Kosterlitz–Thouless theorem (2D XY transition via vortex unbinding)

Context

The 2D XY model has a continuous $U(1)$ symmetry (see continuous symmetry ), so by the Mermin–Wagner theorem it cannot have conventional spontaneous magnetization at $T>0$. Nevertheless, it exhibits a genuine phase transition of topological character, driven by vortices (see vortices as topological defects ).

A standard nearest-neighbor XY Hamiltonian on the 2D square lattice is

with equilibrium described by a finite-volume Gibbs measure (see finite-volume Gibbs measure ) and its infinite-volume limits (see infinite-volume Gibbs measure ).

Theorem (Kosterlitz–Thouless; standard qualitative conclusions)

There exists a critical temperature $T_{KT}>0$ such that:

1. High-temperature phase ($T>T_{KT}$): correlations decay exponentially, i.e. there is a finite correlation length $\xi(T)<\infty$ with

   $$\langle e^{i(\theta_0-\theta_x)}\rangle \approx e^{-|x|/\xi(T)}.$$

2. Low-temperature phase ($0<T<T_{KT}$): there is quasi-long-range order: correlations decay as a power law,

   $$\langle e^{i(\theta_0-\theta_x)}\rangle \approx |x|^{-\eta(T)}, \qquad \eta(T)\in(0,1/4],$$

   so the two-point function decays slowly but still tends to $0$ as $|x|\to\infty$ (consistent with Mermin–Wagner).

3. Topological mechanism: at $T_{KT}$, vortex–antivortex pairs unbind (see vortex unbinding picture ), producing a transition without a conventional symmetry-breaking order parameter.

4. Characteristic singularities: the correlation length diverges with an essential singularity as $T\downarrow T_{KT}$ from above,

   $$\xi(T)\sim \exp\!\left(\frac{b}{\sqrt{T-T_{KT}}}\right) \quad (T\downarrow T_{KT}),$$

   for some $b>0$ (model-dependent).

5. Universal jump (stiffness/helicity modulus): the spin-wave stiffness drops discontinuously at $T_{KT}$, with the universal value

   $$\Upsilon(T_{KT}^-)=\frac{2}{\pi}T_{KT},$$

   where $\Upsilon$ is the helicity modulus (an equilibrium linear response quantity related to fluctuation–dissipation ideas).

Connections and interpretations

• This theorem is the mathematical backbone of the Kosterlitz–Thouless transition concept and motivates renormalization ideas (see renormalization group transformations , RG fixed points ).
• The transition is a central example where symmetry-based order parameters fail, but topology and defects govern phases.

Prerequisites and connections (cross-links)

• Equilibrium and ensembles: canonical ensemble , partition function , ensemble averages .
• Low-dimensional constraints: Mermin–Wagner theorem .
• Phase diagnostics: equivalent phase-transition indicators , structure factor .