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Ivan I. Smalyukh

Publications and source records attributed to Ivan I. Smalyukh.

At least 19 recordsLinked to original sources
arXiv2026

Curvature-guided topology and self-assembly in chiral nematics and liquid-crystal colloids

In soft condensed matter, curvature does more than simply distort an ordered medium: it helps select defect structures, redistribute elastic stress, bias chirality, and guide self-assembly. This review examines how curved, multiply connected, and knotted boundaries in liquid-crystal colloids and confined nematics generate topological defects and localized solitonic textures, and how these structures mediate interactions between mesoscale building blocks. We introduce a unifying framework based on genus, Euler characteristic, anchoring, and chirality, and use it to discuss spherical, handlebody, and boundary-bearing colloids, together with droplets and polymer-dispersed nematics of nontrivial topology. Particular emphasis is placed on the interplay of geometry and topology in determining boojums, disclination loops, hedgehog charges, and linked and knotted defect structures. We then turn to chiral systems hosting skyrmions, torons, hopfions, and related localized textures, highlighting how chirality and confinement stabilize three-dimensional topological states. Finally, we discuss how these concepts translate into design principles for controlled self-assembly, templating, and functional composite materials. More broadly, we argue that liquid-crystal colloids and confined nematics provide experimentally accessible model systems in which curvature, topology, and chirality can be harnessed as programmable tools for designing organized soft matter.

cond-mat.soft↗
arXiv2026

From Knots to Crystals: Machine-Learned Potentials for Self-Assembling Topological Solitons in Liquid Crystals

Knotted fields in classical and quantum systems have long been recognized for their non-trivial topologies and particle-like behavior, but practical applications have been limited by the difficulty of stabilizing them. Recently, stable knotted solitonic textures--heliknotons--were discovered in chiral liquid crystals, forming adaptive crystal assemblies via elastic distortion-mediated interactions. We use machine learning to develop single-site coarse-grained potentials that accurately capture these chiral anisotropic effective interactions. The resulting potentials accurately reproduce experimentally observed heliknoton assemblies and enable simulations at length and time scales far beyond the range of fine-grained continuum models. This general framework is readily transferable to other topological solitons, providing a powerful route to understand, predict, and ultimately control their collective behavior and dynamics.

cond-mat.soft↗
arXiv2026

Analysis of Hopf solitons as generalized fold maps

The Hopf index, a topological invariant that quantifies the linking of preimage fibers, is fundamental to the structure and stability of hopfions. In this work, we propose a new mathematical framework for modeling hopfions with high Hopf index, drawing on the language of singularity theory and the topology of differentiable maps. At the core of our approach is the notion of a generalized Hopf map of order $n$, whose structure is captured via fold maps and their Stein factorizations. We demonstrate that this theoretical construction not only aligns closely with recent experimental observations of high-Hopf-index hopfions, but also offers a precise classification of the possible configurations of fiber pairs associated to distinct points. Our results thus establish a robust bridge between the geometry of singular maps and the experimentally observed topology of complex field configurations of hopfions in materials and other physical systems.

cond-mat.soft↗
arXiv2026

Landau-de Gennes numerical simulation of nematic liquid crystals utilizing radial basis functions

Numerical simulations based on radial basis functions have been developed for systems with complex geometries and have been successfully applied across various fields, including seismology, coastal hydrodynamics, and biology. However, examples in liquid crystal modeling are limited. In this study, we present a Landau-de Gennes numerical simulation of nematic liquid crystals utilizing radial basis functions, emphasizing its advantages over traditional cubic grid calculations, such as enhanced geometric flexibility and improved computational efficiency. Through simulations of liquid crystal-colloid systems with diverse geometries, we demonstrate that our approach effectively captures the essential topological and energetic features of liquid crystal equilibrium structures. Additionally, we introduce an adaptive node refinement scheme that is crucial for resolving the fine structure of singular defects in nematic liquid crystals.

cond-mat.soft↗
arXiv2026

The 2026 Skyrmionics Roadmap

Magnetic skyrmions and related topological spin textures have emerged as a central topic in condensed-matter physics, combining fundamental significance with potential for transformative applications in spintronics, magnonics, and beyond. Over the past decade, advances in material platforms, imaging techniques, theoretical modeling, and device concepts have established skyrmionics as a rapidly expanding field. At the same time, challenges remain in stabilizing, controlling, and integrating such textures into functional architectures, while novel phenomena such as antiskyrmions, higher-order skyrmions, hopfions, and antiferromagnetic textures arise. The 2026 Skyrmionics Roadmap represents a collective effort of many authors, providing a comprehensive perspective on the current state-of-the-art and the outlook for the coming years. In 33 focused sections, each co-authored by two researchers, we chart progress in theory and modeling, material systems, skyrmion dynamics, and skyrmion technologies. By offering a consolidated vision, this Roadmap aims to guide both fundamental research and application-driven efforts, accelerating the transition of skyrmionics from conceptual breakthroughs toward practical technologies.

cond-mat.mes-hall↗
arXiv2025

Fusion and Fission of Particle-like Chiral Nematic Vortex Knots

Vortex knots have been seen decaying in many physical systems. Here we describe topologically protected vortex knots, which remain stable and undergo fusion and fission while conserving a topological invariant analogous to that of baryon number. While the host medium, a chiral nematic liquid crystal, exhibits intrinsic chirality, cores of the vortex lines are structurally achiral regions where twist cannot be defined. We refer to them as "dischiralation" vortex lines, in analogy to dislocations and disclinations in ordered media where, respectively, positional and orientational order is disrupted. Fusion and fission of these vortex knots, which we reversibly switch by electric pulses, vividly reveal the physical embodiments of knot theory's concepts like connected sums of knots. Our findings provide insights into related phenomena in fields ranging from cosmology to particle physics and can enable applications in electro-optics and photonics, where such fusion and fission processes can be used for controlling light.

cond-mat.soft↗
arXiv2025

Multiscale geometrical and topological learning in the analysis of soft matter collective dynamics

Understanding the behavior and evolution of a dynamical many-body system by analyzing patterns in their experimentally captured images is a promising method relevant for a variety of living and non-living self-assembled systems. The arrays of moving liquid crystal skyrmions studied here are a representative example of hierarchically organized materials that exhibit complex spatiotemporal dynamics driven by multiscale processes. Joint geometric and topological data analysis (TDA) offers a powerful framework for investigating such systems by capturing the underlying structure of the data at multiple scales. In the TDA approach, we introduce the $Ψ$-function, a robust numerical topological descriptor related to both the spatiotemporal changes in the size and shape of individual topological solitons and the emergence of regions with their different spatial organization. The geometric method based on the analysis of vector fields generated from images of skyrmion ensembles offers insights into the nonlinear physical mechanisms of the system's response to external stimuli and provides a basis for comparison with theoretical predictions. The methodology presented here is very general and can provide a characterization of system behavior both at the level of individual pattern-forming agents and as a whole, allowing one to relate the results of image data analysis to processes occurring in a physical, chemical, or biological system in the real world.

cond-mat.soft↗
arXiv2025

Space-time crystals from particle-like topological solitons

Time crystals are unexpected states of matter that spontaneously break time translation symmetry either in a discrete or continuous manner. However, spatially-mesoscale space-time crystals that break both the space and time symmetries have not been reported. Here we report a continuous space-time crystal in a nematic liquid crystal driven by ambient-power, constant-intensity unstructured light. Our numerically constructed 4-dimensional configurations exhibit good agreement with these experimental findings. While meeting the established criteria to identify time-crystalline order, both experiments and computer simulations reveal a space-time crystallization phase formed by particle-like topological solitons. The robustness against temporal perturbations and spatiotemporal dislocations shows the stability and rigidity of the studied space-time crystals, which relates to their locally topological nature and many-body interactions between emergent spontaneously-twisted, particle-like solitonic building blocks. Their potential technological utility includes optical devices, photonic space-time crystal generators, telecommunications, and anti-counterfeiting designs, among others.

cond-mat.soft↗
arXiv2025

Emergent discrete space-time crystal of Majorana-like quasiparticles in chiral liquid crystals

Time crystals spontaneously break the time translation symmetry, as recently has been frequently reported in quantum systems. Here we describe the observation of classical analogues of both 1+1-dimensional and 2+1-dimensional discrete space-time crystals in a liquid crystal system driven by a Floquet electrical signal. These classical time crystals comprise particle-like structural features and exists over a wide range of temperatures and electrical driving conditions. The phenomenon-enabling period-doubling effect comes from their topological Majorana-like quasiparticle features, where periodic inter-transformations of co-existing topological solitons and disclinations emerge in response to external stimuli and play pivotal roles. Our discrete space-time crystals exhibit robustness against temporal perturbations and spatial defects, behaving like a time-crystalline analogues of a smectic phase. Our findings show that the simultaneous symmetry breaking in time and space can be a widespread occurrence in numerous open systems, not only in quantum but also in a classical soft matter context.

cond-mat.soft↗
arXiv2025

Hybridization of colloidal handlebodies with singular defects and topological solitons in chiral liquid crystals

Topology can manifest itself in colloids when quantified by invariants like Euler characteristics of nonzero-genus colloidal surfaces, albeit spherical colloidal particles are most often studied, and colloidal particles with complex topology are rarely considered. On the other hand, singular defects and topological solitons often define the physical behavior of the molecular alignment director fields in liquid crystals. Interestingly, nematic liquid crystalline dispersions of colloidal particles allow for probing the interplay between topologies of surfaces and fields, but only a limited number of such cases have been explored so far. Here, we study the hybridization of topological solitons, singular defects, and topologically nontrivial colloidal particles with the genus of surfaces different from zero in a chiral nematic liquid crystal phase. Hybridization occurs when distortions separately induced by colloidal particles and LC solitons overlap, leading to energy minimization-driven redistribution of director field deformations and defects. As a result, hybrid director configurations emerge, combining topological features from both components. We uncover a host of director field configurations complying with topological theorems, which can be controlled by applying electric fields. Rotational and translational dynamics arise due to the nonreciprocal evolution of the director fields in response to alternating electric fields of different frequencies. These findings help define a platform for controlling topologically hyper-complex colloidal structures and dynamics with electrically reconfigurable singular defects and topological solitons induced by colloidal handlebodies.

cond-mat.soft↗
arXiv2025

Emergent dimer-model topological order and quasi-particle excitations in liquid crystals: combinatorial vortex lattices

Liquid crystals have proven to provide a versatile experimental and theoretical platform for studying topological objects such as vortices, skyrmions, and hopfions. In parallel, in hard condensed matter physics, the concept of topological phases and topological order has been introduced in the context of spin liquids to investigate emergent phenomena like quantum Hall effects and high-temperature superconductivity. Here, we bridge these two seemingly disparate perspectives on topology in physics. Combining experiments and simulations, we show how topological defects in liquid crystals can be used as versatile building blocks to create complex, highly degenerate topological phases, which we refer to as 'Combinatorial Vortex Lattices' (CVLs). CVLs exhibit extensive residual entropy and support locally stable quasi-particle excitations in the form of charge-conserving topological monopoles, which can act as mobile information carriers and be linked via Dirac strings. CLVs can be rewritten and reconfigured on demand, endowed with various symmetries, and modified through laser-induced topological surgery - an essential capability for information storage and retrieval. We demonstrate experimentally the realization, stability, and precise optical manipulation of CVLs, thus opening new avenues for understanding and technologically exploiting higher-hierarchy topology in liquid crystals and other ordered media.

cond-mat.soft↗
arXiv2024

Topological Rigidity and Non-Abelian defect junctions in chiral nematic systems with effective biaxial symmetry

We study topologically stable defect structures in systems where the defect line classification in three dimensions and associated algebra of interactions (the fundamental group) are governed by the non-Abelian 8-element group, the quaternions Q_8. The non-Abelian character of the defect algebra leads to a topological rigidity of bound defect pairs, and trivalent junctions which are the building blocks of multi-junction trivalent networks. We realize such structures in laboratory chiral nematics and analyze their behavior analytically, along with numerical modeling.

cond-mat.soft↗
arXiv2024

Chiral, Topological, and Knotted Colloids in Liquid Crystals

The geometric shape, symmetry, and topology of colloidal particles often allow for controlling colloidal phase behavior and physical properties of these soft matter systems. In liquid crystalline dispersions, colloidal particles with low symmetry and nontrivial topology of surface confine-ment are of particular interest, including surfaces shaped as handlebodies, spirals, knots, mul-ti-component links, and so on. These types of colloidal surfaces induce topologically nontrivial three-dimensional director field configurations and topological defects. Director switching by electric fields, laser tweezing of defects, and local photo-thermal melting of the liquid crystal host medium promote transformations among many stable and metastable particle-induced director configurations that can be revealed by means of direct label-free three-dimensional nonlinear op-tical imaging. The interplay between topologies of colloidal surfaces, director fields, and defects is found to show a number of unexpected features, such as knotting and linking of line defects, often uniquely arising from the nonpolar nature of the nematic director field. This review article high-lights fascinating examples of new physical behavior arising from the interplay of nematic mo-lecular order and both chiral symmetry and topology of colloidal inclusions within the nematic host. Furthermore, the article concludes with a brief discussion of how these findings may lay the groundwork for new types of topology-dictated self-assembly in soft condensed matter leading to novel mesostructured composite materials, as well as for experimental insights into the pure-math aspects of low-dimensional topology.

cond-mat.soft↗
arXiv2024

Liquid crystal torons in Poiseuille-like flows

Three-dimensional (3D) simulations of the structure of liquid crystal (LC) torons, topologically protected distortions of the LC director field, under material flows are rare but essential in microfluidic applications. Here, we show that torons adopt a steady-state configuration at low flow velocity before disintegrating at higher velocities, in line with experimental results. Furthermore, we show that under partial slip conditions at the boundaries, the flow induces a reversible elongation of the torons, also consistent with the experimental observations. These results are in contrast with previous simulation results for 2D skyrmions under similar flow conditions, highlighting the need for a 3D description of this LC soliton in relation to its coupling to the material flow. These findings pave the way for future studies of other topological solitons, like hopfions and heliknotons, in flowing soft matter systems.

cond-mat.soft↗
arXiv2024

"Hall" transport of liquid crystal solitons in Couette flow

Topology establishes a unifying framework for a diverse range of scientific areas including particle physics, cosmology, and condensed matter physics. One of the most fascinating manifestations of topology in the context of condensed matter is the topological Hall effect, and its relative: the Skyrmion Hall effect. Skyrmions are stable vortex-like spin configurations in certain chiral magnets, and when subject to external electric currents can drift in the transverse direction to the current. These quasi-particles are characterised by a conserved topological charge which in the Skyrmion Hall effect plays the role of electric charges in the ordinary Hall effect. Recently, it has been shown that liquid crystals endowed with chiral properties serve as an ideal testbed for the fundamental investigation of topological solitons, including their two- and three-dimensional realisations. Here, we show experimentally and numerically that three-dimensional solitons aka "torons" exhibit a Hall-like effect when driven by shear flows: the torons are deflected in the direction perpendicular to the shear plane. The experimental results are rationalised in terms of the dynamic Ericksen-Leslie equations, which predict the emergence of the transverse component of the net mass flow, the magnitude of which scales as the 3rd power of the shear rate. The perturbation analysis highlights an interplay of the viscous and chiral elastic torques as the mechanism for the emergence of net transverse currents. Numerical simulations demonstrate, however, that torons are not merely dragged by the flow but move with their own transverse speed, much larger than the average flow velocity in the transverse direction. Our findings may enable responsive microfluidic applications relying on soft topological solitons.

cond-mat.soft↗
arXiv2023

Out-of-equilibrium interactions and collective locomotion of colloidal spheres with squirming of nematoelastic multipoles

Many living and artificial systems show a similar emergent behavior and collective motions on different scales, starting from swarms of bacteria to synthetic active particles, herds of mammals and crowds of people. What all these systems often have in common is that new collective properties like flocking emerge from interactions between individual self-propelled or externally driven units. Such systems are naturally out-of-equilibrium and propel at the expense of consumed energy. Mimicking nature by making self-propelled or externally driven particles and studying their individual and collective motility may allow for deeper understanding of physical underpinnings behind the collective motion of large groups of interacting objects or beings. Here, using a soft matter system of colloids immersed into a liquid crystal, we show that resulting so-called nematoelastic multipoles can be set into a bidirectional locomotion by external periodically oscillating electric fields. Out-of-equilibrium elastic interactions between such colloids lead to collective flock-like behaviors, which emerge from time-varying elasticity-mediated interactions between externally driven propelling particles. The repulsive elastic interactions in the equilibrium state can be turned into attractive interactions in the out-of-equilibrium state under applied electric fields. We probe this behavior at different number densities of colloidal particles and show that particles in a dense dispersion collectively select the same direction of a coherent motion due to elastic interactions between near neighbors. In our experimentally implemented design, their motion is highly ordered and without clustering or jamming often present in other colloidal transport systems, which is promising for technological and fundamental-science applications, like nano-cargo transport, out-of-equilibrium assembly and microrobotics.

cond-mat.soft↗
arXiv2023

Field-controlled dynamics of skyrmions and monopoles

Magnetic monopoles, despite their ongoing experimental search as elementary particles, have inspired the discovery of analogous excitations in condensed matter systems. In chiral condensed matter systems, emergent monopoles are responsible for the onset of transitions between topologically distinct states and phases, like in the case of transitions from helical and conical phase to A-phase comprising periodic arrays of skyrmions. By combining numerical modeling and optical characterizations, we describe how different geometrical configurations of skyrmions terminating at monopoles can be realized in liquid crystals and liquid crystal ferromagnets. We demonstrate how such complex structures can be effectively manipulated by external magnetic and electric fields. Furthermore, we discuss how our findings may hint at similar dynamics in other physical systems, and their potential applications.

cond-mat.soft↗
arXiv2023

Photosynthetically-powered phototactic active nematic fluids and gels

One of the most ancient forms of life dating to ~3.5 billion years ago, cyanobacteria are highly abundant organisms that convert light into energy and motion, often within conjoined filaments and larger colonies. We study how gradients of light intensity trigger orderly phototactic motions and dense bacterial communities, which remained quantitatively unexplored despite being among the oldest forms of active living matter on Earth. The phototaxis drives a transition from initially polar motions of semiflexible long filaments along complex curved spatiotemporal trajectories confined within illuminated areas to their bipolar motility in the ensuing crowded environment. We demonstrate how simply shining light causes a spontaneous self-assembly of two- and three-dimensional active nematic states of cyanobacterial filaments, with a plethora of motile and static topological defects. We quantify light-controlled evolutions of orientational and velocity order parameters during the transition between disordered and orientationally ordered states of our photosynthetic active matter, as well as the subsequent active nematic's fluid-gel transformation. Patterned illumination and foreign inclusions with different shapes interact with cyanobacterial active nematics in nontrivial ways, while inducing soft interfacial boundary conditions and quasi-boojum-like defects. Commanding this cyanobacterial collective behavior could aid inhibiting generation of toxins or enhancing production of oxygen and biomaterials.

cond-mat.soft↗