Kosterlitz-Thouless transition: Difference between revisions
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The '''Kosterlitz-Thouless transition''' (also known as the Berezinskii-Kosterlitz-Thouless (BKT) phase transition)<ref>[http://www.jetp.ac.ru/cgi-bin/e/index/e/32/3/p493?a=list 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>[http://www.jetp.ac.ru/cgi-bin/e/index/e/34/3/p610?a=list 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== | ||
<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. 1–742, Vol. II, "STRESSES AND DEFECTS", pp. 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]] | [[category: phase transitions]] | ||
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The Kosterlitz-Thouless transition (also known as the Berezinskii-Kosterlitz-Thouless (BKT) phase transition)[1] [2] [3] [4] is a phase transition found in the two-dimensional XY model. Below the transition temperature, , the system plays host to a 'liquid' of vortex-antivortex pairs that have zero total vorticity. Above these pairs break up into a gas of independent vortices.
For the XY model the critical temperature is given by (Eq.4 in [4]):
where is the spin-spin coupling constant. This can be obtained as (Eq.58 in [4]):
See also[edit]
References[edit]
- ↑ 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)
- ↑ 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)
- ↑ 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)
- ↑ 4.0 4.1 4.2 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)
- Related reading
- B. I. Halperin and David R. Nelson "Theory of Two-Dimensional Melting", Physical Review Letters 41 pp. 121-124 (1978)
- A. P. Young "Melting and the vector Coulomb gas in two dimensions", Physical Review B 19 pp. 1855-1866 (1979)
- David R. Nelson and B. I. Halperin "Dislocation-mediated melting in two dimensions", Physical Review B 19 pp. 2457-2484 (1979)
- Farid F. Abraham "Melting in Two Dimensions is First Order: An Isothermal-Isobaric Monte Carlo Study", Physical Review Letters 44 pp. 463-466 (1980)
- Farid F. Abraham "Two-dimensional melting, solid-state stability, and the Kosterlitz-Thouless-Feynman criterion", Physical Review B 23 pp. 6145-6148 (1981)
- Katherine J. Strandburg "Two-dimensional melting", Reviews of Modern Physics 60 pp. 161-207 (1988)
- Hagen Kleinert Gauge Fields in Condensed Matter, Vol. I, " SUPERFLOW AND VORTEX LINES", pp. 1–742, Vol. II, "STRESSES AND DEFECTS", pp. 743–1456, World Scientific (Singapore, 1989); Paperback ISBN 9971-5-0210-0 (also available online: Vol. I and Vol. II)
- 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
- Nobel Prize in Physics 2016 'Scientific Background'