the energetic costs of splay, twist, bend, and saddle-splay deformations, respectively.When the LC is confined and surface anchoring is present, Equation ( 1) is supplemented by a surface anchoring potential at the confining substrates, where W is the surface anchoring strength and n 0 is the easy-axis orientation.To model the effect of electric field E on n r ð Þ, the dielectric coupling term can be included as where ε 0 is vacuum permittivity and Δε is the dielectric anisotropy of the LC.In a chiral LC ferromagnet where magnetically monodomain nanoplates are uniformly dispersed in the hosting chiral LC, n r ð Þ responds to weak magnetic fields due to the coupling between the magnetic moment of the nanoplates and the magnetic field, and the coupling between n r ð Þ and magnetic nanoplates [31].In the case of strong homeotropic (perpendicular) surface anchoring of LC molecules to the magnetic nanoplates, the local magnetisation M r ð Þ is assumed to be collinear with n r ð Þ and vectorises the director field (Figure 1e).The magnetisation unit vector field m r ð Þ;M r ð Þ= M r ð Þ j j can thus replace n r ð Þ in Equations ( 1)-(3), and an additional linear magnetic coupling term between m r ð Þ and an applied magnetic field H reads where μ 0 is vacuum permeability and M r ð Þ j j ¼ ρm p is the magnetisation of the material (approximated to be uniform) as the product of the nanoplate density ρ and the average magnetic moment of a nanoplate m p .The material parameters for a few commercially available LC mixtures are summarised in Table 1.
It is important to note that the vectoral approach in Frank-Oseen free energy does not account for the nonpolar nature of LC director field or the reduction in scalar order parameter near defects.In these cases, the tensorial approach and the Landau-de Gennes expansion is advised for a more complete description of free-energy cost in the perturbation of order in LCs [22,32].Nevertheless, Frank-Oseen free energy is adequate for modelling orientable field configurations, especially when the detailed structures around defects are not the primary focus.

1D twist walls
1D solitonic walls represent the simplest of topological solitons.In condensed matter systems such as LCs and magnets, solitonic walls are rather common [21,[33][34][35].In chiral materials, these solitonic walls are also referred to as twist walls and can be stabilised by the frustration between the material's chirality, surface energy, and coupling to external fields.In material systems where the order-parameter is a unit vector, an integer number of full-revolution (2π) twists of the order-parameter field is embedded in the uniform farfield, characterised by the π 1 S 1 À � ¼ Z homotopy group (Figure 1h,i).This is different for systems with a nonpolar order-parameter field, such as LCs.In a solitonic LC twist wall, the nonpolar director n r ð Þ is confined in a 2D plane (Figure 2b) and an integer number of half-revolution (π) twists can be embedded in the non-polar far-field (Figure 2a,b).The integer number of π-twists identifies with the topological charge.The order-parameter space is S 1 =Z 2 , which is homeomorphic to a circle, or the 1-sphere S 1 (Figure 2c).Therefore, nonpolar LC twist walls are also classified by the first homotopy group of S 1 , π 1 S 1 À � ¼ Z.In LC samples where boundary conditions are often imposed at confining substrates, the configuration of twist walls depends on the effective anchoring strength.With perpendicular boundary conditions imposed by strong homeotropic anchoring, LC twist walls exist in the form of cholesteric fingers of the 3 rd type, or CF-3 [36].In a CF-3, The director orientations are confined approximately in the 2D plane perpendicular to the direction that the finger extends (Figure 2d, top), and a 1D π 1 S 1 À � topological soliton in R 1 terminates at two π 1 S 1 À � half-integer defects near the substrates (Figure 2e).The π 1 S 1 À � defects extend along the translationally invariant direction and form disclination lines.We note that, without the approximation that director orientations are confined in 2D, both twist walls and disclination lines are described by π 1 S 2 =Z 2 À � .Notably, the director field around the two twist disclinations is non-orientable and the tangent of the director forms Möbius strips [37].Therefore, numerical modelling of configurations including twist disclinations requires the tensorial approach in Landau-de Gennes free energy.
Under boundary conditions with weak anchoring energy, the defect lines in a CF-3 escape beyond the confining surfaces, rendering a twist wall configuration translationally invariant across the thickness (Figure 2a,f).The structural stability between CF-3, twist wall, and the uniform state depends on the sample thickness-to-pitch ratio (d=p) and the effective anchoring strength Wp=L, where W is the surface anchoring energy density and L is the average elastic constant in Landau-de Gennes free  2g).In brief, CF-3s are stable at large thicknesses and anchoring strengths, while in the weak anchoring regime, twist walls become the structurally stable configuration.The numerically derived stability between CF-3 and twist wall is consistent with experimental observations (Figure 2e,f), where CF-3s are observed in common LC nematics (Table 1) with p , 10 µm and W , 10 À 4 Jm −2 , corresponding to strong effective anchoring, while 1D twist walls are ubiquitously observed when p , < 1 µm [3,12,33,36,37].The cross-sectional 3PEF-PM images of a twist wall and a CF-3 reveal their detailed structural difference across the thickness (Figure 2e,f).The absence of high-energy defect lines in twist walls also allow them to be more abundant within the sample, lowering the overall elastic energy through twisting, while a CF-3 has a tendency to shrink due to the defect-induced tension.Overall, these results show the stability between 1D twist wall, CF-3, and the uniform state can be controlled by a set of material and sample parameters.eq .S eq is the equilibrium scalar order parameter.Reproduced with permission from [3].

2D skyrmions
2D skyrmions (sometimes referred to as baby skyrmions due to their reduced dimensionality compared to the ones originally proposed by Tony Skyrme [8]) are the topological solitons labelled by elements of the second homotopy group, π 2 ðS 2 Þ ¼ Z for systems described by unit vector fields and π 2 ðS 2 =Z 2 Þ ¼ Z for systems described by nonpolar director fields.Since and π 2 ðS 2 Þ are homeomorphic, unit vector fields may be used in place of the nonpolar LC director field for convenience.Conceptually, a 2D skyrmion in chiral materials can be considered as configured by looping a twist wall into a circle [37].The topological charge of a 2D skyrmion, also called the skyrmion number, counts the integer number of times the entire field wraps around its target space S 2 , and is defined by shows an elementary N sk ¼ 1 skyrmion that covers S 2 exactly once.
Observed and studied extensively in LCs with strong perpendicular boundary conditions, a toron is a 2D skyrmion dressed by two point defects at its ends, with the existence of the defects mandated by the transformation of topology from a skyrmion into the uniform trivial state near the confining substrates (Figure 3b).Torons can therefore be understood as fragments of skyrmions terminating on singular point defects characterised by the same homotopy group, π 2 ðS 2 Þ ¼ Z: Notably, the two oppositely charged π 2 S 2 À � point defects are also a monopole and anti-monopole pair in the emergent magnetic field [38].Torons are common in both LCs and magnets, where the termination of skyrmions at point defects is related to confinement and other stabilising mechanisms [3,12,[39][40][41][42][43] While torons are commonly observed as particle-like solitons in confined LCs with a sample thickness comparable to the helical pitch (d~p) and with strong perpendicular conditions [39,44], skyrmions with translationally invariant π 2 ðS 2 Þ topology emerge when the strength of the surface anchoring or the size of the overall structure is reduced (Figure 3c,d) [12,33].The stability of skyrmions and torons, and their potential inter-transformations have been systematically studied as depending on both the thickness-to-pitch ratio (d=p) and the dimensionless effective anchoring strength Wp=K (Figure 3e,f) [3].Torons are stable compared to skyrmions or topologically trivial states when the surface anchoring is sufficiently strong and the thickness is sufficiently thick.At a smaller thickness, torons' stability is lost to the uniform state.Skyrmions, on the other hand, are stable at weak anchoring, and the transition between torons and skyrmions happens at 10 , < Wp=K , < 30.Depending on the anchoring strength, the transformation between torons and skyrmions can occur at cell thicknesses of 10s of µm or sub-µm.Out of the nonexhaustive competing states, skyrmion or toron lattices can be the lowest energy state within certain parameter regions (Figure 3e,f), pointing at potential topological phases of matter analogous to the skyrmion lattice phase is solid-state chiral magnets [10].In practice, torons and skyrmions can inter-transform and co-exist at p � 1 µm and d � 1 µm, under strong homeotropic anchoring in a chiral LC based on the nematic 5CB.The structural difference between torons and skyrmions can be revealed by the distinct 2D patterns in the polarizing optical microscopy images.Nonlinear fluorescence images further reveal the detailed difference in their 3D structures (Figure 3c,d).Notably, the two point defects near the substrates are present in a toron but not in a skyrmion.

Helical-background skyrmions
Skyrmions can also be embedded in a helical background, contrasting the more commonly known axisymmetric, whirling configuration in the uniform background.The configuration of a helical-background skyrmion can be considered as splitting and splicing the uniformbackground skyrmion while extending the helical region into a uniformly helical background with constant helical axis χ 0 , which defines the axis that n r ð Þ twists around in the far field (Figure 3g,h).Helical-background skyrmions can be stabilised in LC cells with planar boundary condition by an electric field along χ 0 .The stabilised skyrmion extends in the in-plane direction and terminates at point defects, or monopoles in the emergent magnetic field [38].The energetics of in-plane helical-background skyrmions can be controlled by electric fields through the quadratic dielectric coupling.Additionally, when the LC is doped with ferromagnetic nanoplates with strong perpendicular boundary condition and forms a monodomain LC ferromagnet, the magnetisation field m r ð Þ and the collinear director field n r ð Þ additionally couple linearly to the applied magnetic field.This allows the length of the skyrmion and the separation velocity of the defect (monopole-antimonopole) pair to be controlled by the external fields (Figure 3i-k).The controlled dynamics of skyrmions and monopoles can lead to applications utilising the regions around monopoles with high-degree distortions in n r ð Þ as energetic traps for cargo transport of micro-and nanoparticles.ð Þ within the skyrmion, respectively.(l) polarizing optical micrographs of skyrmion bags S(1) to S(4), two stable conformations of the S(13) bag, and the S(59) bag.N sk ¼ 0; 1; 2; 3; 12; 12; 58, respectively.(a-f) Reproduced from ref [3].(g-K) Reproduced from ref [38].(l) Reproduced from ref [4].All with permission.

Skyrmion bags
Beyond elementary skyrmions with unity skyrmion number, high-charge topological field configurations with arbitrary skyrmion number have recently been realised in LCs and modelled in solid-state magnets [4,45].Dubbed 'skyrmion bags', these stable composite structures consist of antiskyrmions contained within a larger skyrmion, with the latter acting like a confining bag (Figure 3l).Skyrmion bags demonstrate a similar pairwise interaction to their elementary counterparts and can be used to encode information in their topological degree of freedom.Conventional proposals of racetrack memory involve encoding bits of information by separations between elementary skyrmions, which allows for limited density and is susceptible to data loss due to their repulsive interaction [20].On the other hand, skyrmion bags utilise topology-encoded information to overcome this problem.For example, a train of skyrmion bags, S(N 1 ), S(N 2 ), . . ., encode information by the integer N j , the number of (anti)skyrmions in each bag, enabling a new paradigm-changing approach in data storage and processing devices [46].

3D hopfions
In 1931, German mathematician Heinz Hopf discovered the Hopf fibration (also called Hopf map or Hopf bundle) and demonstrated how S 3 space can be composed of fibres of circles, with each circle mapping to a single point in S 2 [47].This shows that a continuous map from a higher-dimensional sphere to a lowerdimensional sphere can be nontrivial, an unexpected result to the mathematical community at the time.Indeed, the third homotopy group of S 2 is identified with the group of integers, i.e. π 3 ðS 2 Þ ¼ Z (Figure 1h).3D Hopf solitons, or hopfions, are physical realisations of Hopf fibration, with each closed-loop in the 3D configuration space corresponding to a single distinct orientation in the order-parameter space S 2 for systems described by unit vectors [5,6,[48][49][50][51][52][53] (Figure 4a).Here the 3D configuration space can be R 3 given that it can be compactified to S 3 without introducing discontinuities.An example is the one-point compactification by identify the infinity of R 3 as one point when the far-field is uniform.Realisation of stable 3D hopfions had been challenging, as the Derrick-Hobart theorem shows the absence of localised stable solutions in linear filed theories in dimensions three or higher [27,28].After longpreceded theoretical interest, stable hopfions were recently demonstrated in LCs [54], LC ferromagnets [13], and solid-state magnets [55].In LCs where the director field is nonpolar and the order-parameter space is S 2 =Z 2 , π 3 S 2 =Z 2 À � is homeomorphic to π 3 ðS 2 Þ, and unit vector fields may be used in place of the nonpolar LC director field.
Static and stable hopfions were first experimentally realised in chiral LCs and chiral LC ferromagnets where chirality and confinement help evade the prediction of Derrick-Hobart theorem and contribute to stability [13,54].Topologically equivalent to the Hopf fibration, in a hopfion, the preimage (inverse image) of each point in the order-parameter space S 2 forms a loop in the 3D configuration space, and the preimages of all points in S 2 fill the 3D space (Figure 4b,c).Here preimage indicates the inverse image of the map n : R 3 7 !S 2 , and can be obtained by finding the regions in R 3 that correspond to a given unit vector orientation in S 2 (Figure 4b; note that m r ð Þ is used interchangeably with n r ð Þin Figure 4).The associated topological charge, Hopf index, can be geometrically obtained by the linking number of a pair of preimages of any two points in the order-parameter space S 2 [56].For elementary Hopf solitons with unity charge, the preimages are linked exactly once.A numerical integration method can also be applied to obtain the Hopf index [57,58].Hopfions of Hopf indices Q ¼ �1 and Q ¼ 0 spontaneously emerge in samples of chiral LCs and chiral LC ferromagnets where the confining substrates impose perpendicular boundary conditions and the thickness over pitch ratio d=p � 1, while they can also be controllably generated by laser tweezers, as detailed in Refs [13].and [54] (Figure 4d,e).The n r ð Þ configuration and topology of LC hopfions were unambiguously identified by 3PEF-PM imaging, where experimentally reconstructed n r ð Þ and preimages are consistent with numerical simulations based on minimisation of free energy (Figure 4fi).LC hopfions show hard-particle-like elastic interactions and form 2D hexagonal lattice in the presence of a lateral confinement (Figure 4e).

Topological transformation of hopfions
Topological solitons hosted in chiral LC ferromagnets with equally strong facile responses to electric and magnetic fields provide an ideal platform for studying the interplay between external fields and topological solitons [31].In LC display industry, switching of pixels involves topologically trivial states [23].Understanding switching of topological configurations could additionally lead to multistable energy-efficient electro-optic and data devices.To prepare for topological transformation of solitons controlled by external fields, two elementary doughnut-shaped hopfions in a chiral LC with negative dielectric anisotropy (Δε) were coaxially arranged to create a composite hopfion with Q ¼ À 2 [58] (Figure 5a).The preimage of each point in S 2 is two separate unlinked loops 0 2  1 (Alexander-Briggs notation of knots [5]), and preimages of two distinct S 2 points form a pair of Hopf links, giving rise to the total linking number of −2 (Figure 5b).Upon application of an intermediate electric field perpendicular to the confining substrates, uniform far-field regions that separate the knotted regions of two elementary hopfions disappear, and n r ð Þ smoothly morphs into a single solitonic configuration.The partial merging of two elementary hopfion is demonstrated by the change in the topology of preimages, where a critical polar angle θ ¼ θ c in S 2 separates two subspaces: (a) single-loop (0 1 ) preimages of points with θ < θ c and (b) preimages of points with θ > θ c in the form of two separate unlinked loops (0 2 1 ).This  gives rise to a rich linking topology between preimage pairs depending on their respective polar angle in S 2 (Figure 5c,e).Importantly, while the topology of individual preimage depends on θ and changes with E, the overall field topology of n r ð Þ is preserved, as evidenced by the overall linking number of the reconstructed preimages from 3PEF-PM images and numerical integration of Hopf index (Figure 5f,g).Above a critical E, the interplay between the elastic and dielectric coupling energy causes n r ð Þ to transform discontinuously from Q ¼ À 2 to Q ¼ À 1 through a series of short-lived singular transient states (Figure 5a,d).These processes demonstrate both complex morphing of preimage induced by electric field, while preserving the n r ð Þ field topology, and the field-induced topological transformation of n r ð Þ. Notably, similar topological transformations are also demonstrated for a Q ¼ 0 hopfion consists of two oppositely charged elementary hopfions [58].Magnetic field also causes topological transformation of solitons hosted in chiral LC ferromagnets.Unlike the quadratic dielectric coupling to E, the response of magnetisation field m r ð Þ to the external magnetic field H is polar, leading to rich transformation of both preimage and soliton topology.A structural stability diagram shows a wealth of relaxed knotted field configurations when a complex Q ¼ 0 soliton is subject to a variable magnetic field H.The equilibrium solitonic structures include those with and without singular defects, i.e. torons and hopfions, respectively, and with simple and complex topology of preimage (Figure 5h-j).

Heliknotons
Heliknotons are a new class of 3D knot solitons realised in the helical (cholesteric) background of chiral LCs [14].The LC helical field is ubiquitous in chiral materials (often the ground state) and can be considered as comprising a triad of orthonormal fields, namely the material director field n r ð Þ and two immaterial line fields, the helical axis field χ r ð Þ and τ r ð Þ, which is the field perpendicular to both n r ð Þ and χ r ð Þ (Figure 6a).Heliknotons were found to display field topology resembling Hopf fibration in n r ð Þ and knotted half-integer singular vortex lines in χ r ð Þ and τ r ð Þ (Figure 6b).In conventional LCs such as 5CB, heliknotons spontaneously form after quenching from isotropic to chiral nematic phase when a moderate electric field E was applied along the far-field helical axis χ 0 , though they were later demonstrated to be stable without E in materials with proper elastic anisotropy [59].Heliknotons can also be controllably generated using laser tweezers.Heliknotons are stable in the bulk of chiral LCs and display properties of 3D-localised particles including Brownian motion and orientationcorrelated motion along χ 0 (Figure 6c).This contrasts with 3D LC hopfions in a uniform far-field background, in which case confinement is required for stability.The pair-wise interaction between the heliknotons is anisotropic and highly tunable from 10 s to 1000 s of k B T, depending on the LC material, E and sample thickness (Figure 6d).The inter-heliknoton interaction arises from sharing long-range perturbations in the background helical field and minimising the overall free energy, much like the interaction between nematic colloids [60,61].The interactions give rise to heliknoton crystals with different symmetries through self or guided assembly, including 2D rhombic latices, 2D kagome lattices, and 3D triclinic lattices (Figure 6e-h).Since LCs respond strongly to electric fields, LC heliknoton crystals exhibit giant anisotropic electrostriction, where two lattice parameters extend by 44% and 4% with less than 1.5 V change in voltage (Figure 6i).
The structure of heliknotons were revealed by consistent results of 3PEF-PM imaging and numerical simulations (Figure 6m-r); n r ð Þ in heliknotons from interlinked closed-loop preimages with linking number + 1 (Figure 6j), and χ r ð Þ and τ r ð Þ exhibit singular half-integer vortex lines that close into a trefoil torus knot, labelled as T(2,3) or 3 1 .This demonstrates a heliknoton as dual embodiments of a Q ¼ 1 Hopf soliton and a vortex knot.The knot of singular vortex line in χ r ð Þ and τ r ð Þ also co-localise with a knotted tube of meron (fractional skyrmion) in n r ð Þ [14].Unlike transient vortex lines that shrink and vanish due to energetically costly cores and distorted order around them, vortex-meron knots in heliknotons are energetically favourable and stable because of being nonsingular in the material field n r ð Þ.The twisted structures with the same handedness of the hosting chiral material also aids in stability.Under the same condition that stabilises the elementary heliknoton, solitonic structures consistent with higher-degree heliknotons were also observed (Figure 6k,l), where axially symmetric composite Q ¼ 0 soliton in a chiral LC ferromagnet at an applied magnetic field magnitude H ¼ 0 and d=p ¼ 2:7.(i) θ c and Q versus μ 0 H at d=p ¼ 2:7.(j) Full stability diagram of the Q ¼ 0 soliton in (h) depending on d=p and H.The diagram shows rich solitonic structures including those without singular defects (hopfions) and those containing singular point defects (elementary torons and 3π torons).Here the applied magnetic field is defined to be positive when it aligns with the far-field m 0 .Reproduced from Ref. [58] with permission.Whether heliknotons with even higher charges can be systematically generated and properties pertaining to their stability and many-body interactions require further study.

Heliknoton-hopfion transformation and 3D hopping
Heliknotons were later found to be stabilised in bulk chiral LCs without any electric field applied in nematic mixtures of rod-like and bent-core molecules that exhibit reduced bend elastic constant K 33 [59,62,63].The tunable elastic anisotropy allows the background farfield to be continuously varied between uniform, conical, and helical states with a continuous cone angle θ cone by an electric field when K 33 � K 22 [64].In the uniform background, the hosted Hopf solitons are conventionally referred to as hopfions [13,54,58], and in the helical background, Hopf solitons are identified as heliknotons [14] (Figure 7a-c).Numerical simulations reveal the structural stability diagrams of Hopf solitons depending on elastic anisotropy, electric field, and confinement conditions, in topologically trivial backgrounds with varying θ cone (Figure 7d-e).In the LC bulk without any confinement (Figure 7d), at zero electric field, Hopf solitons are metastable particle-like, spatially localized structures in the helical background with higher energy density than the background when ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi K 33 =K 22 p � 1:2.At larger ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi K 33 =K 22 p , heliknotons are metastable in the presence of an electric field, consistent with the original report that heliknotons require a stabilising E in chiral LCs with ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi K 33 =K 22 p > 1:5 that were used in the study [14].Hopf solitons can also be metastable in the bulk conical background.Notably, between metastable regions and across the helical-uniform and helical-conical boundaries for the background field, stable Hopf solitons with lower energy than the background fill the computational volume and form stable crystalline assemblies.Importantly, regardless of the distinct embedding backgrounds with different θ cone , the Hopf indices of the stabilised solitons remain unchanged.In the presence of confinement and perpendicular boundary conditions (Figure 7e), additional metastability regions of Hopf solitons emerge in the uniform background with θ cone ¼ 0 � (hopfions), differing from the case of no hopfions in the fully unwound LC bulk (Figure 7d,e).This further demonstrates how confinement and boundary conditions helps stabilise hopfions [54,65,66].
The structural stability diagram with confinement suggests an in situ pathway for inter-transformation between a heliknoton and a hopfion by varying E (Figure 7e), which was verified experimentally.The continuous inter-transformation provides further evidence that heliknotons and hopfions are indeed geometric embodiments of the same n r ð Þ field topology (Figure 7f).The heliknoton-hopfion inter-transformation draws analogy with the transformation between geometric shapes of identical topological invariants, e.g. the surface of a coffee mug and a doughnut both have genus one (Figure 1a).Remarkably, the reversible fullcycle transformation is accompanied by a displacement of approximately half a pitch along the long axis of the heliknotons (Figure 7f), enabled by the nonreciprocal evolution of n r ð Þ.Such transformation and the concomitant displacement can be repeated with a periodic voltage switching, leading to an activated propelling motion, analogous to the squirming motion of 2D LC skyrmions (Figure 7g) [67].
A close inspection of the translational motion of Hopf solitons under periodic inter-transformation revealed deviation from a linear trajectory (Figure 7g,h).Such deviation arises from the contrasting energetic landscapes between hopfions in a uniform background and heliknotons in a helical background.The former is monotonically increasing when the soliton displaces away from the energy minimum at the sample midplane, while the latter contains multiple energy minima along the far-field helical axis χ 0 and the midplane is an unstable local energy maximum.This is evident from the different long-axis orientations of heliknotons at static equilibrium with no applied voltage (Figure 7i), since the orientation of the soliton correlates with its z position along χ 0 in a helical background [14].Therefore, in each inter-transformation cycle, the combined propelling motion, complex energetic landscape along χ 0 (for heliknoton), and the correlated orientation and z position of heliknoton in a helical background, results in a combined squirming motion in lateral directions and vertical motion along z (or χ 0 ) -an effective hopping motion in the 3D space.Additionally, a Hopf soliton also demonstrates a propelling motion when a hopfion responds to a low-magnitude voltage modulation that perturbs the background between the uniform and the helical state, while not completely inter-transforming between a hopfion and a heliknoton (Figure 7j,k).This demonstrates the universality and richness of nonreciprocal n r ð Þ evolution in activating the propelling motion of solitons in LCs.

Topological solitons beyond LCs
LCs are an ideal material system for studying topological solitons due to their accessibility in the generation, manipulation, and optical characterisation of solitons.To date, a plethora of topological solitons in different dimensions have been realised and their equilibrium and out-of-equilibrium properties analysed, as partially covered in this review.Remarkably, because of the universality of the underlying topology, the insights into topological solitons may be shared between differential materials and physical systems such as liquid crystals, magnets, Bose-Einstein condensates, optics, etc [10][11][12][13][14][15][16][17][18][19].An exceptional example is solid-state chiral magnets and chiral LCs [68], where the micromagnetic Hamiltonian of magnets takes a similar functional form to the Frank-Oseen free energy of LCs under the one-constant approximation by having K 11 ¼ K 22 ¼ K 33 ;K.Notably, the molecular chirality in chiral LCs plays a role analogous to that of the Dzyaloshinskii-Moriya interaction in solid-state chiral magnets [29,68].The close connection between the mean-field energy functional of the two systems has allowed LCs and magnets to mutually inform and enrich the understanding of topological solitons in each system.Examples of topological solitons in both LCs and magnets include 2D skyrmions [3,12,69], 2D skyrmion bags [4,45], 3D Hopf solitons (hopfions and heliknotons) [41,66,70,71], as well as solitonic structures containing defects such as torons and chiral bobbers [3,39,42,54].However, it is important to note that fundamental differences between the two systems exist; the nonlocal dipole-dipole interaction, magneto-crystalline anisotropy, and the intrinsically polar order parameter are unique to solid-state magnets.Interesting properties pertaining to topological solitons in magnets include the skyrmion Hall effect and topological Hall effect [72], Hopf fibration in the emergent magnetic field [70,73], and topological phases of matter [68], though detailing topological solitons in solid-state magnets is beyond the scope of this review.

Conclusion
Topological solitons are particle-like field deformations that are topologically nontrivial.In this review, I give a brief introduction to topological solitons in the nematic alignment (director) field of chiral LCs based on their classification using homotopy groups.The stability of topological solitons arises from their distinct topology that ensues energetic barriers between topologically distinct states -the so-called topological protection.Material properties such as chirality and elastic anisotropy, as well as confinement, surface anchoring, coupling to external fields, etc., also affect the energetic landscape of the system and the stability of solitons.The fascinating properties of LC topological solitons can largely be attributed to the fact that the contrast between the topological quasi-particles and the background is in the field alignment rather than the chemical composition.This enables topological solitons the capacity to be reconfigured and morphed, which determines solitonsoliton interaction, self-assembly, electrostriction, stimuli-induced dynamics, etc.This contrasts with 'real' particles such as colloids in a liquid, which are chemically distinct from their backgrounds.
The unparalleled accessibility of LCs in generating, manipulating, and characterising a rich collection of nontrivial field configurations has allowed LCs to be an ideal testbed for topological defects, topological solitons, and the underlying theories.The facile response of LCs to externally applied electric fields and LC ferromagnets to magnetic fields further allow additional knobs of control.Together with their topologically protected particle-like properties, LC topological solitons can find useful technological applications in electrooptics, data storage, data processing, microfluidics, etc.
Several interesting subjects pertaining to LC topological solitons were not covered in this review.For example, it was recently shown that nanoparticles are enriched in the distorted regions of heliknotons due to free-energy minimisation, making topological solitons a potential building block of templates for nanoparticle assembly and plasmonic metamaterials [73].Topological solitons in birefringent and reconfigurable media such as LCs can also be used to control light propagation or interact with optical solitons [74,75].Moreover, a hierarchical selfassembly between 1D twist walls and defect lines result in a newly discovered family of solitonic structures, Möbiusons, which display Möbius-strip-like director field topology and self-propelled translational and rotational motions when activated with electric pulses [37].
Moving forward, a diverse array of new frontiers of topological solitons in LCs and beyond await further investigations.Can we go beyond the 2D target space S 2 or S 2 =Z 2 associated with uniaxial polar and nonpolar order parameters?A 3D target space such as S 3 or its quotient groups will enable the realisation of π 3 S 3 À � topological solitons in the 3D configuration space, which share the same topology as nucleons in Skyrme's nonlinear field theory [8].It would be interesting to see if such topological solitons can be stably realised in biaxial LC systems [22,76,77].Also, since topological solitons and colloids can both form crystals, it is tantalising to consider the possibility of hybrid colloidalsolitonic crystals in LC colloidal systems [14,61,78].Moreover, recent developments in ferroelectric LCs raise questions about how long-range dipolar interactions alter the stability of topological field configurations, and how hydrodynamic effects driven by ultralow electric field interact with solitons [79].Interestingly, in contrast to inanimate systems, biological systems made up of anisotropically shaped cells, such as tissues, embryos, and bacterial communities exhibit LC ordering and out-of-equilibrium activity from energy consumption [80][81][82][83].It remains largely unknown if topological solitons emerge in these active nematic systems, and whether nontrivial alignment field configurations play a role in their physiological functions.Finally, from a technological point of view, further studies and engineering of the stability, self-assembly, and controlled dynamics of topological solitons in different material systems will facilitate the advent of topologicalsoliton-enabled technologies.Overall, topological solitons in LCs and beyond represent a burgeoning field with interest in fundamental science and potential for practical applications.

Figure 1 .
Figure 1.Topological solitons.(a) A continuous transformation can turn a coffee mug into a donut [2].(b,c) A field (shown by the orientation of cylinders) can be topologically trivial with a uniform orientation (b) or topologically non-trivial (c) hosting a π-twist soliton.(b) and (c) cannot smoothly transform to each other without introducing discontinuities when the orientation at the end points is held fixed.Note the π-twist soliton circled in red in (c) could translate by reorientation of the field.(d) A 2D topological soliton (skyrmion) in a uniform field background shown by arrows colored according to their orientation.(e) The director field n r ð Þ is the average orientation of anisotropic molecules in nematic LCs.A 5CB molecule is shown as an example.n r ð Þ can be vectorized by smoothly decorating the non-polar line field with arrows.Vectorization of n r ð Þ can also be physically achieved by the dispersion of ferromagnetic particles with magnetization vectors m r ð Þ aligning with n r ð Þ.The orientation of vectorized n r ð Þ is visualized by a 3D arrow.Reproduced with permission from ref [3].(f) Schematic of the helical pitch p in the helical state of n r ð Þ shown by ellipsoids.(g)The target space (order-parameter space) of a field is S 1 for a 2D unit vector (left) and S 2 for a 3D unit vector (right).(h) Homotopy groups used for classification of topological solitons.The green, yellow, and blue highlight topologically nontrivial configurations discussed in this review, while the π 3 S 3 À � solitons (red) arise in Skyrme's nucleon model.(i) A 2D unit vector field in R 1 with a 2π rotation forming a π 1 S 1 À � soliton with a topological degree of 1.The uniform far-field in R 1 (represented by the red arrow pointing upwards) allows the configuration space R 1 to be compactified into S 1 .(j) The target space of a 3D director is S 2 =Z 2 , namely, S 2 with diametrically opposite points identified.Reproduced from[4] with permission.

Figure 2 .
Figure 2. 1D twist walls.(a) A twist wall in chiral LCs.(b) Schematic of a twist wall in the director field n r ð Þ with head-tail symmetry going through a full π rotation shown by cylinders.(c) The director of the twist wall in (a) winds around the order-parameter space S 1 =Z 2 exactly once and S 1 =Z 2 ffi S 1 .(d) A translationally invariant CF-3 along the direction perpendicular to the cross-section with strong perpendicular boundary conditions with the 2D director confined in the yz-plane (top) and unconfined 3D director (bottom).The two half-integer defect lines near the substrates are visualized by the reduction of the scalar order parameter S at the defect cores.(e) Experimental polarizing optical micrograph (left) and 3PEF-PM image (right) obtained with circular polarization in the vertical plane between the arrows in the left panel of a CF-3.d ¼ 10 µm and p ¼ 12:5 µm.(f) Experimental polarizing optical micrograph (left) and 3PEF-PM image (right) of twist walls obtained with circular polarization in the mid-plane perpendicular to the far field in a cell with d ¼ 0:8 µm and p ¼ 1 µm.The 3PEF-PM image in vertical cross-section through the yellow dashed line is shown in the bottom right.(g) Structural stability diagram of twist wall, CF-3, and uniform states dependent on the LC film thickness over pitch ratio d=p and effective anchoring strength.L is the elastic constant in Landau-de Gennes theory, which is related to the average frank elastic constant by L ¼ 2K=9S 2eq .S eq is the equilibrium scalar order parameter.Reproduced with permission from[3].

Figure 3 .
Figure 3. 2D skyrmions and skyrmion bags.(a) Skyrmions in R 2 (bottom) can be mapped bijectively from field configurations in S 2 (top) through stereographic projections (p).The Neel-type (bottom-left) and bloch-type (bottom-right) skyrmions are related by a rotation (ℛ) of vectors.The vector orientations are shown as arrows colored according to their orientations in target space S 2 .(b) A translationally invariant skyrmion along its symmetry axis (left) and a skyrmion terminating at point defects due to boundary conditions (right).The detailed n r ð Þ in spheres around the point defects are shown.(c) Experimental polarizing optical micrograph (top) and 3PEF-PM images in horizontal and vertical cross-sections (bottom) of a 2D skyrmion.d ¼ 0:8 µm and p~1 µm.(d) Experimental polarizing optical micrograph (top) and 3PEF-PM images in horizontal and vertical cross-sections (bottom) of a toron.d ¼ 1:05 µm and p ¼ 1 µm.(e,f) Structural stability diagrams of skyrmion, toron, and the uniform state with elasticity of one-constant approximation (e) and 5CB (f).Blue, red, and black squares indicate stable 2D skyrmion, toron, and uniform state.Filled (unfilled)squares indicate the uniform (translationally invariant configuration; TIC) state is the lower energy state, and a yellow (cyan) background indicates the solitons have lower (higher) energies than the uniform unwound state.(g,h) Skyrmions embedded in a uniform far field n 0 (g) and a helical field background with a constant helical axis χ 0 (h), respectively.(i) Snapshots of polarizing optical micrographs of a helical-background skyrmion terminating at a pair of monopoles (defects) at different voltages.(j) Separation velocity of the monopole-antimonopole pair vs. the applied voltage U. (k) Snapshots of polarizing optical micrographs of two monopoles connected by a skyrmion in the helical background of a chiral LC ferromagnet moving close to or away from each other when an external magnetic field H is applied parallel or antiparallel to magnetization m s , the average m r ð Þ within the skyrmion, respectively.(l) polarizing optical micrographs of skyrmion bags S(1) to S(4), two stable conformations of the S(13) bag, and the S(59) bag.N sk ¼ 0; 1; 2; 3; 12; 12; 58, respectively.(a-f) Reproduced from ref[3].(g-K) Reproduced from ref[38].(l) Reproduced from ref[4].All with permission.

Figure 4 .
Figure 4. 3D hopfions.(a) Preimages in R 3 (and S 3 ) corresponding to distinct points in S 2 form Hopf links ð2 2 1 Þ with linking numbers matching their Q ¼ 1 Hopf index.m r ð Þ is a S 3 !S 2 map and is classified by the 3 rd homotopy group π 3 S 2 Þ ¼ Z À .The stereographic projection ϕ relates smooth configurations in S 3 and R 3 when embedded in a uniform far field.(b) Illustration of a preimage of m r ð Þ of a point in R 3 as the region corresponding to a constant orientation in R 3 .(c) In a hopfion, the circle-like preimages of color-coded points in S 2 (shown as cones) form nested tori in R 3 .(d) Polarizing optical micrographs of a Q ¼ 1 hopfion in chiral LC ferromagnets showing the polar response to an applied magnetic field.(e) Polarizing optical micrographs of a 2D array of Q ¼ 1 hopfions.(f) Numerically simulated cross-sections of the Q ¼ 1 hopfion structure taken in the plane orthogonal to the far field m 0 (top) and in the vertical plane parallel to m 0 (bottom).The vector field is shown by cones colored according to orientation in S 2 .(g) A pair of experimentally reconstructed preimages from 3PEF-PM images (left) and numerically simulated (right) preimages from (f). (h,i) Experimental images of north-and south-pole points in S 2 ð�ẑ orientations, h) and points on the equator with different azimuthal angle in S 2 (i).The corresponding simulated preimages are shown in the bottom-left insets from the m r ð Þ configuration shown in (f).(a) Reproduced from Ref. [58].(f-i) Reproduced from Ref. [13].All with permission.

Figure 5 .
Figure 5. Topological transformation of hopfions.(a) Experimental (Exp.) and simulated (Sim.)polarizingoptical micrographs of Q ¼ À 2 hopfions in a negative Δε chiral LC under different applied voltages of 0, 3.6, and 5.0 V. (b-d) Numerically simulated 3D preimages in R 3 of points in S 2 indicated as cones in the top-right insets.(b), (c), (d) correspond to the solitons shown in (a) under applied voltages of 0, 3.6, and 5.0 V, respectively.In (b), preimages of distinct points in S 2 form a pair of Hopf links with a total linking number −2.In (c), a preimage can be a single loop (0 1 ) or two separate closed loops (0 2 1 ) with different subspaces of S 2 separated by a boundary at θ C ¼ 85 � ; for all combinations of distinct points in S 2 , the total linking number is -2.In (d), a preimage is a single loop in S 2 and the linking number is −1 for all pairs of distinct points in S 2 .(e) θ C and Q of the soliton versus U, with abrupt changes of θ C at U ¼ 2:6V and Q at U ¼ 4:2V.Schematics of preimage linking within each voltage range are shown as insets.(f-g) Experimentally reconstructed preimages of diametrically opposite points on the equator and at the poles in S 2 for a Hopf soliton at U ¼ 0 (f) and at U ¼ 3:6V (g).The simulated counterparts are shown in the bottom-right insets.(h) Vertical mid-plane cross-section of a stable

Figure 6 .
Figure 6.Heliknotons -a new class of 3D knot solitons in the helical background of chiral LCs.(a) Helical field comprising a triad of orthonormal fields n r ð Þ,χ r ð Þ, and τ r ð Þ. n r ð Þ can be a polar vector field (left) or a nonpolar line field (right).(b) Schematics showing preimages in n r ð Þ form inter-linked loops with a linking number of 1 for each pair of preimages (top), and knotted half-integer vortex lines in χ r ð Þ and τ r ð Þ (bottom).(c) Two heliknotons interact in 3D while forming a dimer.(d) Pair interaction of heliknotons.Different symbols represent samples with different LC materials or applied voltage.See ref [14] for more details.(e,f) 2D closed rhombic (e) and open (f) lattices of heliknotons.(g) Crystallites with aligned (top) and anticlinically tilted (bottom) heliknotons at U = 1.8 V and 2.3 V, respectively.(h) 3D heliknoton lattice comprising crystallites with perpendicular orientations.The inset is obtained when focusing in different crystalline planes ∼10 μm apart.(i) Electrostriction of a heliknoton crystal.Insets show lattices at different U, with the lattice parameters shown in blue and red.(j) Experimentally reconstructed interlinked preimages of �ẑ orientations in a heliknoton.The inset

Figure 7 .
Figure 7. Geometric transformation and 3D hopping of Hopf solitons.(a) Schematics of n r ð Þ in the uniform, helical, and conical state, respectively.The conical state is at a cone angle θ cone with respect to the helical axis χ 0 .(b) Hopf solitons observed in chiral LCs using bright-field microscopy in a uniform, helical, and conical background, respectively.(c) Preimages of vector orientations (shown in insets as arrows in the order-parameter space S 2 ) of Hopf solitons stabilized in a uniform, helical, and conical background, respectively.(d, e) Structural stability diagrams of LC Hopf solitons in the bulk (d) and within a confined volume with perpendicular boundary condition and d ¼ 3p 0 (e).Here p 0 is the equilibrium pitch at no field.The data points are colored based on the stability of Hopf solitons, and the contour lines of background mid-plane cone angle θ cone are shown in each diagram.(f) Polarizing optical microscopy snapshots of a Hopf soliton switching from a heliknoton at U = 0 V (0 s) to a hopfion at U = 3.85 V (11.73 s), and back to a heliknoton at U = 0 V (48.06 s).(g) Translational and orientational displacement of a Hopf soliton by repeated voltage switching shown at its initial (left) and final (right) position.The 2D trajectory is color-coded by time and the long-axis orientations of the Hopf soliton in helical background at intermediate positions are shown by double arrows.(h) Distance and accumulated change in orientation in each transformation cycle (Δϕ) of a hopping Hopf soliton shown in (g).(i) The different long-axis orientations of Hopf solitons in a helical background with perpendicular confinement.(j) Snapshots of polarizing optical micrographs of a Q ¼ 1 Hopf soliton subject to background modulation by a modulating voltage profile shown in the top.The modulation period T = 2 s. (k) Squirming of a Q ¼ 1 Hopf soliton shown by superimposed polarising optical micrographs at different times.

Table 1 .
Material parameters of nematic LCs.