Order-parameter phase ($\theta =\arg (m_x+i{m}_y)$)

For a two-component order parameter of nonzero magnitude, write $m_x+i{m}_y=\bar{m}{e}^{i\theta }$. Its phase is the orientation angle modulo $2\pi$, not a thermodynamic phase such as an ordered or disordered state. A global continuous symmetry rotates this angle uniformly; spin waves are smooth spatial changes, while a phase vortex has nonzero winding around an excluded core. At a zero of the magnitude the angle is undefined, allowing the topology of an orientation field to change.