Statistical mechanics, structure of matter Lattice spin systems Theorem

Berezinskii–Kosterlitz–Thouless transition

Builds on XY model Correlation function Gibbs measure

Statement

Consider the XY model on $\Z^2$ with nearest-neighbour coupling $J = 1$ at inverse temperature $\beta \ge 0$ . Let $\langle \cdot \rangle_\beta$ be an infinite-volume Gibbs state obtained as a limit of finite-box Gibbs states. Let $G_\beta(x) = \langle \cos(\theta_0 - \theta_x) \rangle_\beta$ be the spin-spin correlation function for $x \in \Z^2$ .

There is $\beta_0 > 0$ such that for every $\beta < \beta_0$ there are constants $C, m > 0$ with $G_\beta(x) \le C e^{-m \abs{x}}$ for all $x$ .
For every $\beta$ , $G_\beta(x) \to 0$ as $\abs{x} \to \infty$ .
(Fröhlich–Spencer, 1981) There is $\beta_1$ such that for every $\beta > \beta_1$ there are constants $C, \eta > 0$ with $G_\beta(x) \ge C \abs{x}^{-\eta}$ for all $x \ne 0$ .

So the correlations decay exponentially at high temperature and only like a power of the distance at low temperature. At no temperature is there spontaneous magnetization.

Remarks

Part 1 follows from a high-temperature expansion of $e^{\beta \cos(\theta_x - \theta_y)}$ . Part 2 is a form of the Mermin–Wagner theorem: slow spin waves cost little energy in two dimensions and destroy long-range order. Part 3 is the hard part. Fröhlich and Spencer rewrite the model as a Coulomb gas of vortices, and a multiscale analysis shows that at low temperature vortices bind into neutral pairs. Berezinskii and, independently, Kosterlitz and Thouless predicted this picture in the early 1970s, with vortex-antivortex unbinding as the mechanism of the transition. The spin-wave approximation predicts the decay $G_\beta(x) \approx \abs{x}^{-1/(2\pi\beta)}$ at low temperature. The dimension matters: on $\Z^d$ with $d \ge 3$ the XY model has true long-range order at low temperature.

Going deeper

For a student

This is the standard example of a phase transition without symmetry breaking. It shows that the Mermin–Wagner theorem does not rule out a transition, only a nonzero magnetization. The Fröhlich–Spencer proof uses a duality to an integer-valued height model and a Coulomb gas, and it is a landmark of rigorous renormalization. The same mechanism appears in thin superfluid films and in the roughening transition of crystal surfaces.

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Boltzmann distribution concept
Correlation function concept
Gibbs measure concept
Graph concept
XY model example
Berezinskii–Kosterlitz–Thouless transition statement

Tag	`MN952HPZ50`
URI	`m2://MN952HPZ50`
Kind	statement, theorem
Writing	W0 stub
MSC	82B26, 82B20
Status	This copy of the site has no build data. Run `m2 status MN952HPZ50`.
About	XY model
Uses	XY model, Correlation function, Gibbs measure
Same as	Wikidata Q898780 Wikipedia Berezinskii–Kosterlitz–Thouless transition
Changed	3 November 2026

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