Editing Kosterlitz-Thouless transition

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The '''Kosterlitz-Thouless transition''' (also known as the Berezinskii-Kosterlitz-Thouless (BKT) phase transition)<ref>V. L. Berezinskii "Destruction of Long-range Order in One-dimensional and Two-dimensional Systems having a Continuous Symmetry Group I. Classical Systems", Journal of Experimental and Theoretical Physics '''32''' pp. 493 (1971)</ref>
<ref>V. L. Berezinskii "Destruction of Long-range Order in One-dimensional and Two-dimensional Systems Possessing a Continuous Symmetry Group. II. Quantum Systems", Journal of Experimental and Theoretical Physics '''34''' pp. 610 (1972)</ref>
<ref>[http://dx.doi.org/10.1088/0022-3719/5/11/002   J. M. Kosterlitz and D. J. Thouless "Long range order and metastability in two dimensional solids and superfluids. (Application of dislocation theory)", Journal of Physics C: Solid State Physics '''5''' pp. L124-L126 (1972)]</ref>
<ref name="KT_1">[http://dx.doi.org/10.1088/0022-3719/6/7/010  J. M. Kosterlitz and D. J. Thouless "Ordering, metastability and phase transitions in two-dimensional systems", Journal of Physics C: Solid State Physics '''6''' pp. 1181-1203 (1973)]</ref> is a [[phase transitions | phase transition]]
found in the two-dimensional [[XY model]]. Below the transition temperature, <math>T_{KT}</math>, the system plays host to a 'liquid' of vortex-antivortex pairs that have zero total vorticity. Above <math>T_{KT}</math> these pairs break up into a gas of independent vortices.

For the XY model the critical temperature is given by (Eq.4 in <ref name="KT_1"></ref>):

:<math>T_c = \frac{\pi J}{k_B}</math>

where <math>J</math> is the spin-spin coupling constant. This can be obtained as (Eq.58 in <ref name="KT_1"></ref>):

:<math>\frac{\pi J}{k_BT_c}-1 \approx \pi \tilde{y}_c(0) \exp\left(\frac{-\pi^2J}{k_BT_c} \right) \approx 0.12</math>
==See also==
*[[Universality classes#XY | XY universality class]]
==References==
<references/>
;Related reading
*[http://dx.doi.org/10.1103/PhysRevLett.41.121     B. I. Halperin and David R. Nelson "Theory of Two-Dimensional Melting", Physical  Review Letters '''41''' pp. 121-124 (1978)]
*[http://dx.doi.org/10.1103/PhysRevB.19.1855  A. P. Young "Melting and the vector Coulomb gas in two dimensions",  Physical Review B '''19''' pp. 1855-1866 (1979)]
*[http://dx.doi.org/10.1103/PhysRevB.19.2457      David R. Nelson and B. I. Halperin "Dislocation-mediated melting in two dimensions", Physical Review B '''19''' pp. 2457-2484 (1979)]
*[http://dx.doi.org/10.1103/PhysRevLett.44.463 Farid F. Abraham "Melting in Two Dimensions is First Order: An Isothermal-Isobaric Monte Carlo Study", Physical Review Letters '''44''' pp. 463-466 (1980)]
*[http://dx.doi.org/10.1103/PhysRevB.23.6145 Farid F. Abraham "Two-dimensional melting, solid-state stability, and the Kosterlitz-Thouless-Feynman criterion", Physical Review B  '''23''' pp. 6145-6148 (1981)]
*[http://dx.doi.org/10.1103/RevModPhys.60.161  Katherine J. Strandburg "Two-dimensional melting", Reviews of Modern Physics '''60''' pp. 161-207 (1988)]
* [[Hagen Kleinert|Hagen Kleinert]] ''Gauge Fields in Condensed Matter'', Vol. I,  " SUPERFLOW AND VORTEX LINES", pp.&nbsp;1–742, Vol. II,  "STRESSES AND DEFECTS", pp.&nbsp;743–1456,  [http://www.worldscibooks.com/physics/0356.html World Scientific (Singapore, 1989)];  Paperback ISBN 9971-5-0210-0 '' (also available online: [http://www.physik.fu-berlin.de/~kleinert/kleiner_reb1/contents1.html Vol. I] and [http://www.physik.fu-berlin.de/~kleinert/kleiner_reb1/contents2.html Vol. II])''
*[http://dx.doi.org/10.1088/0953-8984/14/9/321 Kurt Binder, Surajit Sengupta and Peter Nielaba "The liquid-solid transition of hard discs: first-order transition or Kosterlitz-Thouless-Halperin-Nelson-Young scenario?", Journal of Physics: Condensed Matter '''14''' pp. 2323-2333 (2002)]
* "40 Years of Berezinskii–Kosterlitz–Thouless Theory" (Ed. Jorge V José) World Scientific Publishing (2013) ISBN 978-981-4417-63-1
*[http://www.nobelprize.org/nobel_prizes/physics/laureates/2016/advanced-physicsprize2016.pdf Nobel Prize in Physics 2016 'Scientific Background']

[[category: phase transitions]]