The –Kosterlitz–Berezinskii–Kosterlitz–Thouless (BKT) transition is a phase transition of the two-dimensional (2-D) XY model in statistical physics. It is a transition from bound vortex-antivortex pairs at low temperatures to unpaired vortices and anti-vortices at some critical temperature. The transition is named for condensed matter physicists Vadim Berezinskii, John M. Kosterlitz and David J. Thouless. 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. 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.
Work on the transition led to the 2016 Nobel Prize in Physics being awarded to Thouless and Kosterlitz; Berezinskii died in 1980.
XY model
The XY model is a two-dimensional vector spin model that possesses U(1) or circular symmetry. 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.
Rigorously the transition is not completely understood, but the existence of two phases was proved by and .
Disordered phases with different correlations
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. The transition from the high-temperature disordered phase with the exponential correlation to this low-temperature quasi-ordered phase is a Kosterlitz–Thouless transition. It is a phase transition of infinite order.
Role of vortices
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 of the Kosterlitz–Thouless transition. At temperatures below this, vortex generation has a power law correlation.
Kosterlitz–Thouless transitions is described as a dissociation of bound vortex pairs with opposite circulations, called vortex–antivortex pairs, first described by Vadim Berezinskii. In these systems, thermal generation of vortices produces an even number of vortices of opposite sign. Bound vortex–antivortex pairs have lower energies than free vortices, but have lower entropy as well. In order to minimize free energy, , the system undergoes a transition at a critical temperature, . Below , there are only bound vortex–antivortex pairs. Above , there are free vortices.
Informal description
There is an elegant thermodynamic argument for the Kosterlitz–Thouless transition. The energy of a single vortex is , where is a parameter that depends upon the system in which the vortex is located, is the system size, and is the radius of the vortex core. One assumes . In the 2D system, the number of possible positions of a vortex is approximately . From Boltzmann's entropy formula, (with W is the number of states), the entropy is , where is Boltzmann's constant. Thus, the Helmholtz free energy is
When , the system will not have a vortex. On the other hand, when , 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. Above , the relation will be linear . Just below , the relation will be , as the number of free vortices will go as . 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. to confirm the Kosterlitz–Thouless transition in proximity-coupled Josephson junction arrays.
Field theoretic analysis
The following discussion uses field theoretic methods. Assume a field φ(x) defined in the plane which takes on values in , so that is identified with . That is, the circle is realized as .
The energy is given by
and the Boltzmann factor is .