= Phase vortex
{title2=$q=(2\pi)^{-1}\oint\nabla\theta\cdot d\boldsymbol\ell$}
= Vortex
{synonym}
= Vortices
{synonym}
A vortex is a topological defect around which the phase of a complex <order parameter> has nonzero integer winding. In two dimensions an isolated vortex in a phase-only model has energy proportional to the logarithm of the system size.
The integer $q$ is a <winding number>: a counterclockwise circuit returns the same complex <order parameter> after its angle changes by $2\pi q$. Smooth changes that keep the <order parameter> nonzero on that circuit cannot change $q$. In a continuum <superfluid>, the core suppresses the magnitude, allowing the winding obstruction to end there; in an <XY model> the microscopic core instead lies beyond the smooth-field approximation. The <vortex-antivortex binding> figure illustrates a locally nonzero charge with zero charge on a surrounding large circuit.
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