Diversity of knot solitons in liquid crystals manifested by linking of preimages in torons and hopfions
DD r a f t Diversity of knot solitons in liquid crystals manifested by linking ofpreimages in torons and hopfions
P. J. Ackerman and I. I. Smalyukh Department of Physics, University of Colorado, Boulder, Colorado 80309, USA Department of Electrical, Computer and Energy Engineering, University of Colorado, Boulder, Colorado 80309, USA University of Colorado, Liquid Crystal Materials Research Center and Materials Science and Engineering Program,Boulder, CO 80309, USA Renewable and Sustainable Energy Institute, National Renewable Energy Laboratory and University of Colorado, Boulder,Colorado 80309, USA (Dated: May 9, 2016)
Abstract
Topological solitons are knots in continuous physical fields classified by non-zero Hopf index values. Despite arising intheories that span many branches of physics, from elementary particles to condensed matter and cosmology, they remainexperimentally elusive and poorly understood. We introduce a method of experimental and numerical analysis of suchlocalized structures in liquid crystals that, similar to the mathematical Hopf maps, relates all points of the medium’sorder parameter space to their closed-loop preimages within the three-dimensional solitons. We uncover a surprisinglylarge diversity of naturally occurring and laser-generated topologically nontrivial solitons with differently knotted nematicfields, which previously have not been realized in theories and experiments alike. We discuss the implications of the liquidcrystal’s non-polar nature on the knot soliton topology and how the medium’s chirality, confinement and elastic anisotropyhelp to overcome the constrains of the Hobart-Derrick theorem, yielding static three-dimensional solitons without or withadditional defects. Our findings will establish chiral nematics as a model system for experimental exploration of topologicalsolitons and may impinge on understanding of such nonsingular field configurations in other branches of physics, as wellas may lead to technological applications
When proposing one of the early models of atoms, Kelvinand Tait considered knotted vortices as candidates and at-tempted to explain the diversity of chemical elements in theperiodic table as that arising from a large variety of knots,giving origins to the modern field of mathematical knot the-ory [1-3]. Even before them, Gauss envisaged that spatiallylocalized knots in physical field lines, such as magnetic orelectric field lines, could behave like particles [1], providedthat the crossing of the field lines is prevented, say, due toenergetic reasons. Hopf later rigorously demonstrated that,indeed, inter-knotted closed loops could be smoothly embed-ded in a uniform far-field background, introducing the cele-brated mathematical Hopf fibration [1,3,4]. Finkelstein ap-plied these mathematical concepts and Hopf mappings tothree-dimensional (3D) physical fields [5], so that the 3D topo-logical solitons based on them subsequently started attractinginterest (mostly theoretical) in many branches of physics [6-22]. However, the experimental realizations and demonstra-tions of topological solitons with knotted field lines typicallydeal only with transient phenomena and out-of-equilibriumsystems or are accompanied by observation of additional de-fects [3,6-22] and their detailed explorations are often hin-dered by the need of 3D spatial imaging of the physicalfields. Moreover, according to the Hobart-Derrick theorem[7,8], physical systems cannot host the static 3D solitons incontinuous fields described within the simplest field theoriesbecause of energetic reasons, except for within the nonlineartheories with higher-order derivatives, such as the Skyrme-Faddeev model [6,9,10]. Thus, not surprisingly, the challengeof reliable experimental realization and robust control of topo-logical solitons persisted for decades.Since the 3D topological solitons smoothly embed into the uniform far-field background, their solitonic field configura-tions in the 3D space can be effectively “compactified” toa 3-sphere S and the field topology is then characterized bythe S → S maps, bringing about the beautiful analogy withthe famous mathematical Hopf and Seifert fibrations [1,3,4].Within the homotopy theory classification of topological de-fects, the 3D solitons in vector fields, with a two-dimensionalsphere S as the ground-state manifold, belong to the thirdhomotopy group π ( S ) = Z and are classic examples of non-singular topological defects. In the nonpolar systems withline fields, such as the director field in nematic liquid crystals(LCs) [23] that describes the spatial changes of the local aver-age orientation of rod-like molecules, the ground-state man-ifold is a projective plane R P (or, equivalently, the spherewith antipodal points identified, S /Z ), so that the corre-sponding 3D topological solitons are labeled as π ( R P ) or π ( S /Z ) = Z [14,22,23]. An intriguing and important fea-ture of these topological solitons is that they have closed-looppreimages (regions within the 3D sample that have the sameorientation of the physical field) corresponding to all points ofthe order parameter space (e.g. S for the vector fields) andthat each two distinct preimages are linked with each otheran integer number of times. This linking number is the so-called Hopf index topological invariant, Q , characterizing thetopology of the 3D topological solitons.The scientific and technological potential of 3D topologi-cal solitons can be appreciated by considering the recent ad-vances in the studies of their two-dimensional counterpartscalled “skyrmions” or “baby skyrmions” [24-30], which be-long to the second homotopy group π ( S ) = Z . Being asubject of purely theoretical studies a decade ago or so [24],they have been recently successfully realized in experimentalsystems, both in the form of isolated individual solitons andas novel phases with arrays of such solitons in the ground-1 a r X i v : . [ c ond - m a t . s o f t ] A p r r a f t state [25-30]. These skyrmions attract a great deal of fun-damental interest and define foundations for skyrmionics andother emerging technologies [25-30], albeit their experimen-tal study is still largely restricted to chiral condensed mattersystems such as non-centrosymmetric ferromagnets and chi-ral nematic LCs [29,30]. One condensed matter system thathas been considered for experimental realization of the 3Dtopological soliton structures with non-zero Hopf index valuesis a cholesteric LC. Short-pitch cholesteric LCs are topolog-ically similar to the still elusive biaxial nematic LCs [31,32]and are characterized by three mutually orthogonal nonpolardirector fields [23]. Some of the transient localized directorstructures from experimental observations [33] could be po-tentially interpreted as being nonsingular in one of these threedirector fields while also having nonzero Hopf invariants [31],albeit unambiguous demonstration of this topological natureof such structures remained impossible due to the lack andlimitations of 3D director field imaging capabilities. Further-more, such localized field configurations were found only astransient structures [33] and could be potentially nonsingu-lar only in one out of the three director fields of this rathercomplex condensed matter system [31-33], thus being onlyloosely related to the hopfions envisaged to exist in manyother branches of physics [9,10], albeit realization of variouslocalized structures in biaxial condensed matter systems isalso of great fundamental interest [23,34]. With the adventof 3D director imaging capabilities [13,14,35], the reconstruc-tion of complex field configurations became possible, but onlya small variety of spatially localized 3D solitonic structureshave been found so far in LCs and chiral ferromagnets [13,14, 22, 36-40]. Furthermore, the understanding of topology,structural diversity and physical underpinnings behind thestability of such 3D solitons is still limited and calls for newexperimental and theoretical approaches in their exploration.Most importantly, there is a need for the direct experimen-tal characterization of linking of preimages and Hopf indicescorresponding to different 3D solitonic structures.In this work, we introduce a method of direct experimen-tal and numerical characterization of preimages of the 3Dsolitons. We then realize and study a series of stationary 3Dsolitons in a confined chiral nematic LC [23] with the nonpo-lar director field n ( r ) describing spatial changes of the localaverage molecular alignment direction of the constituent rod-shaped molecules. By using a combination of a direct 3Dnonlinear optical imaging and numerical modeling throughminimization of the free energy that both yield 3D n ( r ) spa-tial patterns, we construct the soliton preimages correspond-ing to all distinct points of the order parameter space. Froma large variety of experimentally realized solitonic structures,we focus on solitons with inter-linked preimages in the formof closed loops. These solitons are characterized by preim-age linking numbers and the corresponding nonzero topolog-ical Hopf index invariants Q , different from the 3D localizedfield configurations of elementary torons with Q = 0 thatwe studied previously [13]. Numerical modeling provides in-sights into the role of the medium’s chirality, confinement, andelastic constant anisotropy in enabling the stability of these3D solitons. We discuss how a combination of these factorshelps to overcome the constraints of the Hobart-Derrick the-orem [7,8] and how our findings may provide insights into the prospects of obtaining stable topological solitons in otherbranches of physics, both within condensed matter and wellbeyond it. Furthermore, the experimental platform we havedeveloped may lead to technological applications building onthe particle-like nature of topological solitons as well as to therealization of topological solitonic condensed matter phases. The ground-state manifold of a physical system with a char-acteristic vector quantity as an order parameter, such as themagnetization in a ferromagnet, is a two-dimensional unitsphere S . An important property of the π ( S ) = Z topo-logical solitons with nonzero Hopf invariant, such as hopfions,is that the preimages of all points on S are closed loops(Fig. 1a), which are topologically equivalent to one-spheres S . A preimage of the north-pole point of S closes into an S − like loop through infinity and is typically associated witha uniform far-field background embedding the soliton. TheHopf index invariant can be determined as the linking num-ber of any two preimages of two distinct points on S andalso describes how many times S is swept in the processof mapping the vector field from the 3D space of the soli-ton (and the corresponding S ) to S . The topological Hopfindex of a soliton is commonly found as the linking number Q = (cid:80) C/ S (Fig. 1) in the case of vectorized n ( r ), where the signof crossings C = ± n ( r ) ≡ − n ( r ), the ground-state manifold S /Z is a spherewith the diametrically opposite points identified. The impli-cation of the LC’s nonpolar symmetry is that the preimageswith all single orientations of n ( r ) cannot be distinguishedfrom those with − n ( r ) (Fig. 1b). Since the director fieldcan be vectorized and since there is a theorem stating thatall nonsingular π ( S /Z ) structures in the director field arealso nonsingular in the two co-linear mutually opposite vec-tor fields decorating it [31], it is often possible to determinethe Hopf index invariant from the linking of the two closed-loops of a single preimage corresponding to a single point on S /Z (which would correspond to the preimages with twosingle orientations n and − n in the anti-parallel vector fieldsdecorating the director field, as shown in Fig. 1b). To explorethe implications of the nonpolar nature of the director field n ( r ) on the 3D soliton topology and to compare the solitonsin director and vector fields, we use a color scheme for S /Z that is consistent with the identification of its diametricallyopposite points (Fig. 1c). We illustrate the spatial patternof the director with the help of appropriately colored doublecones (Fig. 1c), as opposed to the single cones that we usefor visualizing vectors and the vectorized n ( r ).The 3D space within a topological soliton is smoothly filledwith closed loop preimages that reside on nested tori (Fig.1d,e). For the elementary solitons in vector fields or in vec-torized n ( r ) (Fig. 1d), the torus circular axis and the far-fieldbackground correspond to the south pole and the north pole of S , respectively. In the case of a director field n ( r ) ≡ − n ( r )2 r a f t with a nonpolar nature, both the far-field and the circularaxis of the soliton have the same up-down orientation parallelto the vertical z-axis (Fig. 1e). For the simplest solitons, thetopological Hopf invariant Q can be found as a linking num-ber of preimages of any two distinct points on S (Fig. 1d)in the case of vectorized n ( r ) or, equivalently, as the linkingFigure 1: 3D topological solitons in chiral nematic LCs. (a-c)Examples of closed-loop preimages of 3D topological solitonscorresponding to (a) the two distinct points on S (shownusing cones) for vectorized n ( r ) and (b,c) two distinct pointson S /Z and nonpolar n ( r ), with the ground-state mani-fold S /Z depicted as a sphere with diametrically oppositepoints identified through coloring and making the n and − n orientations non-distinguishable. Note that the diametricallyopposite points in (b) appear simultaneously, so that the twopoints on S /Z are four points on S for the vectorized di-rector field. (d,e) Illustrations of Hopf maps of closed-looppreimages of 3D topological solitons embedded in a uniformfar-field n onto the (d) S ground-state manifold for vector-ized n ( r ) and (e) S /Z order parameter space of the LC fornonpolar n ( r ). The schematics show linking of the hopfion’scircle-like preimages that reside on nested tori in the sample’s3D space and correspond to color-coded points (cones) on S or S /Z . (f,g) Polarizing optical micrographs of the stud-ied localized field configurations co-existing within the sameLC sample. The images were obtained between crossed po-larizers (white double arrows). The green circles with crossesvisible in the micrographs show locations of “invisible” in-frared laser traps used to manipulate the naturally occurringlocalized field configurations. The micrographs were obtainedfor a 5CB-based partially polymerizable nematic LC mixturewith chiral additive cholesteryl pelargonate, d = 10 µ m and d/p ∼
1. number of two loops that comprise a preimage of a single pointon S /Z in the case of analyzing the nonpolar n ( r ) ≡ − n ( r )field (Fig. 1e). However, we will also show that there are morecomplex solitons that require a detailed analysis of preimageswith complex linking. For chiral nematic LCs with helicoidal pitch p , the elastic freeenergy cost for producing spatial deformations of n ( r ) reads[13,23] F = (cid:90) dr (cid:26) K ∇· n ) + K n · ( ∇× n )]+ K n × ( ∇× n )] + K q n · ( ∇ × n ) − K [ ∇ · [ n ( ∇ · n ) + n × ( ∇ × n )]] (cid:27) (1)where q = 2 π/p characterizes the LC chirality and the Frankelastic constants K , K , K and K describe the ener-getic costs of the splay, twist, bend, and saddle-splay elasticdeformations, respectively [23]. Within the K = K = K = K one-constant approximation and while neglectingthe energetic cost due to the saddle-splay term and constantterms, Eq. (1) reduces to the form F = (cid:90) dr (cid:26) J ( ∇ n ) + D n · ( ∇ × n ) (cid:27) (2)where J = K/ D = Kq . The free energy within theone-constant approximation is analogous to one of the mostcommon forms of the Hamiltonian describing solid-state chiralferromagnets, where coefficients J and D in this case describethe strengths of exchange energy and the Dzyaloshinskii-Moriya coupling, respectively [23,24,41]. In the case of D = 0(for the non-chiral systems), Eq. (2) further reduces to theHamiltonian in a form that was considered by Hobart andDerrick [7,8] to demonstrate instability of the localized non-singular 3D field configurations within such a model. Forthe solitons to be stable in chiral nematic LCs, they need toemerge as local or global minima of the free energy given byEqs. (1) and (2) [23]. In addition to chirality, which tendsto stabilize the twisted solitonic structures and may help toovercome the constrains of the Hobart-Derrick theorem [7,8],the anisotropy of elastic constants in Eq. (1), confinementand the surface anchoring at confining surfaces may serve asimilar role. Chiral nematic LCs are dielectric and diamag-netic materials that respond to external fields to minimize thecorresponding terms of the free energy. For example, in thecase of the applied electric fields the Eqs. (1) and (2) needto be supplemented by adding the corresponding electric fieldcoupling term of the free energy F electric = − ( (cid:15) ∆ (cid:15)/ (cid:90) dr ( E · n ) (3)where E is the applied electric field, (cid:15) is the permittivityof free space, and the dielectric anisotropy ∆ (cid:15) can be bothpositive and negative, depending on the LC material system.This coupling term can potentially also promote stabilizationof solitonic structures.3 r a f t The analytical treatment of Eqs. (1) and (2) from thestandpoint of view of stability of complex 3D solitonic struc-tures is prohibitively difficult, especially when supplementedby field coupling terms such as the one given by Eq. (3), andwe therefore resort to the numerical minimization of free en-ergy to arrive at a host of field configurations correspondingto local and global energy minima. We note that the freeenergy of chiral nematic LCs could be also described withinthe Landau-de Gennes tensorial approach [23,42], which is amethod of choice in the cases of structures with singular linedefects [42, 43], but the Frank-Oseen free energy descriptionis more suitable for modeling nonsingular field configurationssuch as the hopfions as well as torons with the additionalpoint singularities [13]. This approach not only allows us todo the full treatment of anisotropic elastic properties of LCs,explore the implications of the one-constant-approximationand the relevance to systems such as chiral ferromagnets, butalso to operate with large sizes of grids and samples in thenumerical modeling, which is critically important for eluci-dating the numerical results we present. Furthermore, thisFrank-Oseen approach and modeling based on Eqs. (1) and(2) also allow us to better connect our work to the efforts ofobserving topological solitons in other branches of physics.
To assure a broad impact of our work and its accessibilityto other researchers, we use commercially available LC ma-terials pentylcyanobiphenyl (5CB, Frinton Laboratories, Inc.)and ZLI-2806 (EM Chemicals). To obtain chiral nematic LCs,these nematic hosts are doped with small amounts of chiraladditives, either the left-handed dopant cholesteryl pelarg-onate (Sigma-Aldrich) or the right-handed chiral dopant CB-15 (EM Chemicals). The chiral additive is added to the ne-matic host at a weight fraction calculated as c = ( ξ · p ) − ,allowing us to define the helicoidal pitch p of the ensuingchiral LC, where ξ is the helical twisting power of the addi-tive in the nematic host [30,36]. The obtained equilibriumpitch value is then probed using the Gradjean-Cano method[13,22]. In addition, a polymerizable nematic mixture of 5CB(69%) with 12% of RM-82 and 18% of RM-257 reactive diacry-late nematics and 1% Irgacure 184 photoinitiator (all fromCIBA Specialty Chemicals) [38] is also doped with the sametwo chiral agents to obtain partially polymerizable chiral ne-matic LCs. The initial powder mixture is first dissolved indichloromethane to homogenize, kept at an elevated temper-ature of 85 ◦ C for one day to remove the solvent, and cooleddown to obtain the final chiral nematic mixture. We haveoptimized the polymerization process by using relatively lowlight exposures for cross-linking of the cholesteric films, sothat the 3D structures of torons and hopfions can be “frozen”by polymerization in a solid film without altering their direc-tor configurations [38]. The polymerization is achieved usingrelatively weak ultraviolet exposure by means of a home-builtsetup with a 20 W mercury bulb (obtained from Cinch) [38].The unpolymerized 5CB within this partially cross-linked sys-tem is partially removed by addition of isopropanol and sub-sequently replaced with immersion oil. This allows us to re- duce the medium’s effective birefringence by about an orderof magnitude (estimated to be ≈ n orientation were prepared by sandwiching the LC mix-tures between glass plates with well-defined perpendicular(homeotropic) surface boundary conditions [13]. The thin(150 µ m) or thick (1 mm) glass substrates forming cells weretreated with a homeotropic polyimide SE1211 (obtained fromNissan Chemicals). The preparation of these alignment layersinvolved spin coating the polyimide on substrates at 2700 rpmfor 30 s and then baking it for 5 min at 90 ◦ C, followed byadditional baking for 1 h at 180 ◦ C. This treatment sets thestrong perpendicular boundary conditions for n ( r ) at the LC-glass interface. LC cells with the gap thickness d = 5 − µ mwere produced using glass microspheres of the correspondingdiameter or Mylar films of corresponding thickness interspac-ing the glass substrates. In some cases, wedge-shaped cellswith small dihedron angles ∼ ◦ were prepared and studiedfor a detailed exploration of the effects of cell thickness d relative to p on the soliton stability. To avoid the nematicflow-induced alignment effects, the cells were filled at an el-evated temperature, right above the LC-isotropic transition,and then cooled down to the room-temperature LC phase.The 3D solitons of different types sometimes appeared duringthe temperature quench from isotropic phase spontaneously,but could be also generated and robustly controlled using lasertweezers when in the LC phase, as discussed below. The experimental identification of 3D topological solitons re-lies on the analysis of preimages constructed on the basisof the nonlinear optical imaging of n ( r ) within these struc-tures. This imaging was performed using three-photon exci-tation fluorescence polarizing microscopy (3PEF-PM) setupbuilt around a BX-81 Olympus inverted optical microscope[37,38]. The polarized self-fluorescence from the LC moleculeswas detected within the 400-450 nm spectral range and ex-cited through a process of three-photon absorption using aTi-Sapphire oscillator (Chameleon Ultra II, Coherent) oper-ating at 870 nm with 140 fs pulses at a repetition rate of80 MHz. The 3PEF-PM signal was collected through an oil-immersion 100 × objective with numerical aperture NA=1.4and detected by a photomultiplier tube (H5784-20, Hama-matsu). We scanned the excitation beam through the sam-ple volume with the help of galvano-mirrors (in lateral direc-tions) and a stepper motor (across the sample thickness) andrecorded the 3PEF-PM signal as a function of coordinates,which was then used to construct 3D images by means of theParaView software (freeware obtained from KitwarePublic).The linear polarization of the excitation beam was controlledusing a polarizer and a rotatable half-wave retardation plate.The detection channel utilized no polarizers. The 3PEF-PMintensity scaled as ∝ cos ψ , where ψ is the angle between n ( r ) and the excitation beam’s linear polarization (assumedto remain unchanged despite the beam focusing through di-4 r a f t electric interfaces and the weakly birefringent LC medium,with the sample design minimizing these changes) and wasused to reconstruct the 3D n ( r ) patterns [35, 38].The reconstruction of the 3D solitonic n ( r )-structurestook advantage of the self-fluorescence patterns obtained atdifferent polarizations of excitation light, as described else-where [30, 35-38]. In order to eliminate the ambiguity be-tween the two possible opposite n ( r ) tilts in the analysis of3D images, additional cross-sectional 3PEF-PM images wereobtained at orientations of the LC cell’s normal tilted by ± ◦ with respect to the microscope axis for linear polarizationsof excitation laser light parallel or perpendicular to the planeof the corresponding vertical cross-sectional image. The n ( r )tilt ambiguity was then eliminated based on the ∝ cos ψ scal-ing of the 3PEF-PM signal and the ensuing spatial changesof intensity prompted by the ± ◦ tilts. To further narrow theangular sector of n -orientations corresponding to preimages ofpoints on S /Z with target azimuthal angles φ , we each timeobtained three 3D images with azimuthal orientation of thelinear polarization of excitation beam at φ and φ ± ◦ . These3D images were smoothed using Matlab-based software andthen used in a differential analysis to improve orientationalresolution to better than ± ◦ . Consistent with the nonpolarnature of n ( r ) and the ground-state manifiold of the LC [23],this 3D imaging yields preimages corresponding to a singlepoint on S /Z (without distinguishing the n and − n orienta-tions). These experimentally reconstructed n ( r )-patterns canbe vectorized by exploiting the continuity of the director field,which was done for some types of analysis that we performed.To further probe the topology of 3D solitons with nonzeroHopf indices, we then used an experimental procedure equiv-alent to the mathematical Hopf mapping (Fig. 1d,e) in orderto relate all inter-linked closed-loop preimages in the LC sam-ple’s volume with all corresponding distinct points on S /Z .This new approach allowed us to experimentally probe link-ing of preimages within the 3D solitons and determine theirHopf indices, revealing a surprisingly large variety of topo-logical solitons described below. The 3D nonlinear opticalimaging was supplemented by conventional polarizing opticalmicroscopy observations in the transmission mode by usingthe same multi-modal imaging setup built around the BX-81Olympus inverted microscope (part of the 3PEF-PM setupdescribed above) and a charge coupled device camera (Flea,PointGrey). To reliably control their topology, the 3D solitons were gener-ated with optical tweezers. This optical generation utilized anYtterbium-doped fiber laser (YLR-10-1064, IPG Photonics,operating at 1064 nm) and a phase-only spatial light modula-tor (P512-1064, Boulder Nonlinear Systems) integrated intoa holographic laser tweezers setup capable of producing arbi-trary 3D patterns of laser light intensity within the sample[13,22,30]. The laser tweezers were also integrated with the3D imaging setup described above [30], enabling fully opticalgeneration, control, and nondestructive imaging of the soli-tons. The physical mechanism behind the laser generationof solitons is the optical Fr´eedericksz transition, the realign- ment of the LC director away from the far-field background n caused by its coupling to the optical-frequency electric fieldof the laser beam, which is described by a corresponding termof free energy [13], similar to that given by Eq. (3) and dis-cussed above for the case of low-frequency electric fields. Thiscoupling, enriched by holographically generated patterning ofthe trapping laser beam’s intensity, phase singularities andtranslational motion of the individual traps [13,14,22,30,35-38], prompts complex director distortions that then relax toglobal or local elastic free energy minima, some of which arethe topologically nontrivial solitons of interest to this study.For example, the 3D solitons with Hopf indices Q = ± n by moving the laser focus of the holographic optical trapalong a circular trajectory within the LC cell’s mid-plane.By limiting the laser power to about 50 mW and by con-trolling the winding direction and depth of the circular lasersbeam motion, we pre-selected generation of the Q = 1 or Q = − >
200 mW, so that the spontaneous appearance ofsolitons could be then prompted upon quenching it back tothe LC phase in a way similar to that after an initial entire-sample quench during the sample preparation. By repeat-ing this laser-induced heating and subsequent cooling manytimes, we generated the desired structures despite the lowprobability/yield of inducing topological solitons. Althoughthe studied 3D solitons could be generated using both of theabove approaches and under multiple types of different con-ditions, to assure that other researchers can reproduce ourresults, we provide specific experimental details correspond-ing to all solitons that we study experimentally in the figurecaptions. Examples of polarizing optical micrographs of sev-eral localized 3D solitonic structures with different topologiesand laser-generated next to each other are shown in Fig. 1f,g.Although the conventional polarizing optical imaging does notprovide insights into the complex and beautiful topology ofthese structures (Fig. 1f,g), we are able to analyze it boththrough computer simulations and experimentally by usingthe 3D nonlinear optical imaging.All solitonic structures could be further optically manip-ulated using the laser tweezers at powers of 2 − n [36]. With thetrapping beam parallel to n , the refractive index of the soli-ton’s surrounding is the ordinary refractive index of the LCwhile that within the soliton is changing with coordinates be-tween the ordinary and extraordinary values, depending onthe orientation of n ( r ) (which is the LC’s optical axis). Theensuing effective refractive index contrast approaches the op-tical refractive index anisotropy, which is ≈ ≈ n ( r ), without any foreign inclusions inthe LC medium, their optical manipulation resembles that of5 r a f t particles and (at higher laser powers >
10 mW) can be furtherenhanced by elastic interactions between the laser-induced di-rector distortions and those due to the localized n ( r )-patternsof the solitons themselves [44]. Numerical modeling of the energy-minimizing n ( r )-structureswas performed using a relaxation routine applied to Eqs. (1)and (2). The Frank elastic constants K , K , and K thatdescribe the energetic cost of splay, twist, and bend defor-mations, respectively, as well as the average constant K , arebased on literature data for the two nematic hosts used inour study (Table 1). We assume that the surface free energydoes not need to be included in the minimization problembecause of the strong surface boundary conditions for n. Thenumerical relaxation routine calculates the spatial derivativesof n ( r ) on a computational grid using the 2nd-order finite dif-ference scheme [36]. Commonly, the periodic boundaries areimplemented along the lateral directions of the computationbox while fixed perpendicular surface boundary conditions areapplied at the top/bottom confining substrates to define theuniform far-field director n =(0,0,1). In some of these simula-tions, the vertical conditions n =(0,0,1) were also enforced atthe lateral edges of the 3D simulation box. Both the analyti-cal ansatz configurations [45] and the random fields were usedas initial conditions in the free energy minimization, yieldingsimilar results. At each time step ∆t of the numerical sim-ulation, we computed the functional derivatives correspond-ing to the Lagrange equation ∂F/ ( ∂n i ) = 0 and then alsothe resulting elementary displacement ∂n i = − ∆ t∂F/ ( ∂n i ),where the subscript i denotes orientations along the x, y, andz axes. The maximum stable time step used in the relax-ation routine is determined as ∆ t = 0 . min ( h i )) /max ( K ),where min ( h i ) is the smallest computational grid spacing and max ( K ) is the largest elastic constant appearing in Eqs. (1)or (2) (Table 1). The steady-state stopping condition is de-termined through monitoring the change of the spatially av-eraged functional derivative with respect to time. When this value asymptotically approaches zero, the system is assumedto be in a state corresponding to a local or global energy min-imum and the relaxation procedure is terminated. The 3Dspatial discretization was performed on fairly large 3D grids,such as the 112 × ×
32 grid. This allowed us to excludediscretization-related artifacts influencing the structural sta-bility of solitons.For the grid spacing of h x = h y = h z = 1 µ m and 32 gridpoints across the cell, the effective LC cell thickness d = 32 µ m was tuned to match that used in experiments; for LC sam-ples of other thickness d mimicking that of experimental cells,the h-values were adjusted accordingly. In order to speed upthe relaxation of energy-minimizing n ( r )-configurations corre-sponding to local or global energy minima, the minimizationwas additionally performed with a relaxation method for atwo-dimensional grid of equally spaced points that was usedassuming the axial symmetry and then rotated around theaxial symmetry axis to obtain a volume of equally spacedvoxels on a 3D grid with corresponding n ( r )-orientations.The grid spacing in this case was equal in all directions anddiscretized into 192 × ×
64 points. The free-energy-minimizing computer-simulated n ( r )-configurations obtainedusing different grids and discretization approaches were ana-lyzed and compared to each other and to experiments throughthe generation of 3D iso-surfaces (including the preimages ofsingle director orientations). This comparison allowed us toassure that our findings are independent of the type of griddiscretization. In order to construct preimages within the 3Dvolume of the static topological solitons, we calculated themagnitude of the difference between a unit director (vector)defining a target point on S /Z ( S ) and the solitonic 3D n ( r ), which was then visualized with the help of an isosur-face of a small value in the ensuing scalar field that confinesa 3D volume of the preimage. Combined with the experi-mental method of construction of preimages described above,this procedure allowed us to unambiguously assure the corre-spondence between numerically simulated and experimentalsolitonic structures.Table 1. Elastic constants of the nematic LC hosts and helical twisting power ξ of the used chiral additives in nematic hosts.Nematic LC host K , pN K , pN K , pN K , pN ξ of cholesterol pelargonate, µm − ξ of CB-15, µm − Q = 0 Our integrated numerical and experimental approach of imag-ing preimages and identification of the 3D topological soli-tons is analogous to the Hopf mapping from the 3D space, R , to the ground-state manifold of the LC, S /Z , or S forthe case of vectorized n ( r ). The LC sample’s spatial regionswith director field n ( r ) orientations corresponding to targetpoints on S /Z (the preimages) are imaged sequentially byvarying linear polarization of the 3PEF-PM excitation light,using the differential analysis to improve orientational sen- sitivity of this approach, and eliminating the director tiltambiguity (see the methods section above). This nonlinearoptical imaging of preimages within the solitons is based onorientation-dependent self-fluorescence of rod-like moleculesof the LC that (on average) locally align with n ( r ). Consis-tent with the nonpolar symmetry of the LC, this 3PEF-PMbased imaging approach simultaneously yields pairs of preim-ages corresponding to n and − n , i.e. to a single point onpoints on S /Z and to two diametrically opposite points on S in the case when this line field is vectorized along n or − n . The far-field vertical alignment of n =(0,0,1) set by thestrong boundary conditions at the confining surfaces, alongwith the continuity of n ( r ) evidenced by absence of singular6 r a f t light-scattering defects, provide the foundations for analyz-ing n ( r ) structures based on the strong ∝ cos ψ orientationaldependence of the 3PEF-PM signal. The numerical analogsof the experimentally reconstructed preimages are obtainedfrom analyzing the 3D director patterns that minimize theelastic free energy, as described above.The simplest observed 3D solitonic structure is shown inFig. 2. The in-plane and vertical cross-sections of its spa- tially localized n ( r ) are presented with the help of double-cones, which are colored according to two different schemes(Fig. 2a-d), both designed to be consistent with the nonpo-lar nature n ≡ − n of the director field. The coloring schemethat show the two cones within each double-cone in differentcolors corresponding to that of respective diametrically oppo-site points on S also allows one to analyze the two mutuallyopposite vector fields vectorizing n ( r ) (Fig. 2a,c). TheFigure 2: 3D solitons with Q = 0. (a-d) Computer simulated (a,b) in-plane and (c,d) vertical cross-sections of n ( r ) depictedusing double cones and two different color schemes that establish correspondence between director orientations and the pointson S /Z (top-right insets). (e,f) Computer-simulated preimages of a Q = 0 3D soliton for two sets of the diametricallyopposite points on S /Z marked by double cones in the top-right insets for two different coloring schemes of the S /Z .(g,h) Comparison of representative (g) computer-simulated and (h) experimental preimages of the Q = 0 soliton for twodiametrically opposite points on the “equator” of S /Z (top-right insets). Gray arrows in (e-h) indicate the consistentlydetermined circulations of the preimages. (i,j) Perspective views of the 3D computer-simulated isosurfaces of normalized (i)handedness and (j) free energy density for this axially symmetric solitonic structure, with the used color schemes provided inthe right-side insets. The experimental preimages were reconstructed based on 3PEF-PM images obtained for structures in a5CB-based partially polymerizable nematic LC mixture with chiral additive CB-15, d = 10 µ m and d/p ∼
1; before imaging,the unpolymerized 5CB was replaced by immersion oil. Computer simulations were preformed for 5CB elastic constants andassuming K = 0. 7 r a f t nonsingular axially symmetric structure of this soliton intro-duces a spatially localized twisted region embedded in theuniform far-field n . Mapping the vectorized n ( r ) from thesoliton’s cross-section (Fig. 2c) onto S does not fully coverit, albeit the similar mapping of nonpolar n ( r ) onto S /Z covers this ground-state manifold fully, with some parts of S /Z covered twice (Fig. 2d). A similar analysis can bedone for preimages of vectorized and nonpolar n ( r ) in the en-tire 3D volume of the soliton. The vectorized n ( r )-patternof these solitons has closed-loop preimages for a majority ofpoints on S (see examples in Fig. 2e), except for the vicin-ity of its south pole. In the case of the nonpolar n ( r ) andthe S /Z ground-state manifold, the preimages of most ofthe points are pairs of unlinked closed loops (Fig. 2f), albeitone of the two loops shrinks into a disc and disappears forpoints near the north/south pole of S /Z . The computer-simulated preimages closely match their experimental coun-terparts, as shown by comparing examples of two-loop preim-ages of the same single point (marked by the blue doublecone) on S /Z in Fig. 2g,h, with the consistently chosencirculation directions marked using curved gray arrows. Thenumerical energy-minimizing n ( r )-configuration allows us toplot the handedness of the director twist [37,46], defined as H = − n · ( ∇ × n ) and normalized by q , to observe that H matches the intrinsic chiral nematic LC’s handedness withinmost of the volume of the soliton, except for a small regionclose to to one of the confining substrates, where it reverses and becomes opposite (Fig. 2i). This observation hints thatsuch a 3D soliton is stabilized by the chiral LC medium’s ten-dency to twist, which is confirmed by a 3D plot of free energydensity isosurfaces of the soliton (Fig. 2j). Importantly, thefree energy within the soliton is below that of the uniformunwound state almost everywhere except for the small local-ized region matching the region of H opposite to that of LC’schirality (Fig. 2i,j).The nonsingular 3D soliton shown in Fig. 2 is rather dif-ferent from the simplest toron structure that we introduced inRef. [13], with π -twist from its central axist to the peripheryin all radial directions (Fig. 3). The midplane cross-sectionof the toron is a two-dimensional skyrmion (baby skyrmion),which belongs to the second homotopy group π ( S ) = Z [30].Mapping the vectorized director from the elementary toron’smidplane (Fig. 3a) onto the S order parameter space coversthe sphere once, indicating that the skyrmion number is equalunity [30]. In the overall 3D configuration of the elementarytoron, the localized twisted region is capped with two hyper-bolic point defects near confining surfaces, as we discussed indetail in Ref. [13]. The preimages of this simplest toron haveshapes of bands (two per single point on S /Z and a singleband in the case of a vectorized field and the S ground-statemanifold) terminating on the point defects. Interrupted bythe two point singularities, the two-band preimages of pointson S /Z of these torons do not form closed loops (Fig. 3c-e),albeit this behavior is very different for the new types ofFigure 3: Elementary toron structure in a confined chiral nematic LC. (a,b) Computer simulated (a) in-plane and (b) verticalcross-sections of n ( r ) of the toron depicted using double cones and the color scheme that establishes correspondence betweendirector orientations and the points on S /Z (top-right insets). (c) Computer-simulated preimages of the toron for two sets ofthe diametrically opposite points on S /Z marked by double cones in the top-right inset. (d,e) Comparison of representative(d) computer-simulated and (e) experimental preimages of the toron for two diametrically opposite points on the “equator” of S /Z (top-right insets). The experimental preimages in (e) were reconstructed based on 3PEF-PM images of structures in a5CB-based partially polymerizable nematic LC mixture with chiral additive CB-15, d = 10 µ m and d/p ∼
1; before imaging,the unpolymerized 5CB was washed out and replaced by immersion oil. Gray arrows indicate the consistently determinedcirculations of the preimages. Computer simulations were preformed for 5CB elastic constants while also assuming K = 0.8 r a f t torons with larger amounts of twist and nontrivial closed-looppreimages that we discuss below. The torons, ensembles of thenonsingular solitonic twisted structures and singular point de-fect or disclination loop entities, can be classified not only bythe types of the self-compensating singular defects [13], butalso by the linking of preimages of their solitonic parts [36],as we discuss in detail below for the cases of new, complextoron structures. Q = ± andlinking of preimages The solitons shown in Figs. 4 and 5 also embed the axiallysymmetric twisted regions into the uniform background n ,but their solitonic n ( r ) twists by 2 π in all radial directionsfrom the 3D soliton’s central axis (parallel to n ) to the far-field periphery (Fig. 4a,b and Fig. 5a,b). For both of theseFigure 4: 3D topological soliton with Q = 1. (a,b) Computer simulated (a) in-plane and (b) vertical cross-sections ofthe axially symmetric n ( r )-structure of the hopfion depicted using double cones and the color scheme that establishescorrespondence between director orientations and the points on S /Z (top-right insets). (c) Computer-simulated preimagesof the hopfion for two sets of the diametrically opposite points on S /Z marked by double cones in the top-right inset. (d,e)Comparison of representative (d) computer-simulated and (e) experimental preimages of the hopfion for two diametricallyopposite points on the “equator” of S /Z (top-right insets). Computer simulations were preformed for elastic constantsof 5CB while also assuming K = 0. The preimages were reconstructed based on 3PEF-PM images of structures in a5CB-based partially polymerizable nematic LC mixture with a chiral additive CB-15, in a cell with d = 10 µ m and d/p ∼ Q = 1, as discussed in the text. Gray arrows in (c-e) indicate the consistently determined circulations of the preimages.(f,g) Perspective views of the 3D computer-simulated isosurfaces of normalized (f) handedness and (g) free energy densityfor this axially symmetric hopfion, with the corresponding color schemes provided in the right-side insets.9 r a f t Figure 5: 3D topological soliton with Q = −
1. (a,b) Computer simulated (a) in-plane and (b) vertical cross-sections of theaxially symmetric n ( r )-structure of the Q = − S /Z (top-right insets). (c) Computer-simulated preimagesof the hopfion for two sets of the diametrically opposite points on S /Z marked by double cones in the top-right inset. (d,e)Comparison of representative (d) computer-simulated and (e) experimental preimages of the hopfion for two diametricallyopposite points on the “equator” of S /Z (top-right insets). Linking of the two closed loops establishes the Hopf index Q = −
1, as discussed in the text. Computer simulations were preformed for elastic constants of 5CB while also assuming K = 0. The preimages were reconstructed based on 3PEF-PM images of structures in a 5CB-based partially polymerizablenematic LC mixture with a chiral additive CB-15, in a cell with d = 10 µ m and d/p ∼
1; before imaging, the unpolymerized5CB was replaced by immersion oil. Gray arrows in (c-e) indicate the consistently determined circulations of the preimages.(f,g) Perspective views of the 3D computer-simulated isosurfaces of normalized (f) handedness and (g) free energy densityfor this axially symmetric hopfion, with the corresponding color schemes shown in the right-side insets.solitons, all points on S /Z have preimages in the form oftwo inter-linked closed loops (Fig. 4c-e and 5c-e). Experi-mental closed-loop preimages closely match their theoreticalcounterparts (Figs. 4d,e and 5d,e) and all wind around eachother to link once. In the case of a vectorized n ( r ), for bothsolitons, the preimages of every single point on S are singleclosed loops and such preimages of any two distinct points on S are linked once. As discussed above, the Hopf index of sucha soliton Q = (cid:80) C/ n ( r ) or, equivalently, the linking number of the two closed loopsthat form a single preimage of any point on S /Z [31]. Byvectorizing n ( r ) of the two types of observed solitons (Figs.4 and 5), so that the n points in the same directions forboth of them and so that the corresponding circulations of thefar-field preimages define circulations of all other preimagesthrough the requirement of continuity, we find the opposite Q = ± S /Z for these two different solitons in the case10 r a f t of nonpolar n ( r ), the corresponding Hopf links of single-looppreimages of any two points on S for vectorized n ( r ) and theLC in which these solitons are hosted are all chiral in nature.By using a vectorized n ( r ) and choosing the circulation ofthe preimage of the north pole on S to be along n throughthe soliton’s center, we consistently define circulations of allother preimages while smoothly exploring S . We find thatthus determined linking number stays conserved for all pairsof vectorized- n ( r ) preimages of S -points within the same so-lition, yielding its Hopf index, which is Q = 1 for the solitonshown in Fig. 4 and Q = − Q -values stay the same upon inverting thevectorization direction n ( r ) → − n ( r ), which is different fromthe case of hedgehog charges of point defects in LCs [47] thatchange sign in response to the n ( r ) → − n ( r ) operation. How-ever, taking a mirror image negates the linking numbers of allHopf links and the corresponding Q values while also trans-forming a left-handed LC host into its right-handed counter-part, which is again different from the hedgehog charges ofpoint defects that would stay unchanged during this opera-tion [14,47]. These Q = ± Q (cid:54) = 0 as metastable or sta-ble field configurations, albeit the main physical mechanismresponsible for their stability is the same as for the solitonswith Q = 0 (Figs. 2 and 3) and is related to the LC medium’stendency to twist n ( r ) with the spatial periodicity compara-ble to its intrinsic helicoidal pitch p . Prevailing parts of thesoliton’s volume have twist handedness matching that of theLC hosting them (Figs. 4f and 5f), although there are somesmall regions within the solitons with H opposite to that ofthe LC’s intrinsic chirality. The 3D isosurfaces of free energydensity convincingly show that such a 3D topological solitonembeds a large region of low-energy twisted n ( r ) (lower thanthat of the surrounding unwound uniform background) andthat its stability is helped by the medium’s chirality (Figs. 4gand 5g). The elementary Q = ± n ( r )-twist embedded in the uniform un-wound background n . For example, n ( r ) twists by 4 π inall radial directions from the localized configuration’s centralaxis (parallel n ) to the periphery of the 3D solitons shownin Figs. 6, 7 and 8. These axially symmetric solitons (Figs.6a,b,j,k, 7a,b and 8a,b) have two-closed-loops preimages of all S points in the case of a vectorized n ( r ) while the preimagesof each point on S /Z of the nonpolar director field com-prise four individual closed loops (Figs. 6d,e,m,n, 7c,d and8c,d). The preimages of S -points for the same polar angle θ of vectorized n ( r ) with respect to n but corresponding to itsdifferent azimuthal orientations tile into tori surfaces (Figs.6f,o, 7e and 8e). There are always two such tori for a given θ (Figs. 6f,o, 7e and 8e), which is different from the case ofelementary hopfions (Figs. 4 and 5), for which there is onlyone torus surface for each θ -value. Although all four exam-ples of solitons shown in Figs. 6-8 have preimages in the formof two separate closed loops for every single point on S forvectorized n ( r ), the nature and topology of inter-linking ofthese closed-loop preimages is different, as we discuss below.For all S -points in the case of the vectorized n ( r ) ofthe soliton shown in Fig. 6a-i, the individual preiamges areformed by two separate unlinked closed loops while preimagesof two separate S -points form two Hopf links with the linkingnumber +1 for each of them (Fig. 6d,e). The sloitons shownin Fig. 6j-r also have preimages comprised of two separate un-linked closed loops and preimages of two separate S -pointsforming two Hopf links, but the linking number characteriz-ing these two individual Hopf links is − θ characterizing orienta-tion of n ( r ), the series of preimages with different azimuthalorientations of n ( r ) tile into two separate tori surfaces (Fig.6f,o). By scanning the n ( r ) orientations from n ( r )=(0,0,-1) tothe far n ( r )=(0,0,1)= n , we find that the two tori formed bypreimages of constant θ remain separate until merging withthe far field background when n ( r ) becomes parallel to n (Fig. 6g,p). The behavior of the individual S /Z -preimagescorresponding to nonpolar n ( r ) is reminiscent to that of pairsof preimages of S -points for vectorized n ( r ) (Fig. 6d-g andm-p). The preimages of S -points in the vicinity of the northpole are two separate tori that characterize n ( r )-orientationssmoothly transforming to n , with the far-field preimages con-nected through infinity (Fig. 6g,p). Thus, one can interpretthe two solitonic structures shown in Fig. 6 as being formedby a coaxial arrangement of two separate hopfions of Q = 1(Fig. 6a-i) and Q = − Q = 1hopfion in the interior and Q = − r a f t Figure 6: 3D topological solitons formed by coaxial arrangement of two different elementary hopfions of the same signs Q = ±
1. (a,b) Computer simulated (a) in-plane orthogonal to n and (b) vertical containing n cross-sections of the axiallysymmetric n ( r )-structure of a soliton with two-loop preimages depicted using colored double cones; the color scheme thatestablishes the correspondence between director orientations and the points on S /Z (top-right insets). (c) A polarizingoptical micrograph of such a 3D soliton in a confined chiral nematic LC; white double arrows show crossed polarizers. (d)Computer-simulated and (e) experimental preimages of the soliton for the diametrically opposite points on S /Z markedby double cones in the right-side inset. By analyzing such preimages of all points on S /Z , we find no inter-linking betweenthe preimages of the two separate hopfions and their Hopf indices Q = 1. (f) For a constant polar angle value (inset), theclosed-loop preimages of individual points on S tile into two separate tori surfaces sharing the same vertical axis parallelto n . (g) preimages of the north and south poles of S for the vectorized director field. (h,i) Perspective views of the 3Dcomputer-simulated isosurfaces of normalized (h) handedness and (i) free energy density for this axially symmetric soliton,with the corresponding color schemes shown in the insets above them. (j,k) Computer simulated (j) in-plane and (k) verticalcross-sections of an axially symmetric n ( r )-structure of a soliton with two-loop preimages depicted using colored doublecones and the same color scheme as in (a,b) that establishes the correspondence between director orientations and the S /Z -points. (l) A polarizing optical micrograph of such a 3D soliton. (m) Computer-simulated and (n) experimental preimagesof the hopfion for the diametrically opposite points on S /Z marked by double cones in the right-side inset. (o) For aconstant polar angle value, the closed-loop preimages of the individual points on S tile into two separate tori that share thevertical axis parallel to n . By analizing preimages of all points on S /Z , we find no inter-linking between the preimagesof the two separate hopfions and also their Hopf indices Q = −
1. (p) preimages of the north and south poles of S for thevectorized director field. (q,r) Perspective views of the 3D computer-simulated isosurfaces of normalized (q) handedness and(r) free energy density for this soliton, with the corresponding color schemes shown in the insets. Computer simulations werepreformed for elastic constants of 5CB while also assuming K = 0. The polarizing optical micrographs in (c) and (l) wereobtained for structures in a 5CB-based partially polymerizable nematic LC mixture with a chiral additive CB-15 in a cellwith thickness d = 10 µ m and d/p ∼
1. Gray arrows in (d,e,m,n,g,p) indicate the consistently determined circulations of thepreimages. The 3D preimages were reconstructed based on 3PEF-PM images of these structures after the unpolymerized5CB was replaced by immersion oil. 12 r a f t A series of other solitons with 4 π twist in radial directionshave a very different linking of preimages (Figs. 7 and 8 andsupplementary videos S1 and S2). To analyze them, we firstobserve that the so-called “Pontryagin-Thom constructions” [14], isosurfaces of zero z -component of the director, n z = 0(corresponding to θ = π/ n ( r ) (Figs. 7f-h and 8f-h),Figure 7: 3D soliton with complex linking and 4 π twist from its central axis to the far-field periphery. (a,b) Computersimulated (a) in-plane and (b) vertical cross-sections of the axially symmetric n ( r )-structure of the 3D soliton depictedusing double cones and the color scheme that establishes correspondence between director orientations and the points on S /Z (top-right insets). (c) A representative computer-simulated preimage of the hopfion for the diametrically oppositepoints on S /Z marked by double cones in the top-right inset. The preimage is comprised of four inter-linked closed loops.(d,e) Computer-simulated preimages of the 3D soliton in a vectorized n ( r ) (d) for two diametrically opposite points on S corresponding to its north and south poles (top-right inset) and (e) for a set of points characterized by a constant polar angle θ and forming a circle parallel to the spheres equator (top-right inset). Note that the preimages in (e) reside on two nestedtori surfaces. A large variety of preimages of this soliton are shown in the supplementary video S1. (f-h) Three representativeviews on the isosurfaces of θ = π/ n z = 0) colored by azimuthal orientations of n ( r ) according to the scheme shownin the right-side inset of (f). The numbers on top of the tori shown in (f) indicate the linking numbers that characterizethe inter-linking of colored closed-loop bands and preimages of n ( r ) corresponding to points on the equator of S /Z . (i,j)Perspective views of the 3D computer-simulated isosurfaces of normalized (i) handedness and (j) free energy density for thisaxially symmetric 3D soliton, with the corresponding color schemes shown in the right-side insets. Gray arrows in (c,d) andon the green bands of (g) indicate the consistently determined circulations of the preimages. Computer simulations werepreformed for elastic constants of ZLI-2806 (Table 1) and d/p = 2.13 r a f t Figure 8: 3D soliton with complex linking and 4 π twist from its central axis to the far-field periphery. (a,b) Computersimulated (a) in-plane and (b) vertical cross-sections of the axially symmetric n ( r )-structure of the 3D soliton depictedusing double cones and the color scheme that establishes correspondence between director orientations and the points on S /Z (top-right insets). (c) A representative computer-simulated preimage of the hopfion for the diametrically oppositepoints on S /Z marked by double cones in the top-right inset. The preimage is comprised of four inter-linked closed loops.(d,e) Computer-simulated preimages of the 3D soliton in a vectorized n ( r ) (d) for two diametrically opposite points on S corresponding to its north and south poles (top-right inset) and (e) for a set of points characterized by a constant polar angle θ and forming a circle parallel to the spheres equator (top-right inset). Note that the preimages in (e) reside on two nestedtori surfaces. A large variety of preimages of this soliton are shown in the supplementary video S . (f-h) Three representativeviews on the isosurfaces of θ = π/ n z = 0) colored by azimuthal orientations of n ( r ) according to the scheme shownin the right-side inset of (f). The numbers on top of the tori shown in (f) indicate the linking numbers that characterizethe inter-linking of colored closed-loop bands and preimages of n ( r ) corresponding to points on the equator of S /Z . (i,j)Perspective views of the 3D computer-simulated isosurfaces of normalized (i) handedness and (j) free energy density for thisaxially symmetric 3D soliton, with the corresponding color schemes shown in the right-side insets. Gray arrows in (c,d) andon the green bands of (g) indicate the consistently determined circulations of the preimages. Computer simulations werepreformed for elastic constants of ZLI-2806 and d/p = 2.also form two separate tori. The like-colored closed-loopbands of constant azimuthal n ( r )-orientation on these sur-faces of two separate tori link with each other once, with theconsistently determined circulation directions shown with the gray curved arrows on the green bands. These colored bandscovering the θ = π/ S (or S /Z ) and are shownusing the color scheme chosen to be consistent with the non-14 r a f t polar nature of n ( r ) (Figs. 7f-h and 8f-h) (note that the n and − n bands are shown using the same color). The linkingnumber of each pair of like-colored closed-loop bands is ± θ = π/ n ( r ) link differently from preim-ages of the north-pole point on S and the S -points in itsvicinity (Figs. 7c-e and 8c-e). Moreover, the two-tori surfacesof constant θ are nested one in another for θ < θ c , but be-come separate from each other within θ c < θ < ◦ , wherethe critical polar angle is θ c ≈ ◦ for the soliton shown inFig. 7 and θ c ≈ ◦ for that in Fig. 8. Most interestingly,such transformation of the two-tori surfaces upon changing θ takes place without compromising the soliton’s nonsingularnature and the pre-images on the different tori align with each other during the re-linking (supplementary videos S1 and S2).For the vectorized n ( r ), the two-loop preimages of single S -points are linked once at θ < θ c but unlinked at θ > θ c . Thelinking of preimages of different S -points of such 3D solitonsdepends on the locations of these points on S (Figs. 7c-h and8c-h) and is not a conserved quantity. The nature of linkingof four-loop preimages of distinct points on S /Z for thesesolitons in a nonpolar n ( r ) is even more complex. To charac-terize it, we use simplified-topology and graph presentations(Table 2) of the closed-loop preimages and their linking. Inthese graphs, the closed-loop components of preimages areshown as filled circles colored according to the positions ofcorresponding points on the ground-state manifold and theindividual links are indicated by black edges connecting thecorresponding circles. This presentation allows us to providean exhaustive set of possibilities for inter-linking of preimagesof two distinct points on the S /Z or S depending on theirrelative locations (Table 2). Moreover, the summary of thepreimage linking in Table 2 reveals differences in topology ofthe two solitons shown in Figs. 7 and 8, which is manifested bythe differences in consistently defined preimage circulations.Table 2. Linking diagrams and graphs of complex 3D solitons. The table presents the analysis of linking of preimages of twoseparate points on S and S /Z for composite 3D topological solitonic field configurations on the basis of both nonpolar andvectorized n ( r ) of the studied structures. The insets in the red boxes at the top of the columns “linking diagrams” depict the15 r a f t order parameter spaces of vectorized (top) and nonpolar (bottom) n ( r ), with arrows or double arrows indicating the pointsfor which the preimage linking is analyzed. The dashed lines on the S and S /Z schematics separate the fragments of the S and S /Z with θ < θ c (top parts) and θ > θ c (bottom parts). The locations of the points corresponding to preimages,shown using single and double arrows on S and S /Z are the same for all solitons within the same column. In the graphs,the individual links are indicated by black or gray lines connecting the corresponding colored filled circles that representclosed-loop preimages (the black lines indicate positive signs of Linking of preimages as determined by circulations while thegray lines correspond to the negative ones). The colors of the filled circles are indicative of the points on S (for schematicsshown above the horizontal dashed lines of the table) or S /Z (for schematics shown below the horizontal dashed lines of thetable); for n ( r ) at θ < θ c , two out of eight filled circles of the graphs are shown as red and the rest as orange to distinguishthem on the basis of the number of times the corresponding preimages are linked. The mutually linked preimage rings in thesimplified topology presentations are also shown in colors corresponding to their locations on S or S /Z and have arrowsdenoting circulation consistent with the far-field preimage. The point defects of torons within the topological skeletons areshown using black stars. Both the topological skeleton and graph representations of the preimage structures are constructedfor the same solitonic field configurations and are provided next to each other for the case of vectorized n ( r ).A detailed analysis of the linking diagrams (Table 2) showsthat Pontryagin-Thom construction does not fully reveal thetopology of complex 3D solitons, which requires directly an-alyzing preimages of all points on the ground-state manifoldand their interlinking, not just a subset of them. Indeed, thelinking numbers for the n and − n preimages forming two sep-arate tori at θ > θ c (marked on the Pontryagin-Thom surfacesin Figs. 7f and 8f for θ = π/
2) change with varying θ (Ta-ble 2). The supplementary videos S1 and S2 show that thischange in the linking of preimages is directly related to thetransformation of the two tori corresponding to constant θ -values, which occurs at θ c . This transformation is manifestedby a transition from two separate concentric tori (similar tothe ones shown in Figs. 7f-h and 8f-h) to two inter-nestedtori like the ones depicted in Figs. 7e and 8e. The alignmentand merging of preimages residing on two different tori thatleads to the change of linking of constituent two rings com-prising the individual preimages is interesting and calls forexploration of similar field configurations in other branchesof physics. Finally, similar to the case of elementary hopfions(Figs. 4 and 5), the comparison of 3D isosurface plots of nor-malized handedness and free energy density (Figs. 6h,i,q,r,7i,j and 8i,j) shows that the stability of such 3D solitons isgreatly enhanced by the medium’s chiral nature and tendencyto form twisted structures consistent with the intrinsic pitch p of the used chiral LC. In addition to the elementary torons with π -twist of n ( r ) fromtheir central axis to the n -periphery in all radial directions(Fig. 3) [13], torons with larger amounts of such twist also ex-ist (Fig. 9a,b, 10a,b and 11a,b) [36]. For example, the toronsshown in Figs. 9 and 10 have 3 π - and the ones in Fig. 11have 5 π -twist of n ( r ) in all radial directions from the toron’scentral axis (parallel to n ) to the far-field periphery (Table2), respectively. The preimages of the distinct points on the S /Z or S are either closed loops or bands terminating onthe two hyperbolic point defects (Figs. 9c-e, 10c-h and 11c-h).Some of the high-twist torons exhibit re-linking of preimages(Figs. 10c-h and 11c-h and supplementary video S3 and S4),similar to that observed for the solitons shown in Figs. 7 and8. The critical polar angles of re-linking are θ c ≈ ◦ and θ c ≈ ◦ for the structures shown in Figs. 10 and 11, respec-tively (Table 2). For the toron structure shown in Fig. 9, on the other hand, θ c ≈ ◦ and, therefore, they effectivelycan be thought of as separate elementary hopfion (Figs. 4and 5) and an elementary toron (Fig. 3) arranged coaxiallyin such a way that their vertical axes coincide and are par-allel n . The analysis of linking of closed-loop preimages forsingle and distinct points on S /Z and S reveals that suchcomplex toron structures could be thought of as hybrids ofelementary torons (Fig. 3) and different nonsingular solitonsthat we discussed above. Interestingly, the multicomponentpreimages are comprised of closed loops and half-loop bandsthat terminate on the point singularities. The components ofpreimages inter-transform between one another, revealing alarge diversity of torons (Table 2). These findings show an un-expected large diversity of torons and that the torons shouldbe classified not only based on the types of the constituentself-compensating singular defects [13], but also by the typesand linking of preimage components and different preimagesof the nonsingular solitonic part of the torons (Table 2). Our method of analyzing preimages can be also applied totwistions, localized structures that embed twisted regions intoa uniform background of the far-field but lack axial symmetryand (unlike torons) contain more than two self-compensatingpoint defects [37] (Fig. 12). Although we provide here anexample of a twistion with the amount of twist from itsinterior to periphery by ∼ π , the analogs of twistions withlarger amounts of twist in the director field can exist tooand will be a subject of our future studies. The configu-ration of such a twistion with a stretched closed loop of π -twist of n ( r ) and four self-compensating hyperbolic point de-fects is shown with the help of in-plane and vertical cross-sections in Fig. 12a-c. The preimages of single points on S /Z ( S for the vectorized director) are bands spanning be-tween the four point singularities (Fig. 12d,e). This exampleof the twistion shows that the localized toron- and hopfion-like field configurations in confined chiral nematic LCs arenot restricted to hosting none (as in hopfions) or only pairs(as in the torons) of self-compensating singular defects, butthat such self-compensation can occur in a number of othermore complex ways, e.g. through the co-existence of fourself-compensating hyperbolic point defects shown in additionto various solitonic components with band-like or closed-loop16 r a f t preimages (Fig. 12). In addition to the number and types ofsingular defects, another source of diversity of solitonic struc-tures can emerge from the large variety of nonsingular twisted regions and preimages that they can exhibit, which we will ex-plore elsewhere.Figure 9: 3D toron-hopfion hybrid solitons formed by coaxial arrangement of an elementary toron and elementary hopfionof Q = 1. (a,b) Computer simulated (a) in-plane orthogonal to n and (b) vertical containing n cross-sections of theaxially symmetric n ( r )-structure of a soliton depicted using colored double cones; the color scheme that establishes thecorrespondence between director orientations and the points on S /Z (top-right insets). (c) Computer-simulated and (d)experimental preimages of the soliton for the diametrically opposite points on S /Z marked by double cones in the right-sideinsets. Gray arrows indicate the consistently determined circulations of the preimages. By analyzing such preimages of allpoints on S /Z , we find no inter-linking between the preimages of the hopfion and toron, as well as the hopfion’s Hopf index Q = 1. The 3D preimages were reconstructed based on 3PEF-PM images of these structures after the unpolymerized 5CB wasreplaced by immersion oil. Computer simulations were preformed for elastic constants of 5CB while also assuming K = 0.(e) For a constant polar angle value, the closed-loop preimages of individual points on S tile into a torus and a spheresurfaces sharing the same vertical axis parallel to n , with the sphere having two small holes at the poles corresponding tothe location of the hyperbolic point defects. (f) A polarizing optical micrograph of such a 3D soliton, with the white doublearrows showing crossed polarizers. The polarizing optical micrograph in (f) was obtained for a structure in a 5CB-basedpartially polymerizable nematic LC mixture with a chiral additive CB-15 in a cell with thickness d = 10 µ m and d/p ∼ r a f t Figure 10: Toron with complex linking and 3 π twist from its central axis to the far-field periphery. (a,b) Computer simulated(a) in-plane and (b) vertical cross-sections of the axially symmetric n ( r )-structure of the toron depicted using double conesand the color scheme that establishes correspondence between director orientations and the points on S /Z (top-rightinsets). The two small regions of discontinuity in orientation of the double cones are the hyperbolic point defects. (c) Arepresentative computer-simulated preimage of the hopfion for the diametrically opposite points on S /Z marked by doublecones in the top-right inset. The preimage is comprised of two closed loops and two half-loop bands terminating on the pointdefects. (d,e) Computer-simulated preimages of the toron in a vectorized n ( r ) (d) for two diametrically opposite points on S corresponding to its north and south poles (top-right inset) and (e) for a set of points characterized by a constant polarangle θ and forming a circle parallel to the spheres equator (top-right inset). Note that the closed-loop preimages in (e)reside on a torus surface while half-loop bands form another surface spanning between the point defects. A large variety ofpreimages of this solitonic configuration are shown in the supplementary video S3. (f-h) Three representative views on theisosurfaces of θ = π/ n z = 0) colored by azimuthal orientations of n ( r ) according to the scheme shown in the right-sideinset of (f). The “-1” on top of the torus shown in (f) indicates the linking number that characterizes the inter-linking ofcolored closed-loop bands and preimages of n ( r ) corresponding to points on the equator of S /Z . Gray arrows in (c,d)and on the green bands of (g) indicate the consistently determined circulations of the preimages. (i,j) Perspective views ofthe 3D computer-simulated isosurfaces of normalized (i) handedness and (j) free energy density for this axially symmetrictoron, with the corresponding color schemes shown in the right-side insets. Computer simulations were preformed for elasticconstants of ZLI-2806 and d/p = 2. 18 r a f t Figure 11: Toron with complex linking and 5 π twist from its central axis to the far-field periphery. (a,b) Computer simulated(a) in-plane and (b) vertical cross-sections of the axially symmetric n ( r )-structure of the toron depicted using double conesand the color scheme that establishes correspondence between director orientations and the points on S /Z (top-rightinsets). The two small regions of discontinuity in orientation of the double cones are the hyperbolic point defects. The insetin (a) shows a polarizing optical micrograph of such a structure. The optical micrograph was obtained for a 5CB-basedpartially polymerizable LC in a cell of d = 10 µ m. (c) A representative computer-simulated preimage of the hopfion for thediametrically opposite points on S /Z marked by double cones in the top-right inset. The preimage is comprised of fourinter-linked closed loops and two half-loop bands terminating on the point defects. (d,e) Computer-simulated preimages ofthe toron in a vectorized n ( r ) (d) for two diametrically opposite points on S corresponding to its north and south poles(top-right inset) and (e) for a set of points characterized by a constant polar angle θ and forming a circle parallel to thespheres equator (top-right inset). Note that the closed-loop preimages in (e) reside on two nested tori surfaces while half-loopbands form another surface spanning between the point defects. A large variety of preimages of this solitonic configurationare shown in the supplementary video S4. (f-h) Three representative views on the isosurfaces of θ = π/ n z = 0) coloredby azimuthal orientations of n ( r ) according to the scheme shown in the right-side inset of (f). The numbers on top of thetori shown in (f) indicate the linking numbers that characterize the inter-linking of colored closed-loop bands and preimagesof n ( r ) corresponding to points on the equator of S /Z . Gray arrows in (c,d) and on the green bands of (g) indicate theconsistently determined circulations of the preimages. (i,j) Perspective views of the 3D computer-simulated isosurfaces ofnormalized (i) handedness and (j) free energy density for this axially symmetric 3D soliton, with the corresponding colorschemes shown in the right-side insets. Computer simulations were preformed for elastic constants of ZLI-2806 and d/p = 2.19 r a f t Figure 12: A twistion structure in a chiral nematic LC. (a-c) Computer simulated (a) in-plane and (b,c) vertical cross-sectionsof the 3D n ( r )-structure of the twistion depicted using double cones and the color scheme that establishes correspondencebetween director orientations and the points on S /Z (top-right insets). The locations of vertical cross-sections (b) and(c) are depicted in (a) using arrows. (d) Computer-simulated preimages of the twistion for two sets of the diametricallyopposite points on S /Z marked by double cones in the top-right inset. (e) Computer-simulated preimages of the twistionfor points on the “equator” of S /Z (top-right inset). Computer simulations were preformed for elastic constants of 5CBand d/p = 0 .
85. (f) A polarizing optical micrograph of the twistion, with the white double arrows showing crossed polarizers.The polarizing micrograph was obtained for a structure in a 5CB-based partially polymerizable nematic mixture doped witha chiral additive CB-15 in a cell with d/p ∼ Although the 3D solitons with nonzero Hopf invariants aretheoretically predicted to exist in many branches of science,ranging from particle physics to cosmology, their experimen-tal identification and detailed study is often prohibitively dif-ficult. Even in the case of solid-state chiral ferromagnets, inwhich the two-dimensional counterparts of the 3D solitonsare recently extensively studied [25-29], experimental imag-ing techniques are lacking the ability to resolve details of fieldconfigurations within the nanometer-sized localized structureswith high resolution in 3D. So, in fact, the 3D topological soli-tons may (under certain circumstances, considering the simi-lar description by Eq. (2) within the simples models [24,25])exist in the solid-state chiral ferromagnet systems but thelack of appropriate imaging and analysis techniques prohibitstheir experimental identification and classification. Chiralnematic LCs provide an experimental advantage of hostingmicrometer-sized 3D solitonic structures, so that their struc-ture is accessible to the direct 3D nonlinear optical imag-ing [35]. The experimental 3D configurations of n ( r ) withinthe studied solitons closely agree with numerical modeling,allowing us to robustly identify and classify the nonsingular π ( S /Z ) = Z topological defects with different Hopf indices. This will provide important insights needed for the realizationof topological solitons in other physical systems. We also en-visage that chiral LCs will serve as a test bed for theories of3D topological solitons.The possibility of realizing 3D localized field configura-tions embedded in a uniform far-field background as staticsolitons is a subject of active studies in different branches oftheoretical physics and applied mathematics [3]. The Hobart-Derrick theorem states that the static 3D solitons cannot havefinite energy for the free energy functional resembling the firstterm in Eq. (2) [7,8]. Indeed, our numerical modeling con-firms that all computer-simulated 3D solitons discussed abovebecome unstable after removing the chiral terms of free en-ergy in Eqs. (1) and (2) for the nonchiral nematic LC with q = 0 while using all other parameters within the experi-mentally accessible ranges, consistent with the correspondingexperiments at otherwise identical conditions. However, thechiral LC medium’s tendency to twist n ( r ) in the frustratedconfined geometry of unwound homeotropic cells helps to em-bed energetically favorable twisted regions of solitons into theuniform unwound background of n . When the LC cell thick-ness d is comparable to the intrinsic helicoidal pitch of thechiral nematic medium, with d/p = 0 . −
2, a large numberof spatially localized structures with twisted n ( r ) can become20 r a f t embedded in the uniform far-field background (Figs. 2-12) tolocally relieve the frustration imposed by the incompatibilityof homeotropic boundary conditions and the helicoidal struc-ture of the chiral LC’s ground-state. Interestingly, all of thestudied 3D solitons emerge as local minima of the free energyfunctionals given by both Eqs. (1) and (2), albeit at some-what different d/p ratios and other parameters. The analysisof 3D isosurfaces of the twist handedness H reveals how LC’schirality mediates appearance of the twisted solitons with fi-nite dimensions comparable to p by showing that H within thelocalized structures is mostly the same as that of the ground-state chiral nematic LC in which they are hosted. However,an interesting, unexpected observation is that the localizedstructures also posses small energetically costly regions with H opposite to that of the ground-state LC medium. This find-ing may imply that the twisted solitons require reversal of H to match their internal field configurations with the uniformfar-field n , albeit this aspect will require separate detailedstudies.A comparison of Eqs. (1) and (2) helps to identify elasticconstant anisotropy as (unique to LC systems) an additionalmechanism for stabilizing the 3D knotted solitons. Indeed,although we find that the studied solitons and torons are solu-tions of both Eqs. (1) and (2), the parameter space of stabilityis different. For example, the studied structures of elementaryhopfions (Figs. 4 and 5) tend to be stable at smaller d/p val-ues roughly within 0 . − . − . K = 0. In addition to the chirality and elas-ticity, the stability of 3D solitonic structures can likely befurther controlled by varying the strength of vertical surfaceboundary conditions for the director and by applying exter-nal fields that would dielectrically or diamagnetically coupleto n ( r ). For example, in the case of LCs with positive dielec-tric anisotropy, the applied low-voltage fields (1 − d/p values. Since the focus of our study ison topology of the 3D solitonic structures, the exploration ofdetailed structural diagrams of hopfion and toron stability isbeyond the scope of this work, but will be a subject of ourfuture studies.An interesting observation is that fully nonsingular soli-tons in n ( r ) emerge when the twist of director from the centralaxis of these axially symmetric structures to the periphery isan even integer of π (2 π and 4 π in the provided examplesin Figs. 4-8) while several different torons with point singu-larities have an odd number of π of such twist ( π , 3 π and5 π in the provided examples in Figs. 3,9-11). This obser-vation provides insights into one of the sources of diversityof studied solitonic structures in confined chiral LCs, whichstems from the amount of the director twist in radial direc-tions of the axially symmetric structures of solitons, as wellas insights into how such structures can be generated on de-mand experimentally. Another interesting observation is thatthe far-field distortions of n ( r ) corresponding to most of thestudied solitonic configurations are symmetric with respect tothe sample midplane, a plane crossing centers of the localizedstructures orthogonally to n , except for the topologically trivial soliton with Q = 0 shown in Fig. 2 and the twistionstructure presented in Fig. 12. These structural features canbe analyzed based on vertical cross-sections of the field con-figurations in the planes containing n (Figs. 1-12) and arerelated to the multitude of different ways of embedding local-ized twisted regions within the uniform far-field background n . Consistently with the peculiarities of the up-down sym-metry of the solitonic configurations in the homeotropic cells,we find that the solitons exhibit richness of elastic interactionsand self-assembly. We previously explored such interactionsand self-assembly for the elementary torons [36], but will alsoextend such studies to the other solitonic structures, whichwill be reported elsewhere.We have demonstrated that our method of preimage anal-ysis provides a comprehensive way of exploring the topologyof 3D solitons, providing insights into the nature of both non-singular topologically nontrivial structures and the singulardefects such as the point singularities found in the studiedtorons. This approach is further expanding the capabilitiesof the method of Pontryagin-Thom construction that we alsoused for this purpose [14]. A detailed analysis of preimagesallows us to uncover a number of rather unexpected featuresof 3D solitons in LCs. For example, the 3D soliton with Q = 0shown in Fig. 2 has closed-loop preimages for the majorityof S -points (for the vectorized director field), except for thevicinity of the south pole of the S -sphere. This informationcould have been missed had we not analyzed preimages ofall S -points but only some of them, as in the case of thePontryagin-Thom construction. A comparison of topologicallinking of preimages of the solitons shown in Figs. 7 and 8based on the summary presented in Table 2 additionally em-phasizes the need to analyze both vectorized and nonpolar n ( r ). Indeed, the different topological nature of these twosolitons could not be revealed based on the Pontryagin-Thomconstructions, which are rather similar for the two structures(compare Fig. 7f-h and Fig. 8f-h). Moreover, we find exactlythe same linking of the preimages of all pairs of points on S and S /Z (Table 2 and Figs. 7 and 8) of these two differ-ent solitonic structures and the difference between them canbe seen only when the preimage circulations are consistentlydefined. For both nonpolar and vectorized n ( r ), we can seethe difference between topologies of these two solitons on thebasis of circulation directions (Table 2 and Figs. 7 and 8),with the differences in preimage linking apparent when one S -point is at θ < θ c and one at θ > θ c or both are at θ < θ c ,but not when both of the analyzed points are at θ > θ c . Inprinciple, 3D solitons with different structures could have thesame topology of preimage linking, being homeomorphic toeach other, but this is not the case for the solitons shownin Figs. 7 and 8, which cannot be smoothly morphed one toanother. The detailed analysis of preimages allows us to iden-tify and demonstrate such subtle differences between the 3Dsolitons. In a similar way, the important differences betweenthe four different types of torons that we present in this studycould be missed without a detailed analysis of preimages ofall points on S /Z for nonpolar and on S for vectorized n ( r ) (Figs. 3 and 9-11 and Table 2). On the other hand, theanalysis of only the closed-loop preimages but not the onesterminating on point defects would fail to reveal differencesbetween certain types of torons and solitons without point21 r a f t defects (compare Fig. 7 and Fig. 11, as well as the corre-sponding summaries presented in Table 2).The 3D solitons that we discuss in this work constitutea non-exhaustive, illustrative set of examples of topologicallynontrivial field configurations that can be stabilized in chiralliquid crystals and ferromagnets, but a much larger varietyof π ( R P ) ≡ π ( S /Z ) = Z and π ( S ) = Z topologicaldefects can exist in these condensed matter systems and willbe the subject of our future studies. Among many interest-ing questions that remain to be answered, one concerns find-ing the different ways in which various stable and metastablestates with the same topology (and Q ) can be realized in thestudied system. For example, the uniform unwound state andthe 3D solitons shown in Figs. 2 and 3 are all characterized by Q = 0, but it remains to be found whether other solitons with Q = 0 can be realized experimentally and as local free energyminima in modeling. Importantly, our approach of imagingpreimages is ideally suited to reveal the diversity and complex-ity of the 3D solitons in LCs. Indeed, we note that despitethe fact that polarizing optical micrographs (Figs. 1f,g, 6c,l,9f, 11a, and 12f) of different solitons differ from each other,they do not allow for determining the type of preimages andtheir interlinking, which is only possible to do on the basisof the 3D nonlinear optical imaging and with the help of themethod of preimages that we have introduced.Although 3D topological solitons have been studied as partof dynamic and transient phenomena in many different phys-ical systems (for example, see [14,16,22,48]), chiral nematicLCs are perhaps the only system in which these solitons arerealized as long-term stable configurations accessible to de-tailed experimental studies of their 3D structure and topology,which makes them ideally suited to serve as model systems forthe study of π ( S /Z ) = Z and π ( S ) = Z topological de-fects. For example, the initial interest in 3D topological soli-tons emerged in the fields of particle physics and cosmology [1,3, 6-10, 48], where they continue to play important roles [3].Although the Skyrme model (and related models such as theSkyrme-Faddeev model [9]) was initially proposed as a modeldescribing strong interactions of hadrons [6], Witten and col-leagues later demonstrated that similar ideas could be derivedon the basis of quantum chromodynamics (QCD) [49]. In thelow-energy pion dynamics model, certain elementary particles(including protons) can be thought of as π textures/solitons[48,49,50]. In addition, the 3D topological solitons are alsopredicted to occur in cosmology [48,50] and in many otherphysical systems [3], in which their detailed study is typicallyinaccessible to the direct experimentation. We thus expectthat the use of chiral nematic LCs as a model system to study3D topological solitons may impinge on their understandingin contexts of physics phenomena in other branches of physics. Furthermore, since the solitonic structures with different Hopfindices are topologically distinct from each other, transforma-tions between them are discontinuous and involve energeticbarriers. Thus, different types of solitonic structures can beobtained as long-lived states corresponding to locally differ-ent optical and other properties. Since the different solitonsincorporate different patterns of the effective refractive indexdistribution, they could serve as means of realizing recon-figurable phase gratings [44], pixels for bi-stable and multi-stable displays, etc. If the 3D topological solitons discussedhere can be also discovered in solid ferromagnets [24-29], theycan potentially revolutionize the field of Skyrmionics currentlyenabled by their two-dimensional counterparts, the so called“baby skyrmions” [28,29].To conclude, we have introduced an approach for experi-mental and numerical analysis of 3D topological solitons withnonzero Hopf invariants. Within this approach, inspired bythe mathematical Hopf maps, point-by-point, we experimen-tally scan the order parameter space (the S sphere or S /Z )and find regions within the sample that have orientationsof the director/vector corresponding to the point of S or S /Z . The same procedure was implemented numericallybased on field configurations arising from the minimizationof free energy, allowing for the unambiguous characterizationof the topology of 3D solitons. We have applied this analy-sis of experimental and numerical preimages and Hopf mapsas a means of uncovering an unexpectedly large diversity of3D spatially localized solitonic structures in confined chiralnematic LCs. We have revealed a host of torons, hopfions,and other solitons with complex linking of closed-loop preim-ages and both with and without singular point defects. Self-assembly of such 3D solitons with different topological char-acteristics may result in emergence of new condensed matterphases with rich phase diagrams and unusual physical behav-ior. A comparison of nematic and ferromagnetic hopfions andtorons that we studied recently [22] will allow for probing therole of field polarity in the topology of knotted solitons. Fi-nally, the experimental and theoretical frameworks that wehave introduced will help establishing chiral nematic LCs asa test-bed for the study of 3D topological solitons, which areabundant in theories in practically all branches of physics. We are grateful to D. Broer for providing the RM-82 andRM-257 reactive mesogens. We acknowledge discussions withS. de Alwis, T. DeGrand, N. Clark, R. Kamien, T. Lubensky,H. Osman, and M. Tasinkevych, as well as the support of theUS National Science Foundation Grant DMR-1410735.
References [1] L. H. Kauffman, Knots and Physics. (World Scientific Publishing, 2001).[2] W. Thomson, On vortex atoms. Philos. Mag. 34, 15-24 (1867).[3] N. Manton and P. Sutcliffe, Topological Solitons. (Cambridge University Press, 2004).[4] H. Hopf, ¨Uber die Abbildungen der dreidimensionalen Sph¨are auf die Kugelfl¨ache. Math. Ann. 104, 637 −
665 (1931).22 r a f t [5] D. Finkelstein, Kinks. J. Math. Phys. 7, 1218 (1966).[6] T. H. R. Skyrme, A Non-Linear Field Theory. Proc. R. Soc. A 260, 127 −
138 (1961).[7] R. H. Hobart, On the Instability of a Class of Unitary Field Models. Proc. Phys. Soc. London 82, 201 −
203 (1963).[8] G. H. Derrick, Comments on Nonlinear Wave Equations as Models for Elementary Particles. J. Math. Phys. 5, 1252(1964).[9] L. Faddeev, and A. J. Niemi, Stable knot-like structures in classical field theory. Nature 387, 58 −
61 (1997).[10] R. A. Battye, and P. M. Sutcliffe, Knots as Stable Soliton Solutions in a Three-Dimensional Classical Field Theory.Phys. Rev. Lett. 81, 4798 − − −
720 (2008).[13] I. I. Smalyukh, Y. Lansac, N. A. Clark, and R. P. Trivedi, Three-dimensional structure and multistable optical switchingof triple-twisted particle-like excitations in anisotropic fluids. Nat. Mater. 9, 139 −
145 (2010).[14] B. G. Chen, P. J. Ackerman, G. P. Alexander, R. D. Kamien, and I. I. Smalyukh, Generating the Hopf FibrationExperimentally in Nematic Liquid Crystals. Phys. Rev. Lett. 110, 237801 (2013).[15] D. S. Hall, et al. Tying quantum knots. Nat. Phys. (2016). doi:10.1038/nphys3624[16] D. M. Kleckner, and W. T. Irvine, Creation and dynamics of knotted vortices. Nat. Phys. 9, 253 −
258 (2013).[17] S. Bolognesi, and M. Shifman, Hopf Skyrmion in QCD with adjoint quarks. Phys. Rev. D 75, 065020 (2007).[18] A. Gorsky, M. Shifman, and A. Yung, Revisiting the Faddeev-Skyrme model and Hopf solitons. Phys. Rev. D 88, 045026(2013).[19] A. Acus, E. Norvaias, and Y.Shnir, Hopfions interaction from the viewpoint of the product ansatz. Phys. Lett. B 733,15 −
20 (2014).[20] A. Thompson, A., Wickes, J. Swearngin, and D. Bouwmeester, Classification of electromagnetic and gravitationalhopfions by algebraic type. J. Phys. A 48, 205202 (2015).[21] M. Kobayashi and M. Nitta, Torus knots as Hopfions. Phys. Lett. B 728, 314 −
318 (2014).[22] Q. Zhang, P. J. Ackerman, Q. Liu, and I. I. Smalyukh, Ferromagnetic Switching of Knotted Vector Fields in LiquidCrystal Colloids. Phys. Rev. Lett. 115, 097802 (2015).[23] P. M. Chaikin, and T. C. Lubensky, Principles of Condensed Matter Physics. (Cambridge Univ. Press, 2000).[24] A. N. Bogdanov, and A. J. Hubert, Thermodynamically stable magnetic vortex states in magnetic crystals. J. Magn.Magn. Mater. 138, 255 −
269 (1994).[25] U. K. Rler, A. N. Bogdanov, and C. Pfleiderer, Spontaneous skyrmion ground states in magnetic metals. Nature 442,797 −
801 (2006).[26] N. Romming, et al. Writing and Deleting Single Magnetic Skyrmions. Science 341, 636 −
639 (2013).[27] X. Z. Yu, et al. Real-space observation of a two-dimensional skyrmion crystal. Nature 465, 901 −
904 (2010).[28] A. Fert, V. Cros, and J. Sampaio, Skyrmions on the track. Nat. Nanotech. 8, 152 −
156 (2013).[29] N. Nagaosa, and Y. Tokura, Topological properties and dynamics of magnetic skyrmions. Nat. Nanotech. 8, 899 − r a f t [33] Y. Bouligand. Research on the textures of mesomorphic states. V. Nuclei, threads and Moebius ribbons in nematics andweakly twisted cholesterics. J. Physique 35, 959 (1974).[34] Q. Liu, P. J. Ackerman, T. C. Lubensky, and I. I. Smalyukh, Biaxial ferromagnetic liquid crystal colloids. Proc. Natl.Acad. Sci. (2016). 10.1073/pnas.1601235113[35] R. P. Trivedi, I.I. Klevets, B.I. Senyuk, T. Lee, and I.I. Smalyukh. Multi-scale interactions and three-dimensionalpatterning of colloidal particles and defects in lamellar soft media, Proc. Nat. Acad. Sci. U.S.A. 109, 4744 − − − − −
263 (2014).[44] I. I. Smalyukh, D.S. Kaputa, A.V. Kachynski, A.N. Kuzmin, and P.N. Prasad. Optical trapping of director structuresand defects in liquid crystals using laser tweezers. Opt. Exp. 15, 4359 − −
67 (1999).[46] E. Efrati and W. T. M. Irvine, Orientation-Dependent Handedness and Chiral Design. Phys. Rev. X 4, 011003 (2014).[47] B. Senyuk, Q. Liu, S. He, R. D. Kamien, R. B. Kusner, T. C. Lubensky, and I. I. Smalyukh. Topological colloids. Nature493, 200 −
205 (2013).[48] A. Chuang, A. Durrer, N. Turok, and B. Yurke. Cosmology in the laboratory: defect dynamics in liquid crystals. Science251, 1336 − − −−