Disclination Loops, Hedgehogs, and All That

Abstract

The homotopy theory of topological defects is a powerful tool for organizing and unifying many ideas across a broad range of physical systems. Recently, experimental progress has been made in controlling and measuring colloidal inclusions in liquid crystalline phases. The topological structure of these systems is quite rich but, at the same time, subtle. Motivated by experiment and the power of topological reasoning, we review and expound upon the classification of defects in uniaxial nematic liquid crystals. Particular attention is paid to the ambiguities that arise in these systems, which have no counterpart in the much-storied XY model or the Heisenberg ferromagnet.

Colloquium, 19 pages, 9 figures

References in corpus (5)

• Non-Abelian Anyons and Topological Quantum Computation
• Spontaneous Skyrmion Ground States in Magnetic Metals
• Reconfigurable knots and links in chiral nematic colloids
• Nematic braids: topological invariants and rewiring of disclinations
• Smectics: Symmetry Breaking, Singularities, and Surfaces

Cited by in corpus (105)

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• Topological Classification of Non-Hermitian Hamiltonians
• Observation of non-Abelian topological semimetals and their phase transitions
• Geometric approach to fragile topology beyond symmetry indicators
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• Stable nematic droplets with handles
• Chiral structures and defects of lyotropic chromonic liquid crystals induced by saddle-splay elasticity
• Review: Topology of solitons and singular defects in chiral liquid crystals
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• Non-abelian topology of nodal-line rings in $\mathcal{P}\mathcal{T}$-symmetric systems
• Nematics, Knots and Non-orientable Surfaces
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• Knotted Defects in Nematic Liquid Crystals
• Exploiting imperfections: Directed assembly of surface colloids via bulk topological defects
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• Weaving knotted vector fields with tunable helicity
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• Geometry and mechanics of disclination lines in 3D nematic liquid crystals
• Umbilic Lines in Orientational Order
• Global Defect Topology in Nematic Liquid Crystals
• Topology of three-dimensional active nematic turbulence confined to droplets
• Particle-resolved topological defects of smectic colloidal liquid crystals in extreme confinement
• Introduction to topological defects: from liquid crystals to particle physics
• Spin Nematics Meet Spin Liquids: Exotic Quantum Phases in the Spin-$1$ Bilinear-Biquadratic Model with Kitaev Interactions
• Braid Protected Topological Band Structures with Unpaired Exceptional Points
• Quantized Integrated Shift Effect in Multigap Topological Phases
• Micro-Bullet Assembly: Interactions of Oriented Dipoles in Confined Nematic Liquid Crystal
• Sculpting stable structures in pure liquids
• Line Singularities and Hopf Indices of Electromagnetic Multipoles
• Optical manifestations and bounds of topological Euler class
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• Floquet Non-Abelian Topological Insulator and Multifold Bulk-Edge Correspondence
• The Topology of Dislocations in Smectic Liquid Crystals
• Quaternions and hybrid nematic disclinations
• Singularity identification for the characterization of topology, geometry, and motion of nematic disclination lines
• Spectral energy analysis of bulk three-dimensional active nematic turbulence
• Aspects of Defect Topology in Smectic Liquid Crystals
• Non-Abelian Hopf-Euler insulators
• Topological elasticity of non-orientable ribbons
• Berry phase induced dimerization in one-dimensional quadrupolar systems
• Intradomain Textures in Block Copolymers: Multizone Alignment and Biaxiality
• Point Defects, Chirality and Singularity Theory in Cholesteric Liquid Crystal Droplets
• Edge pinning and transformation of defect lines induced by faceted colloidal rings in nematic liquid crystals
• Weak-strong uniqueness for measure-valued solutions to the Ericksen-Leslie model equipped with the Oseen-Frank free energy
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• Kinematics and dynamics of disclination lines in three-dimensional nematics
• Experimental demonstration of non-adjacent band topology connecting multiple nodal links
• Measure-valued solutions to the Ericksen-Leslie model equipped with the Oseen-Frank energy
• Topological-defect-induced surface charge heterogeneities in nematic electrolytes
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• Disorder-induced topological quantum phase transitions in multi-gap Euler semimetals
• Alice string in Non-Hermitian Systems
• Discharging dynamics of topological batteries
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• A nematic liquid crystal with an immersed body: equilibrium, stress, and paradox
• Three-dimensional vortex structures and dynamics in hexagonal manganites
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• Approximation and optimal control of dissipative solutions to the Ericksen--Leslie system
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• Coexistence of defect morphologies in three dimensional active nematics
• Contact Topology and the Classification of Disclination Lines in Cholesteric Liquid Crystals
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• Berry curvature inside parity-time-symmetry protected exceptional surface
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• Topological atom optics and beyond with knotted quantum wavefunctions
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• A computational study of nematic core structure and disclination interactions in elastically anisotropic nematics
• Emergent Quadrupolar Order in the Spin-$1/2$ Kitaev-Heisenberg Model
• Unavoidable emergent biaxiality in chiral molecular-colloidal hybrid liquid crystals
• Topological Description of the Solidification of Undercooled Fluids and the Temperature Dependence of the Thermal Conductivity of Crystalline and Glassy Solids Above Approximately 50 K
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• Coarse-grained theory for motion of solitons and skyrmions in liquid crystals
• Topological fine structure of smectic grain boundaries and tetratic disclination lines within three-dimensional smectic liquid crystals
• Liquid Structures: a novel Computational Fluid Dynamics (CFD) inspired metamaterial