Schools of skyrmions with electrically tunable elastic interactions

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


Introduction.
Soft matter and living systems are commonly described as close cousins 1 , both with properties stemming from interactions between the constituent building blocks that are comparable in strength to thermal fluctuations. Active soft matter systems 2,3 are additionally out-of-equilibrium in nature, like everything alive. They exhibit emergent collective dynamics that closely mimic such behavior in living systems [2][3][4] . For example, mechanically-agitated fluidized monolayers of rods form a dynamic granular liquid crystal (LC) 5 . Coherent motion emerges in many systems where particles communicate through collisions or short-range interactions like screened electrostatic repulsions [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] . However, these interactions typically cannot be tuned in strength or switched from attractive to repulsive. Moreover, with a few exceptions [5][6][7]10,11,[19][20][21][22] , including the ones in which electric energy is used to power motions 6,7,11 , most active matter systems have biological origins and either chemical or mechanical energy conversion within the constituent building blocks. This poses the grand challenge to develop versatile reconfigurable active matter formed by inanimate, man-made particles both as models of biological systems and for technological uses 21 .
We describe an emergent collective dynamic behavior of skyrmions [23][24][25] , particle-like twodimensional (2D) topological analogs of Skyrme solitons used to model atomic nuclei with different baryon numbers 26,27 . In LCs 1,28 , these skyrmions are elements of the second homotopy group 29 and contain smooth but topologically-nontrivial, spatially-localized structures in the alignment field of constituent rod-like molecules, the director field n(r). They are characterized by integer-valued topological invariants, the skyrmion numbers 29 . Depending on the applied voltage, the internal n(r)-structures within our skyrmions adopt different orientations relative to the 2D sample plane and the far-field alignment. Thousands-to-millions of skyrmions start from random orientations and motions while each individually converting energy due to oscillating voltage, but then synchronize motions and develop polar ordering within seconds. The ensuing schools of topological solitons differ from all previously studied systems  because skyrmions have no physical boundaries, membranes, chemical composition or density gradients, or singularities in the order parameter at the level of the host fluid [30][31][32][33][34][35][36] , even though they exhibit giant number fluctuations in terms of the skyrmionic, topologically protected, localized structures of n(r). Although the LC medium is nonpolar, spontaneous symmetry breaking and many-body dynamic interactions lead to polar ordering of skyrmions, which is characterized by near-unity values of polar and velocity order parameters. Electrically tunable interactions stemming from the orientational elasticity of LCs 30,31 provide a versatile means of controlling this behavior while probing order and giant number fluctuations within the schools. The dynamic multi-skyrmion assemblies echo formation of high-baryon-number skyrmions in nuclear physics due to addition of charge-one topological invariants, which in equilibrium condensed matter could be only achieved when forming skyrmion bags 29 , very differently from the behavior of singular active matter topological defects that conserve the net winding number to always add to zero 2,33-36 . Our findings highlight the interplay between nonsingular topology of field configurations and out-ofequilibrium behavior and promise a host of technological uses.

Results.
Skyrmion schooling. Schooling of fish (Fig. 1a), like many other forms of collective motions 2 , is accompanied by inhomogeneities and dynamic local clustering. Similar behavior is observed in our rather unusual schools formed by thousands-to-millions of localized particle-like skyrmionic orientational structures of n(r) within LCs (Fig. 1b-f). While moving and bypassing obstacles, these skyrmions exhibit dynamically self-reconfigurable assembly ( Fig. 1f and Supplementary Movie 1). In our experiments, under conditions and sample preparation similar to that in LC displays, such skyrmions are controllably mass-produced at different initial densities (Methods) 32 and also generated one-by-one using laser tweezers 23,29 . Skyrmion stability is enhanced by soft perpendicular boundary conditions on the inner surfaces of confining glass plates (Fig. 1d) 23 and the LC's chirality, which prompts n(r) twisting 23,24 . At no fields, the structure of each skyrmion is axisymmetric (Fig. 1g) 23 , with -twist of n(r) from the center to periphery in all radial directions and containing all possible n(r)-orientations within it. Due to the used LC's negative dielectric anisotropy, electric field E applied across the cell tends to align n(r)⊥E (Fig. 1d), so that n(r) around the skyrmions progressively tilts away from the cell normal with increasing voltage U, whereas n(r) within the skyrmions morphs from an originally axisymmetric structure (Fig. 1g) to a highly asymmetric one (Fig. 1j,k) that matches this tilted director surrounding. Since the tilting of n(r) in E with respect to the sample plane breaks the nonpolar symmetry of the resulting effectively-2D structure, we vectorize n(r) and visualize it with arrows colored by orientations and corresponding points in the two-sphere order parameter space (Fig. 1g) 29 . The asymmetric skyrmion is described by a preimage vector connecting preimages of the south and north poles of ( Fig. 1k), the regions where n(r) points into and out of the sample plane, respectively. The skyrmion number, a topological invariant describing how many times n(r) within the single skyrmion wraps around , remains equal to unity, indicating topological stability with respect to smooth n(r)-deformations prompted by changing U. This is confirmed by experimental polarizing optical micrographs of individual skyrmions at different U (Fig. 1i,m) that closely match their computer-simulated counterparts (Fig. 1h,l). Skyrmions exhibit only Brownian motion at no fields and at high frequencies (like 1kHz) of applied field, at which n(r) cannot follow the temporal changes of E and responds to its time average. 24 At frequencies for which the voltage oscillation period TU =1/f is comparable or larger than the LC response time, spatial translations of individual skyrmions 24 in an oscillating electric field arise because the temporal evolution of asymmetric n(r) is not invariant upon reversal of time with turning U on and off within each TU (Figs. 1g-m and 2a,b). Because of the initial axisymmetric structure of skyrmions and homeotropic n(r) background around them, the spontaneous symmetry breaking leads to random motion directions of individual skyrmions within the sample plane. This feature of our system indicates that the coherent unidirectional motion of many skyrmions within the thousands-to-million schools is an emergent phenomenon with a physical mechanism that relies on inter-skyrmion interactions, which we explore in detail below.
Collective motions of two-to-hundred skyrmions. To gain insights into the emergent schooling with electrically reconfigurable clusters of topological solitons ( Fig. 1g-m), we first probe how the dynamics of skyrmions change with increasing number density (Fig. 2). Using laser tweezers, we set up a "race" by arranging skyrmion pairs along a straight line together with single skyrmions  Fig. 1). Since spatial translations of skyrmions arise from temporal evolution of asymmetric n(r) not invariant upon reversal of time with turning U on and off, skyrmions within large assemblies tend to share these asymmetric distortions to reduce the elastic free energy 30,31 , which impedes their translations while in tightly packed reducedasymmetry assemblies. However, with the oscillating field's modulation changing to f = 1 Hz, the solitons tend to spread apart, so that the motion of well-spread pairs is then nearly as fast as that of single skyrmions (Fig. 2b). The dynamic evolution of n(r) during motions of skyrmion pairs is enriched by elastic interactions that tend to minimize the ensuing elastic free energy costs due to arranging skyrmions at different relative positions and can be controlled from attractive to repulsive (Fig. 2c). Tuning U and f alters these inter-skyrmion interactions and reconfigures larger that this complex behavior can be understood by taking into account the out-of-equilibrium elastic interactions between skyrmions that arise from minimizing the elastic free energy due to partial sharing of n(r)-distortions associated with multiple moving skyrmions, as we detail below.
Out-of-equilibrium elastic interactions. Elastic interactions between skyrmions emerge to reduce the free energy costs of n(r)-distortions around these topological solitons, like in nematic colloids [29][30][31][32] , albeit typically without the dynamic n(r) fully reaching equilibrium because of the voltage modulation and soliton motions. The elastic interactions between skyrmions confined to a 2D plane have dipolar nature 29 , though the complex temporal evolution of n(r) in periodically modulated U makes these elastic dipoles effectively change their tilt relative to the 2D sample plane within TU and self-propel while they interact. Such dynamic dipolar skyrmions mutually repel at small U, but exhibit anisotropic interactions (Fig. 2c), including attractions, when oscillating E prompts their n(r) symmetry breaking (Fig. 1j,k) and motions. Oscillating E rotates preimage dipoles from pointing orthogonally to the sample plane at U=0 (Fig. 1g) to being tilted or in-plane (Fig. 1j,k) when U increases, with the effective tilt periodically changing with a voltage modulation period (Fig. 1d)  and vs is the absolute value of velocity of a coherently-moving school. Both S and Q increase from zero to ~0.9 within seconds (Fig. 4d), indicating the emergence of coherent unidirectional motion of polar skyrmionic particles, like that of fish in schools 2,3 . At relatively low initial packing fractions (~0.1 by area) we observe no clustering of skyrmions as they move coherently within the schools, repelling each other at short distances and weakly attracting at larger distances (Fig. 4c).
This emergent behavior is different from pair interactions and dynamics at similar voltages ( Fig.   2c), where moving skyrmions tend to attract to form chains at shorter inter-skyrmion distances.
The presence of such short-range repulsive and long-range attractive interactions is consistent with the formation of coherently-moving schools and results from many-body interactions (Fig. 4c), where elastic interactions between skyrmions with periodically-evolving n(r) are further enriched by backflows and electro-kinetic effects. 28  Edges of schools are well defined regardless of the internal clustering within the schools ( Fig. 6a-d). Individual skyrmions, which happen to be slowed down by imperfections (Fig. 6e), move faster than the clusters and thus "catch up" to the school's edge. Although the speed decreases as the number of skyrmions within the clusters increases, this reduction is less than 50% even for very large clusters containing over 100 skyrmions (Fig. 6f), also showing how the dynamic behavior under schooling conditions differs from that of individual skyrmions and their small-to-medium clusters ( Supplementary Fig. 1).

Diagram of dynamic and static states.
We summarize the schooling behavior of skyrmions using a structural diagram (Fig. 7), where we present results for square-wave electric fields oscillating at frequency f (we note that the diagram of states changes when other electric signal waveforms and various carrier frequencies are used, although exploration of all these parameter spaces is outside the scope of our present work). Being unstable at high U and low f (Fig. 7), skyrmions at oscillating U revealed by numerical modeling (Fig. 5f-m) and qualitatively discussed above by using the dipolar elastic interactions. Fine details of clusters, like inter-particle distances ( Fig. 5n) vary along the f-axis and would be difficult to capture within a single diagram of states, but the simplified three-dimensional diagram in Fig. 7 overviews the tendencies and helps to emphasize the physical underpinnings of the observed rich out-of-equilibrium behavior, which we summarize below.

Discussion.
Playing response time is longer than TU, so that the skyrmions are always asymmetric with periodicallychanging preimage dipole tilts (Fig. 1j,k). When the instantaneous U within TU drops to zero, the elastic torque tends to relax the n(r) of all skyrmions to an axially symmetric state shown in Fig.   1g, but well before this happens, competing electric and elastic torques re-morph the skyrmions back to the highly-asymmetric structures (Fig. 1j,k). While a viscous torque resists changes of

Sample preparation
Chiral nematic LC mixtures with negative dielectric anisotropies were prepared by mixing a chiral  23 The LCs were mixed with ~0.1wt% of cationic surfactant Hexadecyltrimethylammonium bromide (CTAB, purchased from Sigma-Aldrich) in order to allow for inducing electrohydrodynamic instability with low-frequency applied field 32 . The CTAB doping allowed for facile generation of large numbers (thousands to millions) of skyrmions at different initial packing fractions upon relaxation of the cells from electrohydrodynamic instability (Fig. 9), though the presence of CTAB is not required for skyrmion schooling as similar dynamics could also be obtained in samples without CTAB and

Generation, manipulation, pinning and control of skyrmions
Skyrmions can be formed spontaneously upon thermally quenching the sample from the isotropic to the chiral nematic phase, by relaxing the LC from electrohydrodynamic instability (Fig. 9), or by a direct one-by-one optical generation using holographic laser tweezers 23,32 . Spontaneous  Fig. 9), whereas control of distortions in these two cases allows for defining initial densities and locations of skyrmions. In order to morph skyrmions and power their motions via macroscopically-supplied energy, electric field was applied across the cell using a homemade MATLAB-based voltage-application program coupled with a data-acquisition board (NIDAQ-6363, National Instruments) 24 .
The means of controlling dynamics of skyrmions include voltage driving schemes, selection of LCs with different material parameters and chiral additives ( Table 1), design of LC cell geometry, and strength of surface boundary conditions, etc. Collective motion effects could be obtained both when simply using low frequency oscillating field (yielding field oscillations at time scales comparable to the LC's response time) and when modulating high-frequency carrier signals (e.g. at 1kHz) at modulation frequencies that again yield modulation periods comparable to the LC's response time. The large range of possibilities exists in terms of controlling the skyrmion schooling by varying electric signal waveforms, modulation and carrier frequencies and so on, but detailed exploration of all these possibilities is outside the scope of our present study.

Numerical modeling
We  (Table 1). As the applied voltage is modulated, the competing electric and elastic torques are balanced by a viscous torque associated with rotational viscosity, γ, that opposes the fast rotation of the director 1 . The resulting director dynamics is governed by a torque balance equation 1 , γ∂ni/∂t = −δW/δni, from which both the equilibrium n(r) and the effective temporal evolution of the director field towards equilibrium are obtained for the director, ni(t) where ni is the component of n along the i th axis (i = x, y, z). As in experiments, we start our computer simulations from skyrmions embedded in a homeotropic LC background, for which the equilibrium director structure is obtained by minimizing free energy in Eq. (1) at no external fields ( Fig. 1g), as detailed in our previous studies. 24,29 Then, we minimize free energy to obtain skyrmion's field configuration's at various applied voltages (Fig. 1j,k) when starting from the skyrmion structure at no fields (Fig. 1g) within the computational volume as the initial condition.
In a similar way, to obtain multi-skyrmion clusters or chains, we start from minimizing free energy at U=0 for seven (Fig. 5f) or five (Fig. 5j) (Table 1).

Optical microscopy, video characterization and data analysis
Images and videos were obtained using charge-coupled device cameras Grasshopper (purchased     respectively. e, Log-log plots of N versus <N> for clusters (red) and chains (blue); linear fits with slope  values and a black dashed line with  = 0.5 (for reference) are displayed. f,g, Computer-simulated moving cluster of seven quasi-hexagonally-assembled skyrmions at U=3.5V displayed using arrows colored according to the color scheme in Fig.1g (f) and corresponding smoothly-colored representation of n(r) (g). h, Simulated and i, experimental polarizing optical micrographs of a similar moving cluster. j,k, Simulated moving linear chain of five skyrmions at U=4.0V (j) with corresponding smoothly-colored n(r) representation (k). l, Simulated and m, experimental polarizing micrographs of the moving chain. Scale bars in (i) and (m) are 10 µm. Upon turning voltage off and on again while minimizing n(r), the clusters in (f,j) shift laterally within the computational volume by about ~0.5p. n, Experimental normalized distributions for inter-skyrmion distance in chains during motion versus U for f=50, 75 and 100 Hz. Polarizer orientations (double arrows), vs and E are labeled throughout. The LC is the nematic ZLI-2806 doped with the chiral additive CB-15.    , electrohydrodynamic instability at U=7V (2nd image from the left), skyrmions at low number density forming upon switching back to U=0 and relaxing the sample from electrohydrodynamic instability (3rd image from the left), electrohydrodynamic instability in the same sample area upon again applying U=10V (4th image from the left), and, finally, skyrmions at high number density forming upon switching back to U=0 again and allowing the instability to relax (5th image from the left). These images illustrate the control of the initial skyrmion packing fraction (by area, displayed in the top-right of the images), where the amplitude of U used to generate electrohydrodynamic instabilities corelates with the ensuing skyrmion number densities. White double arrows denote crossed polarizer orientations. Elapsed time, skyrmion packing fraction by area, applied voltage, and direction of E are marked in the corners of micrographs. The used electric field frequency was 10Hz.