The following discussion uses field theoretic methods. Assume a field φ(x) defined in the plane which takes on values in S^1, so that \phi(x) is identified with \phi(x) + 2\pi. That is, the circle is realized as S^1 = \mathbb{R}/2\pi\mathbb{Z}. The energy is given by : E = \int \frac{1}{2} \nabla\phi\cdot\nabla\phi \, d^2 x and the
Boltzmann factor is \exp (-\beta E). Taking a
contour integral \textstyle \oint_\gamma d\phi = \oint_\gamma \frac{d\phi}{dx}dx over any contractible closed path \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 \phi). To render the theory well-defined, it is only defined up to some energetic cut-off scale \Lambda, so that we can puncture the plane at the points where the vortices are located, by removing regions with size of order 1/\Lambda. If \gamma winds counter-clockwise once around a puncture, the contour integral \textstyle \oint_\gamma d\phi is an integer multiple of 2\pi. The value of this integer is the
index of the vector field \nabla \phi. Suppose that a given field configuration has N punctures located at x_i, i=1,\dots,N each with index n_i=\pm 1. Then, \phi decomposes into the sum of a field configuration with no punctures, \phi_0 and \textstyle \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 \phi is defined modulo 2\pi, it has no physical consequences. Now, : E = \int \frac{1}{2} \nabla\phi_0\cdot\nabla\phi_0 \, d^2 x + \sum_{1\leq i If \textstyle \sum_{i=1}^N n_i \neq 0, the second term is positive and diverges in the limit \Lambda \to \infty: configurations with unbalanced numbers of vortices of each orientation are never energetically favoured. However, if the
neutral condition \textstyle \sum_{i=1}^N n_i=0 holds, the second term is equal to \textstyle -2\pi \sum_{1\leq i , 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 \pm 1. At low temperatures and large \beta the distance between a vortex and antivortex pair tends to be extremely small, essentially of the order 1/\Lambda. At large temperatures and small \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. == See also ==