In plain words
How a flat, two-dimensional system that cannot have perfect order can still switch between an almost-ordered state and a one. At low temperature, tiny whirlpools in the pattern of its – or of a – exist only in tightly bound pairs; above a they break free and scramble the order.
Going deeper
Order that the rules seem to forbid
The forbids perfect long-range order in two dimensions when the order can point in any direction within a plane: at any temperature above zero, gentle long-wavelength twists of the order cost so little energy that they wander without limit. Yet helium films a few atoms thick become superfluid, and thin superconducting films lose their resistance. Vadim Berezinskii, and independently Michael Kosterlitz and David Thouless, resolved the puzzle in the early 1970s: the order does not have to be perfect. At low temperature correlations decay only as a power of the distance – so slowly that the film behaves as ordered over any realistic size – and superflow survives.
What destroys this near-order is not the gentle twists but a different kind of disturbance, and the transition it drives has no local order parameter at all. Kosterlitz and Thouless shared the 2016 Nobel Prize in Physics, with Duncan Haldane, for this and related work on transitions.
Vortices, bound and free
A vortex is a point around which the direction of the spins, or the phase of a superconductor, turns through a full circle. Its energy grows with the logarithm of the size of the sample, but so does its entropy, because there are that many more places to put it. Below a critical temperature the energy wins and free vortices cannot exist; they appear only as tightly bound pairs with antivortices of opposite winding, whose disturbances cancel at a distance. Above it the entropy wins, the pairs unbind, and free vortices scramble the order across the whole sheet.
The winding number of a vortex is a whole number, which is what makes this a topological transition. It leaves a sharp signature: the stiffness of the order jumps from a universal value, set by the transition temperature alone, straight to zero – measured first in helium films in 1978.
Where it shows up in 2D materials
Superconducting and thin crystals show it as a transition below the temperature at which pairing starts: the resistance falls to zero only at the BKT temperature, and there the current–voltage curve follows a cube law, turning steeper below. In magnets it needs spins that lie in a plane without a preferred axis within it. Monolayer CrCl3 grown on graphene shows the critical behaviour of such an XY magnet, while in NiPS3 the ordering seen in thicker fades in the monolayer.
Real materials are never perfectly planar-symmetric. A preferred axis, in the plane or out of it, eventually turns the transition into an ordinary ordering one near the BKT temperature, and finite flake size, disorder and inhomogeneity blur the signatures, so a BKT claim needs more than a fit of one curve: the cube law at the right temperature, the stiffness jump where it can be measured, and consistency between the two.
For specialists
A phase transition without a local order parameter in two-dimensional systems with a continuous planar symmetry – XY magnets, superfluid and superconducting films – driven by the unbinding of vortex–antivortex pairs. Below T_BKT correlations decay as a power of distance (quasi-long-range order), above it exponentially. The superfluid stiffness jumps to zero from the universal value 2k_BT_BKT/π, and in superconducting films the current–voltage curve follows V ∝ I3 at T_BKT. It evades the Mermin–Wagner theorem, which forbids true long-range order but not this.