Compact phase winding term
= Compact phase winding term
{title2=$i\rho_0\int\partial_\tau\phi$}
A periodic complex field permits $\phi(\beta,\mathbf r)-\phi(0,\mathbf r)=2\pi w(\mathbf r)$. Its uniform-density first-order term is therefore a winding-dependent imaginary action, not generally a removable global constant. It vanishes in the smooth zero-winding sector used for Gaussian <phonons>. Density defects and vortices require more than that smooth-sector approximation.