The transition is named for condensed matter physicists Vadim Berezinskii, John M. Kosterlitz and David J.
[1] BKT transitions can be found in several 2-D systems in condensed matter physics that are approximated by the XY model, including Josephson junction arrays and thin disordered superconducting granular films.
[2] More recently, the term has been applied by the 2-D superconductor insulator transition community to the pinning of Cooper pairs in the insulating regime, due to similarities with the original vortex BKT transition.
The critical density of the BKT transition in the weakly interacting system reads[3] where the dimensionless constant was found to be
[4] Work on the transition led to the 2016 Nobel Prize in Physics being awarded to Thouless and Kosterlitz; Berezinskii died in 1981.
This system is not expected to possess a normal second-order phase transition.
This is because the expected ordered phase of the system is destroyed by transverse fluctuations, i.e. the Nambu-Goldstone modes associated with this broken continuous symmetry, which logarithmically diverge with system size.
This is a specific case of what is called the Mermin–Wagner theorem in spin systems.
In the XY model in two dimensions, a second-order phase transition is not seen.
However, one finds a low-temperature quasi-ordered phase with a correlation function (see statistical mechanics) that decreases with the distance like a power, which depends on the temperature.
In the 2-D XY model, vortices are topologically stable configurations.
It is found that the high-temperature disordered phase with exponential correlation decay is a result of the formation of vortices.
Vortex generation becomes thermodynamically favorable at the critical temperature
At temperatures below this, vortex generation has a power law correlation.
, the system undergoes a transition at a critical temperature,
is a parameter that depends upon the system in which the vortex is located,
In the 2D system, the number of possible positions of a vortex is approximately
, entropic considerations favor the formation of a vortex.
The critical temperature above which vortices may form can be found by setting
and is given by The Kosterlitz–Thouless transition can be observed experimentally in systems like 2D Josephson junction arrays by taking current and voltage (I-V) measurements.
This jump from linear dependence is indicative of a Kosterlitz–Thouless transition and may be used to determine
This approach was used in Resnick et al.[5] to confirm the Kosterlitz–Thouless transition in proximity-coupled Josephson junction arrays.
Assume a field φ(x) defined in the plane which takes on values in
To render the theory well-defined, it is only defined up to some energetic cut-off scale
, so that we can puncture the plane at the points where the vortices are located, by removing regions with size of order
The complex argument function has a branch cut, but, because
: configurations with unbalanced numbers of vortices of each orientation are never energetically favoured.
, which is the total potential energy of a two-dimensional Coulomb gas.
the distance between a vortex and antivortex pair tends to be extremely small, essentially of the order
this distance increases, and the favoured configuration becomes effectively the one of a gas of free vortices and antivortices.