Topological defect: vortex

Definition (vortex in 2D U(1) systems)

Consider a two-dimensional system with a $U(1)$ order parameter that can be represented by an angle field $\theta(x)\in \mathbb{R}/2\pi\mathbb{Z}$ (e.g., planar spins $s(x)=(\cos\theta(x),\sin\theta(x))$). A vortex is a point defect around which $\theta$ winds nontrivially.

For a closed loop $\gamma$ encircling a defect, the winding number (topological charge) is

• $m=+1$ is a vortex, $m=-1$ an antivortex (conventions may vary).
• The integer quantization comes from $\pi_1(S^1)\cong \mathbb{Z}$.

Energetics (spin-wave stiffness picture)

In the continuum XY/superfluid effective energy

a single vortex of charge $m$ in a system of linear size $L$ with microscopic core cutoff $a$ has energy scaling

This logarithmic growth underlies the competition between energy and entropy that drives vortex unbinding at the BKT transition .

Vortex pairs and unbinding

• At low temperature, vortices appear mainly as tightly bound vortex–antivortex pairs, preserving quasi-long-range order.
• At high temperature, unbound vortices proliferate, producing exponential decay of correlations.

Prerequisites / cross-links

• continuous symmetries on spin systems
• Kosterlitz–Thouless (BKT) transition
• Kosterlitz–Thouless theorem
• Mermin–Wagner theorem