Unitary One Matrix Models: String Equations and Flows

We review the Symmetric Unitary One Matrix Models. In particular we discuss the string equation in the operator formalism, the mKdV flows and the Virasoro Constraints. We focus on the \t-function formalism for the flows and we describe its connection to the (big cell of the) Sato Grassmannian \Gr via the Plucker embedding of \Gr into a fermionic Fock space. Then the space of solutions to the string equation is an explicitly computable subspace of \Gr\times\Gr which is invariant under the flows.Comment: 20 pages (Invited talk delivered by M. J. Bowick at the Vth Regional Conference on Mathematical Physics, Edirne Turkey: December 15-22, 1991.

Topological Sound and Flocking on Curved Surfaces

Active systems on curved geometries are ubiquitous in the living world. In the presence of curvature orientationally ordered polar flocks are forced to be inhomogeneous, often requiring the presence of topological defects even in the steady state due to the constraints imposed by the topology of the underlying surface. In the presence of spontaneous flow the system additionally supports long-wavelength propagating sound modes which get gapped by the curvature of the underlying substrate. We analytically compute the steady state profile of an active polar flock on a two-sphere and a catenoid, and show that curvature and active flow together result in symmetry protected topological modes that get localized to special geodesics on the surface (the equator or the neck respectively). These modes are the analogue of edge states in electronic quantum Hall systems and provide unidirectional channels for information transport in the flock, robust against disorder and backscattering.Comment: 15 pages, 6 figure

The Cosmological Kibble Mechanism in the Laboratory: String Formation in Liquid Crystals

We have observed the production of strings (disclination lines and loops) via the Kibble mechanism of domain (bubble) formation in the isotropic to nematic phase transition of a sample of uniaxial nematic liquid crystal. The probablity of string formation per bubble is measured to be $0.33 \pm 0.01$. This is in good agreement with the theoretical value $1/ \pi$ expected in two dimensions for the order parameter space $S^2/{\bf Z}_2$ of a simple uniaxial nematic liquid crystal.Comment: 17 pages, in TEX, 2 figures (not included, available on request

Defect unbinding in active nematics

We formulate the statistical dynamics of topological defects in the active nematic phase, formed in two dimensions by a collection of self-driven particles on a substrate. An important consequence of the non-equilibrium drive is the spontaneous motility of strength +1/2 disclinations. Starting from the hydrodynamic equations of active nematics, we derive an interacting particle description of defects that includes active torques. We show that activity, within perturbation theory, lowers the defect-unbinding transition temperature, determining a critical line in the temperature-activity plane that separates the quasi-long-range ordered (nematic) and disordered (isotropic) phases. Below a critical activity, defects remain bound as rotational noise decorrelates the directed dynamics of +1/2 defects, stabilizing the quasi-long-range ordered nematic state. This activity threshold vanishes at low temperature, leading to a re-entrant transition. At large enough activity, active forces always exceed thermal ones and the perturbative result fails, suggesting that in this regime activity will always disorder the system. Crucially, rotational diffusion being a two-dimensional phenomenon, defect unbinding cannot be described by a simplified one-dimensional model.Comment: 15 pages (including SI), 4 figures. Significant technical improvements without changing the result

Crystalline Order On Riemannian Manifolds With Variable Gaussian Curvature And Boundary

We investigate the zero temperature structure of a crystalline monolayer constrained to lie on a two-dimensional Riemannian manifold with variable Gaussian curvature and boundary. A full analytical treatment is presented for the case of a paraboloid of revolution. Using the geometrical theory of topological defects in a continuum elastic background we find that the presence of a variable Gaussian curvature, combined with the additional constraint of a boundary, gives rise to a rich variety of phenomena beyond that known for spherical crystals. We also provide a numerical analysis of a system of classical particles interacting via a Coulomb potential on the surface of a paraboloid.Comment: 12 pages, 8 figure