In two dimensions the phase transition that occurs for superfluids, superconductors, and planar classical magnets is qualitatively different from those which occur in higher dimensions. Known as the Berezinskii-Kosterlitz-Thouless (BKT) transition, it involves several unique emergent phenomena.
Novelty
The low-temperature state does not exhibit long-range-order or spontaneous symmetry breaking. Instead, the order parameter has power-law correlations, below a temperature T_BKT. Hence, it is qualitatively different from the high-temperature disordered state, which has correlations that decay exponentially. It is a distinct state of matter, with properties that are intermediate between the low- and high-temperature states normally associated with phase transitions. The power law correlations are similar to those at a conventional critical point, which decay in powers of the critical exponent eta. However, the BKT phase diagram can be viewed as having a line of critical points, consisting of all the temperatures below TBKT. Along this line, the critical exponent eta varies continuously with a value that depends on interaction strength. In contrast, at conventional critical points, eta has a fixed value determined by the universality class.
Sometimes it is stated that the low-temperature state has topological order, but I am not really sure what that means. Has this been made precise somewhere?
The mechanism of the phase transition is qualitatively different from that for conventional phase transitions. It is driven by the unbinding of vortex and anti-vortex pairs by thermal fluctuations. In contrast, conventional phase transitions are driven by thermal fluctuations in the magnitude of the order parameter.
Discontinuity
There is a discontinuity in the stiffness of the order parameter at this transition temperature.
Unlike for conventional phase transitions the specific heat capacity is a continuous function of temperature. This is why the BKT transition is sometimes referred to as a continuous transition.
Toy model
A classical Heisenberg model for a planar spin, also known as the XY model, captures the essential physics.
Modularity at the mesoscale
The quasiparticles of the system that are relevant to understanding the transition are not magnons (for magnets) or phonons (for superfluids), but vortices, i.e., topological defects.
These entities are usually on the mesoscale, i.e, there size is much larger than the lattice spacing. The relevant effective theory is not a non-linear sigma model. Thermal excitation of vortex-antivortex pairs determines the temperature dependence of physical properties and the transition at T_BKT. There is an effective interaction between a vortex and an anti-vortex that is attractive and a logarithmic function of their spatial separation, analogous to a two-dimensional Coulomb gas.
Universality
The BKT transition occurs in diverse two-dimensional models and materials including superfluids, superconductors, ferromagnets, arrays of Josephson junctions, and the Coulomb gas. The discontinuity in the order parameter stiffness at T_BKT has a universal value.
The renormalisation group (RG) equations associated with the transition are the same as those of a multitude of other systems. The classical two-dimensional systems include the Coulomb gas, Villain model, Z_n model for large n, solid-on-solid model, eight vertex model, and the Ashkin-Teller model. They also apply to classical Ising chain with 1/r^2 interactions. Aside: Phil Anderson discovered these RG equations for the Ising chain before BKT derived their own equations.
Quantum models with the same RG equations include the anisotropic Kondo model, spin boson model, XXZ antiferromagnetic Heisenberg spin chain, and the sine-Gordon quantum field theory in 1+1 dimensions. In other words, all these models are in the same universality class.
Singularity
The correlation length of the order parameter is a non-analytic function of the temperature.
This is related to the non-perturbative nature of the corresponding quantum models at their critical point.
Personal aside: I first encountered this singularity (long ago) when working on a spin-Peierls model with quantum phonons.
Two-dimensional crystals
Similar physics is relevant to the solidification of two-dimensional liquids. However, the relevant toy model is not the classical XY model as one needs to include the effect of the discrete rotational symmetry of the lattice of the solid. The low-temperature state exhibits discrete rotational, but not spatial, symmetry breaking, with power-law spatial correlations. This state does not directly melt into a liquid, but into a distinct state of matter, the hexatic phase. It has short-range spatial order and quasi-long-range orientational (sixfold) order. The phase transitions are driven by topological defects, disclinations and dislocations.
Predictability
The BKLT transition, the quasi-ordered low-temperature state, and the hexatic phase were all predicted theoretically before they were observed experimentally. This is unusual for emergent phenomena but shows that unpredictability is not equivalent to novelty.
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