Berezinskii–Kosterlitz–Thouless transition

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 McBryan & Spencer (1977) and Fröhlich & Spencer (1981).

There is an elegant thermodynamic argument for the Kosterlitz–Thouless 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, ${\displaystyle S=k_{\rm {B}}\ln W}$ (with W is the number of states), the entropy is ${\displaystyle S=2k_{\rm {B}}\ln(R/a)}$, where ${\displaystyle k_{\rm {B}}}$ is Boltzmann's constant. Thus, the Helmholtz free energy is

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

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

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

The Kosterlitz–Thouless 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 Kosterlitz–Thouless transition and may be used to determine ${\displaystyle T_{c}}$. This approach was used in Resnick et al.[3] to confirm the Kosterlitz–Thouless transition in proximity-coupled Josephson junction arrays.

The following discussion uses field theoretic methods. Assume a field φ(x) defined in the plane which takes on values in ${\displaystyle S^{1}}$, so that ${\displaystyle \phi (x)}$ is identified with ${\displaystyle \phi (x)+2\pi }$. That is, the circle is realized as ${\displaystyle S^{1}=\mathbb {R} /2\pi \mathbb {Z} }$.

The energy is given by

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

and the Boltzmann factor is ${\displaystyle \exp(-\beta E)}$.

Taking a contour integral ${\displaystyle \oint _{\gamma }d\phi =\oint _{\gamma }{\frac {d\phi }{dx}}dx}$ over any contractible closed path ${\displaystyle \gamma }$, we would expect it to be zero (for example, by the fundamental theorem of calculus. However, this is not the case due to the singular nature of vortices (which give singularities in ${\displaystyle \phi }$).

To render the theory well-defined, it is only defined up to some energetic cut-off scale ${\displaystyle \Lambda }$, so that we can puncture the plane at the points where the vortices are located, by removing regions with size of order ${\displaystyle 1/\Lambda }$. If ${\displaystyle \gamma }$ winds counter-clockwise once around a puncture, the contour integral ${\displaystyle \oint _{\gamma }d\phi }$ is an integer multiple of ${\displaystyle 2\pi }$. The value of this integer is the index of the vector field ${\displaystyle \nabla \phi }$.

Suppose that a given field configuration has ${\displaystyle N}$ punctures located at ${\displaystyle x_{i},i=1,\dots ,N}$ each with index ${\displaystyle n_{i}=\pm 1}$. Then, ${\displaystyle \phi }$ decomposes into the sum of a field configuration with no punctures, ${\displaystyle \phi _{0}}$ and ${\displaystyle \sum _{i=1}^{N}n_{i}\arg(z-z_{i})}$, where we have switched to the complex plane coordinates for convenience. The complex argument function has a branch cut, but, because ${\displaystyle \phi }$ is defined modulo ${\displaystyle 2\pi }$, it has no physical consequences.

Now,

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

If ${\displaystyle \sum _{i=1}^{N}n_{i}\neq 0}$, the second term is positive and diverges in the limit ${\displaystyle \Lambda \to \infty }$: configurations with unbalanced numbers of vortices of each orientation are never energetically favoured.

However, if the neutral condition ${\displaystyle \sum _{i=1}^{N}n_{i}=0}$ holds, the second term is equal to ${\displaystyle -2\pi \sum _{1\leq i<j\leq N}n_{i}n_{j}\ln(|x_{j}-x_{i}|/L)}$, which is the total potential energy of a two-dimensional Coulomb gas. The scale L is an arbitrary scale that renders the argument of the logarithm dimensionless.

Assume the case with only vortices of multiplicity ${\displaystyle \pm 1}$. At low temperatures and large ${\displaystyle \beta }$ the distance between a vortex and antivortex pair tends to be extremely small, essentially of the order ${\displaystyle 1/\Lambda }$. At large temperatures and small ${\displaystyle \beta }$ this distance increases, and the favoured configuration becomes effectively the one of a gas of free vortices and antivortices. The transition between the two different configurations is the Kosterlitz–Thouless phase transition, and the transition point is associated with an unbinding of vortex-antivortex pairs.

• Березинский, В. Л. (1970), "Разрушение дальнего порядка в одномерных и двумерных системах с непрерывной группой симметрии I. Классические системы", ЖЭТФ (in Russian), 59 (3): 907–920. Translation available: Berezinskii, V. L. (1971), "Destruction of long-range order in one-dimensional and two-dimensional systems having a continuous symmetry group I. Classical systems" (PDF), Sov. Phys. JETP, 32 (3): 493–500, Bibcode:1971JETP...32..493B
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• J.V. Jose, 40 Years of Berezinskii–Kosterlitz–Thouless Theory, World Scientific, 2013, ISBN 978-981-4417-65-5
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