302 research outputs found
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 . This is in
good agreement with the theoretical value expected in two dimensions
for the order parameter space 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
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