Symmetry breaking is ubiquitous in nature and represents the key mechanism behind rich diversity of patterns exhibited by nature. One commonly introduces an order parameter field to describe onset of qualitatively new ordering in a system on varying a relevant control parameter driving a symmetry breaking transition. In case of continuous symmetry breaking an order parameter consists of two qualitatively different components: an amplitude and gauge field. The latter component enables energy degeneracy and reveals how symmetry is broken. Inherent degeneracy could in general lead to nearby regions exhibiting significantly different gauge fields. Resulting frustrations can nucleate topological defects (TDs). These represent topologically stable localized nonlinear order parameter solutions. Their key property is a discrete topological charge, which is a conserved quantity. One commonly refers to TDs with positive (negative) charge as defects (antidefects). In general a nearby pair defect-antidefect tends to annihilate each order because presence of TDs is in general energetically costly. The total topological charge in a system is conserved for fixed boundary conditions. Consequently number of topological defects could be varied either my merging of relevant TDs or formations of {defect,antidefect} pairs. There is strong interest in different fields of physics to find conditions where different number of TDs could be generated&stabilized and how to control their positioning. Famous examples of stable configuration of TDs represent Abrikosov lattices in superconductors and Skyrmions in nuclear physics. In particular, the latter example suggests that TDs might represent “particles” if fields are fundamental constituents of nature.
A convenient systems to study system exhibiting complex patterns of TDs are various liquid crystalline (LC) phases. Due to their softness, fluidity and optical anisotroy&transparency TDs could be relatively easily experimentally generated and observed in them. Consequently, in LCs various theoretical predictions could be relatively easily tested. In the lecture we present a numerical study of TDs within effectively two-dimensional (2D) closed soft films exhibiting in-plane orientational ordering. Popular examples of such class of systems are liquid crystalline shells and various biological membranes. We introduce the Effective Topological Charge Cancellation mechanism controlling localized positional assembling tendency of TDs and formation of pairs {defect,antidefect} on curved surfaces and/or presence of relevant impurities (e.g. nanoparticles). For this purpose we define an effective topological charge ?meff consisting of "real" TDs, the smeared curvature topological charge, and virtual topological charges within a surface patch ?A characterized by an spatially averaged local Gaussian curvature K. We demonstrate a strong tendency enforcing ?meff=0 on surfaces composed of ?A exhibiting significantly different values of K. For ?meff?0 we estimate the critical depinning threshold to form pairs {defect,antidefect} using the electrostatic analogy describing an electric field driven formation&stabilization of pairs {electron,positron}. Furthermore, we demonstrate LC analogue of the electrostatic Faraday cavity effect.
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