Topological defects

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1. Thread topological defects and domain walls. In which cases these defects involve new items in Hamiltonians of models? The connection between thread topological defects and first-order phase transitions (for instance in the Nielsen-Olesen model, helium II, in which Chalatnikov quantum vortices appear, in type II superconductors). More exactly, it is known that all the kinds of topological defects imply breaking the thermodynamic equilibrium over the defect regions (in the coordinate space). In which agreement this fact is with above "defects" items in Hamiltonians and with the first-order phase transitions dynamics therein.
2. Ring vortices. Mit which a topological relation they are govern (I mein a relation between the \pi groups of the residual symmetry group space and the appropriate vacuum manifold at breaking the [gauge] symmetry). Is that the same relation as in the case of rectilinear vortices?
 

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