Layering transitions and metastable structures of cholesteric liquid crystals in cylindrical confinement.

Our study of cholesteric lyotropic chromonic liquid crystals in cylindrical confinement reveals the topological aspects of cholesteric liquid crystals. The double-twist configurations we observe exhibit discontinuous layering transitions, domain formation, metastability, and chiral point defects as the concentration of chiral dopant is varied. We demonstrate that these distinct layer states can be distinguished by chiral topological invariants. We show that changes in the layer structure give rise to a chiral soliton similar to a toron, comprising a metastable pair of chiral point defects. Through the applicability of the invariants we describe to general systems, our work has broad relevance to the study of chiral materials.

Geodesic fibrations for packing diabolic domains.

We describe a theory of packing hyperboloid "diabolic" domains in bend-free textures of liquid crystals. The domains sew together continuously, providing a menagerie of bend-free textures akin to the packing of focal conic domains in smectic liquid crystals. We show how distinct domains may be related to each other by Lorentz transformations and that this process may lower the elastic energy of the system. We discuss a number of phases that may be formed as a result, including splay-twist analogues of blue phases. We also discuss how these diabolic domains may be subject to "superluminal boosts," yielding defects analogous to shock waves. We explore the geometry of these textures, demonstrating their relation to Milnor fibrations of the Hopf link. Finally, we show how the theory of these domains is unified in four-dimensional space.

Woven Nematic Defects, Skyrmions, and the Abelian Sandpile Model.

We show that a fixed set of woven defect lines in a nematic liquid crystal supports a set of nonsingular topological states which can be mapped on to recurrent stable configurations in the Abelian sandpile model or chip-firing game. The physical correspondence between local skyrmion flux and sandpile height is made between the two models. Using a toy model of the elastic energy, we examine the structure of energy minima as a function of topological class and show that the system admits domain wall skyrmion solitons.

Composite Dislocations in Smectic Liquid Crystals.

Smectic liquid crystals are characterized by layers that have a preferred uniform spacing and vanishing curvature in their ground state. Dislocations in smectics play an important role in phase nucleation, layer reorientation, and dynamics. Typically modeled as possessing one line singularity, the layer structure of a dislocation leads to a diverging compression strain as one approaches the defect center, suggesting a large, elastically determined melted core. However, it has been observed that for large charge dislocations, the defect breaks up into two disclinations [C. E. Williams, Philos. Mag. 32, 313 (1975)PHMAA40031-808610.1080/14786437508219956]. Here we investigate the topology of the composite core. Because the smectic cannot twist, transformations between different disclination geometries are highly constrained. We demonstrate the geometric route between them and show that despite enjoying precisely the topological rules of the three-dimensional nematic, the additional structure of line disclinations in three-dimensional smectics localizes transitions to higher-order point singularities.

Instabilities and Solitons in Minimal Strips.

We show that highly twisted minimal strips can undergo a nonsingular transition, unlike the singular transitions seen in the Möbius strip and the catenoid. If the strip is nonorientable, this transition is topologically frustrated, and the resulting surface contains a helicoidal defect. Through a controlled analytic approximation, the system can be mapped onto a scalar ϕ^{4} theory on a nonorientable line bundle over the circle, where the defect becomes a topologically protected kink soliton or domain wall, thus establishing their existence in minimal surfaces. Demonstrations with soap films confirm these results and show how the position of the defect can be controlled through boundary deformation.

Knots and nonorientable surfaces in chiral nematics.

Knots and knotted fields enrich physical phenomena ranging from DNA and molecular chemistry to the vortices of fluid flows and textures of ordered media. Liquid crystals provide an ideal setting for exploring such topological phenomena through control of their characteristic defects. The use of colloids in generating defects and knotted configurations in liquid crystals has been demonstrated for spherical and toroidal particles and shows promise for the development of novel photonic devices. Extending this existing work, we describe the full topological implications of colloids representing nonorientable surfaces and use it to construct torus knots and links of type (p,2) around multiply twisted Möbius strips.

Knotted defects in nematic liquid crystals.

We show that the number of distinct topological states associated with a given knotted defect, L, in a nematic liquid crystal is equal to the determinant of the link L. We give an interpretation of these states, demonstrate how they may be identified in experiments, and describe the consequences for material behavior and interactions between multiple knots. We show that stable knots can be created in a bulk cholesteric and illustrate the topology by classifying a simulated Hopf link. In addition, we give a topological heuristic for the resolution of strand crossings in defect coarsening processes which allows us to distinguish topological classes of a given link and to make predictions about defect crossings in nematic liquid crystals.

Biases in inverse Ising estimates of near-critical behavior.

Inverse Ising inference allows pairwise interactions of complex binary systems to be reconstructed from empirical correlations. Typical estimators used for this inference, such as pseudo-likelihood maximization (PLM), are biased. Using the Sherrington-Kirkpatrick model as a benchmark, we show that these biases are large in critical regimes close to phase boundaries, and they may alter the qualitative interpretation of the inferred model. In particular, we show that the small-sample bias causes models inferred through PLM to appear closer to criticality than one would expect from the data. Data-driven methods to correct this bias are explored and applied to a functional magnetic resonance imaging data set from neuroscience. Our results indicate that additional care should be taken when attributing criticality to real-world data sets.

A Björling representation for Jacobi fields on minimal surfaces and soap film instabilities.

We develop a general framework for the description of instabilities on soap films using the Björling representation of minimal surfaces. The construction is naturally geometric and the instability has the interpretation as being specified by its amplitude and transverse gradient along any curve lying in the minimal surface. When the amplitude vanishes, the curve forms part of the boundary to a critically stable domain, while when the gradient vanishes the Jacobi field is maximal along the curve. In the latter case, we show that the Jacobi field is maximally localized if its amplitude is taken to be the lowest eigenfunction of a one-dimensional Schrödinger operator. We present examples for the helicoid, catenoid, circular helicoids and planar Enneper minimal surfaces, and emphasize that the geometric nature of the Björling representation allows direct connection with instabilities observed in soap films.

Global defect topology in nematic liquid crystals.

We give the global homotopy classification of nematic textures for a general domain with weak anchoring boundary conditions and arbitrary defect set in terms of twisted cohomology, and give an explicit computation for the case of knotted and linked defects in [Formula: see text], showing that the distinct homotopy classes have a 1-1 correspondence with the first homology group of the branched double cover, branched over the disclination loops. We show further that the subset of those classes corresponding to elements of order 2 in this group has representatives that are planar and characterize the obstruction for other classes in terms of merons. The planar textures are a feature of the global defect topology that is not reflected in any local characterization. Finally, we describe how the global classification relates to recent experiments on nematic droplets and how elements of order 4 relate to the presence of τ lines in cholesterics.