integral is an integer multiple of . The value of this integer is the index of the vector field . Suppose that a given field configuration has punctures located at each with index . Then, decomposes into the sum of a field configuration with no punctures, and , where we have switched to the complex plane coordinates for convenience. The complex argument function has a branch cut, but, because is defined modulo , it has no physical consequences. Now, If , the second term is positive and diverges in the limit : configurations with unbalanced numbers of vortices of each orientation are never energetically favoured. When however , the second term is equal to , 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 . At low temperatures and large the distance between a vortex and antivortex pair tends to be extremely small, essentially of the order . At large temperatures and small 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. See also KTHNY theory Goldstone boson Ising model Lambda transition Potts model Quantum vortex Superfluid film Hexatic phase Topological defect Notes References . Translation available: . Translation available: B. I. Halperin, D. R. Nelson, Phys. Rev. Lett. 41, 121 (1978) A. P. Young, Phys. Rev. B 19, 1855 (1979) Books J.V. Jose, 40 Years of Berezinskii–Kosterlitz–Thouless Theory, World Scientific, 2013, H. Kleinert, Gauge Fields in Condensed Matter, Vol. I, " SUPERFLOW AND VORTEX LINES", pp. 1–742, World Scientific (Singapore, 1989); Paperback (also available online: Vol. I. Read pp. 618–688); H. Kleinert, Multivalued Fields in Condensed Matter, Electrodynamics, and Gravitation'', World Scientific (Singapore, 2008) (also available online: here) Theoretical physics Statistical mechanics Lattice models Phase transitions