Kosterlitz–Thouless (BKT) transition

A 2D topological phase transition driven by vortex unbinding, featuring quasi-long-range order and an essential singularity in the correlation length.

Kosterlitz–Thouless (BKT) transition

Definition (BKT transition)

A Kosterlitz–Thouless (also called Berezinskii–Kosterlitz–Thouless, BKT) transition is a phase transition in two-dimensional systems with a continuous $U(1)$ -type order parameter (e.g., the 2D XY model) in which:

there is no conventional long-range order at any positive temperature,
the low-temperature phase exhibits quasi-long-range order with algebraic decay of correlations,
the high-temperature phase exhibits exponential decay of correlations,
the transition is driven by the unbinding of vortex–antivortex pairs (topological defects).

Hallmark features

Correlation decay regimes

Let $G(r)=\langle e^{i(\theta(r)-\theta(0))}\rangle$ denote the two-point function in an angle representation.

Low temperature ( $T<T_c$ ):
$G(r)\sim r^{-\eta(T)} \quad \text{(algebraic decay).}$
High temperature ( $T>T_c$ ):
$G(r)\sim e^{-r/\xi(T)} \quad \text{(exponential decay).}$

Essential singularity of the correlation length

As $T\downarrow T_c$ from above, the correlation length diverges faster than any power law:

\xi(T) \sim \exp\!\left(\frac{b}{\sqrt{T-T_c}}\right),

for a nonuniversal constant $b>0$ (model-dependent).

Universal jump (stiffness / helicity modulus)

In the 2D XY model, the superfluid stiffness (helicity modulus) $\Upsilon(T)$ exhibits a universal jump at $T_c$ :

\Upsilon(T_c^-)=\frac{2T_c}{\pi}.

(Precise definitions depend on boundary conditions and how $\Upsilon$ is measured/defined.)

RG perspective (brief)

The BKT transition is associated with an RG flow in a two-parameter space (often expressed in terms of a stiffness and a vortex fugacity), featuring:

a low-temperature line of fixed points (power-law phase),
a separatrix ending at the transition,
flow to a disordered phase when vortices proliferate.