Kibble mechanism
= Kibble mechanism
{c}
The <Kibble mechanism> explains how a <spontaneous symmetry breaking> transition can form <topological defects>. Regions that order without coordinating their choices can settle at different points of a <vacuum manifold>. Smooth relaxation removes many mismatches, but some cannot be removed while the field remains in that <vacuum manifold> everywhere.<Topology of cosmic domains and strings by Thomas Walter Bannerman Kibble>[Section 3, Formation of protodomains, printed pp. 1389–1391]{ref}
The possible defects depend on the <homotopy groups> of the <vacuum manifold>: disconnected components permit <domain walls>, noncontractible loops permit <cosmic strings>, and nontrivial surrounding two-spheres permit <magnetic monopoles> in three spatial dimensions. This is the <homotopy classification of cosmological defects>, rather than a prediction that every permitted defect must occur.<Topology of cosmic domains and strings by Thomas Walter Bannerman Kibble>[Sections 4 and 6, topology and summary of walls, strings and monopoles]{ref}
For a direct winding example, an <order parameter> represented by a <complex scalar field> with fixed nonzero magnitude has a circular <vacuum manifold>. If its <order-parameter phase> changes by $2\pi$ around a closed spatial loop, it cannot extend continuously across a spanning disk while keeping its magnitude nonzero. A zero of the <order parameter> supplies the core; in three dimensions the corresponding defect can be a string.
A formation estimate needs a domain <correlation length> $\xi$ and a model for the chance of trapping a defect. For point defects, a dimensional estimate is $n_{\rm defect}\sim p_{\rm defect}/\xi^3$, with dimensionless formation efficiency $p_{\rm defect}$. Causality restricts how ordering can spread; taking a horizon-sized $\xi$ supplies a scale estimate, not by itself a guaranteed number of defects in every horizon <volume>. Subsequent annihilation or expansion can further change the abundance. Kibble's original discussion separately estimates the initial domain scale from fluctuations and follows later evolution.<Topology of cosmic domains and strings by Thomas Walter Bannerman Kibble>[Sections 3 and 5, initial scale and subsequent evolution]{ref}