# Kosterlitz–Thouless transition

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The Berezinskii–Kosterlitz–Thouless transition (BKT transition) is a phase transition in the two-dimensional (2-D) XY model. 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, Kosterlitz and Duncan Haldane.

## 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 (see Goldstone boson) 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 McBryan & Spencer (1977) and Fröhlich & Spencer (1981).

## KT transition: 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 ${\displaystyle T_{c}}$ of the KT transition. At temperatures below this, vortex generation has a power law correlation.

Many systems with KT transitions involve the dissociation of bound anti-parallel vortex pairs, called vortex–antivortex pairs, into unbound vortices rather than vortex generation.[1][2] 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, ${\displaystyle F=E-TS}$, the system undergoes a transition at a critical temperature, ${\displaystyle T_{c}}$. Below ${\displaystyle T_{c}}$, there are only bound vortex–antivortex pairs. Above ${\displaystyle T_{c}}$, there are free vortices.

## Informal description

There is an elegant thermodynamic argument for the KT transition. The energy of a single vortex is ${\displaystyle \kappa \ln(R/a)}$, where ${\displaystyle \kappa }$ is a parameter that depends upon the system in which the vortex is located, ${\displaystyle R}$ is the system size, and ${\displaystyle a}$ is the radius of the vortex core. One assumes ${\displaystyle R\gg a}$. In the 2D system, the number of possible positions of a vortex is approximately ${\displaystyle (R/a)^{2}}$. From Boltzmann's entropy formula, ${\displaystyle S=k_{B}\ln W}$ (with W is the number of states), the entropy is ${\displaystyle S=2k_{B}\ln(R/a)}$, where ${\displaystyle k_{B}}$ is Boltzmann's constant. Thus, the Helmholtz free energy is

${\displaystyle F=E-TS=(\kappa -2k_{B}T)\ln(R/a).}$

When ${\displaystyle F>0}$, the system will not have a vortex. On the other hand, when ${\displaystyle F<0}$, entropic considerations favor the formation of a vortex. The critical temperature above which vortices may form can be found by setting ${\displaystyle F=0}$ and is given by

${\displaystyle T_{c}={\frac {\kappa }{2k_{B}}}.}$

The KT transition can be observed experimentally in systems like 2D Josephson junction arrays by taking current and voltage (I-V) measurements. Above ${\displaystyle T_{c}}$, the relation will be linear ${\displaystyle V\sim I}$. Just below ${\displaystyle T_{c}}$, the relation will be ${\displaystyle V\sim I^{3}}$, as the number of free vortices will go as ${\displaystyle I^{2}}$. This jump from linear dependence is indicative of a KT transition and may be used to determine ${\displaystyle T_{c}}$. This approach was used in Resnick et al.[1] to confirm the KT 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 ${\displaystyle S^{1}}$. For convenience, we work with the universal cover R of ${\displaystyle S^{1}}$ instead, but identify any two values of φ(x) that differ by an integer multiple of 2π.

The energy is given by

${\displaystyle E=\int {\frac {1}{2}}\nabla \phi \cdot \nabla \phi \,d^{2}x}$

and the Boltzmann factor is exp(−βE).

If we take the contour integral ${\displaystyle \oint _{\gamma }d\phi }$ over any closed path γ, we would expect it to be zero if γ is contractible, which is what we would expect for a planar curve. But here is the catch. Assume the XY theory has a UV cutoff Λ which requires some UV completion (i.e., the integration above is only defined up to E ~ Λ and needs a precise completion above). Then, we can have punctures in the plane (holes so to speak) so that if γ is a closed path which winds once around the puncture, ${\displaystyle \oint _{\gamma }d\phi }$ is only an integer multiple of 2π. These punctures are called vortices and if γ is a closed path which only winds once counterclockwise around the puncture and its winding number about any other puncture is zero, then the integer multiplicity can be attached to the vortex itself. Let's say a field configuration has n punctures at xi, i = 1, ..., n with multiplicities ni. Then, φ decomposes into the sum of a field configuration with no punctures, φ0 and ${\displaystyle \sum _{i=1}^{n}n_{i}\arg(z-z_{i})}$ where we have switched to the complex plane coordinates for convenience. The latter term has branch cuts, but because φ is only defined modulo 2π they are unphysical.

Now,

${\displaystyle E=\int {\frac {1}{2}}\nabla \phi _{0}\cdot \nabla \phi _{0}\,d^{2}x+\int {\frac {1}{2}}\nabla \sum _{i=1}^{n}n_{i}\arg(z-z_{i})\cdot \nabla \sum _{j=1}^{n}n_{j}\arg(z-z_{j})\,d^{2}x}$

Unless ${\displaystyle \sum _{i=1}^{n}n_{i}=0}$ the second term is positive infinite, so configurations with unbalanced numbers of vortices of each orientation zero are never observed.

When ${\displaystyle \sum _{i=1}^{n}n_{i}=0}$, the second term is equal to ${\displaystyle \sum _{1\leq i.

This is exactly the energy function for a two-dimensional Coulomb gas; the scale L contributes nothing but a constant.

Assume the case with only one vortex of multiplicity one and one vortex of multiplicity −1. At low temperatures, i.e. large β, because of the Boltzmann factor, the vortex–antivortex pair tends to be extremely close to one another. In fact, their separation would be around the cutoff scale. With more vortex–antivortex pairs, we have a collection of vortex-antivortex dipoles. At large temperatures, i.e. small β, the probability distribution swings the other way around and we have a plasma of vortices and antivortices. The phase transition between the two is the Kosterlitz–Thouless phase transition.