The geodesic rule for higher codimensional global defects

Abstract

We generalize the geodesic rule to the case of formation of higher codimensional global defects. Relying on energetic arguments, we argue that, for such defects, the geometric structures of interest are the totally geodesic submanifolds. On the other hand, stochastic arguments lead to a diffusion equation approach, from which the geodesic rule is deduced. It turns out that the most appropriate geometric structure that one should consider is the convex hull of the values of the order parameter on the causal volumes whose collision gives rise to the defect. We explain why these two approaches lead to similar results when calculating the density of global defects by using a theorem of Cheeger and Gromoll. We present a computation of the probability of formation of strings/vortices in the case of a system, such as nematic liquid crystals, whose vacuum is $\mathbb{R}{P}^2$.

17 pages, no figures. To be published in Mod. Phys. Lett. A

References in corpus (15)

• Heat kernel expansion: user's manual
• Big bang simulation in superfluid 3He-B -- Vortex nucleation in neutron-irradiated superflow
• The Cosmological Kibble Mechanism in the Laboratory: String Formation in Liquid Crystals
• Formation of Topological Defects at First Order Phase Transitions
• The Thermodynamics of Cosmic String densities in U(1) Scalar Field Theory
• On the Production of Flux Vortices and Magnetic Monopoles in Phase Transitions
• Vortex-Antivortex Pair Production in a First Order Phase Transition
• Relaxing the Geodesic Rule in Defect Formation Algorithms
• Defect formation in the early universe
• Resonant Production of Topological Defects
• Defect-Antidefect pair production via field oscillations
• Topological Defects from First Order Gauge Theory Phase Transitions
• How does the geodesic rule really work for global symmetry breaking first order phase transitions?
• A stochastic derivation of the geodesic rule
• The geodesic rule and the spectrum of the vacuum