 Solution

The physical angle takes values on a circle, so it is defined modulo $2\pi$. Around a closed curve enclosing a phase vortex,The integer winding number cannot change through a smooth single-valued deformation unless a defect crosses the curve. Thus a globally smooth Gaussian spin wave field omits whole topological sectors of the XY model.

Outside a vortex core of radius $a$, an isolated charge-$n$ phase vortex has $\theta =n\phi$ and $|\nabla \theta |=|n|/r$. Its dimensionless energy in a sample of size $L$ isThere are of order $(L/a{)}^2$ positions for its core, so the positional entropy contributes $2\log (L/a)$. The competition for unit charge is therefore $(\pi \bar{K}-2)\log (L/a)$, apart from core terms. Opposite charges have a logarithmic attraction and form a vortex Coulomb gas.

At low temperature, vortex-antivortex binding leaves no free charges on large scales. Spin waves still destroy true long-range order, but the correlation function decays algebraically with the renormalized phase stiffness. At the Berezinskii–Kosterlitz–Thouless transition, pairs unbind; free vortices screen the interaction and produce a finite correlation length and exponential decay above the transition. The energy-entropy estimate identifies the threshold stiffness $K_R=2/\pi$, where $K_R$ is the long-distance dimensionless stiffness, not necessarily the bare $\bar{K}$. It agrees with the universal stiffness jump and the limiting algebraic exponent $1/(2\pi K_R)=1/4$. The two-dimensional transition is vortex unbinding between quasi-long-range order and disorder, rather than the onset of nonzero spontaneous magnetization.