Kosterlitz-Thouless transition - Revision history

Carl McBride: Added an internal link

2016-10-04T12:16:58Z

Added an internal link

← Older revision		Revision as of 14:16, 4 October 2016
Line 13:		Line 13:

	:<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>		:<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/>		<references/>

Carl McBride: Added a little more

2016-10-04T12:12:27Z

Added a little more

← Older revision		Revision as of 14:12, 4 October 2016
Line 3:		Line 3:
	<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://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>[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>[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]]		<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.		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>

	==References==		==References==
	<references/>		<references/>

Carl McBride: Added a little text

2016-10-04T11:33:58Z

Added a little text

← Older revision		Revision as of 13:33, 4 October 2016
Line 4:		Line 4:
	<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>[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>[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]]		<ref>[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]].		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.
	==References==		==References==
	<references/>		<references/>

Carl McBride: fixed link

2016-10-04T10:27:06Z

fixed link

← Older revision		Revision as of 12:27, 4 October 2016
Line 17:		Line 17:
	*[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)]		*[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		* "40 Years of Berezinskii–Kosterlitz–Thouless Theory" (Ed. Jorge V José) World Scientific Publishing (2013) ISBN 978-981-4417-63-1
	*[www.nobelprize.org/nobel_prizes/physics/laureates/2016/advanced-physicsprize2016.pdf Nobel Prize in Physics 2016 'Scientific Background']		*[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]]

Carl McBride: Added a link to the 2016 Nobel Prize press release

2016-10-04T10:26:20Z

Added a link to the 2016 Nobel Prize press release

← Older revision		Revision as of 12:26, 4 October 2016
Line 17:		Line 17:
	*[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)]		*[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		* "40 Years of Berezinskii–Kosterlitz–Thouless Theory" (Ed. Jorge V José) World Scientific Publishing (2013) ISBN 978-981-4417-63-1
			*[www.nobelprize.org/nobel_prizes/physics/laureates/2016/advanced-physicsprize2016.pdf Nobel Prize in Physics 2016 'Scientific Background']

	[[category: phase transitions]]		[[category: phase transitions]]

Carl McBride: /* References */ Added a book

2016-10-04T10:03:53Z

References: Added a book

← Older revision		Revision as of 12:03, 4 October 2016
Line 16:		Line 16:
	* [[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])''		* [[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)]		*[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

	[[category: phase transitions]]		[[category: phase transitions]]

Carl McBride: Added external links to references

2016-10-04T09:55:06Z

Added external links to references

← Older revision		Revision as of 11:55, 4 October 2016
Line 1:		Line 1:
	{{Stub-general}}		{{Stub-general}}
	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>		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>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://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>[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>[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]]		<ref>[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]]

130.133.8.114 at 10:01, 28 July 2012

2012-07-28T10:01:52Z

← Older revision		Revision as of 12:01, 28 July 2012
Line 14:		Line 14:
	*[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/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)]		*[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 ~~Kleinet~~\|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])''		* [[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)]		*[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)]

	[[category: phase transitions]]		[[category: phase transitions]]

130.133.8.114 at 10:00, 28 July 2012

2012-07-28T10:00:23Z

← Older revision		Revision as of 12:00, 28 July 2012
Line 14:		Line 14:
	*[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/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)]		*[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 Kleinet\|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)]		*[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)]

	[[category: phase transitions]]		[[category: phase transitions]]

Carl McBride: Slight tidy

2011-03-03T12:34:23Z

Slight tidy

← Older revision		Revision as of 14:34, 3 March 2011
Line 1:		Line 1:
	{{Stub-general}}		{{Stub-general}}
	The '''Kosterlitz-Thouless transition''' (also known as the Berezinskii-Kosterlitz-Thouless (BKT) phase transition) is a [[phase transitions \| phase transition]]		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>[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]].		found in the two-dimensional [[XY model]].
	==References==		==References==
	#V. L. Berezinskii "DESTRUCTION OF LONG-RANGE ORDER IN ONE-DIMENSIONAL AND 2-DIMENSIONAL SYSTEMS HAVING A CONTINUOUS SYMMETRY GROUP 1 - CLASSICAL SYSTEMS", Journal of Experimental and Theoretical Physics '''32''' pp. 493 (1971)		<references/>
	#V. L. Berezinskii "DESTRUCTION OF LONG-RANGE ORDER IN ONE-DIMENSIONAL AND 2-DIMENSIONAL SYSTEMS POSSESSING A CONTINUOUS SYMMETRY GROUP .2. QUANTUM SYSTEMS", Journal of Experimental and Theoretical Physics '''34''' pp. 610 (1972)		;Related reading
	~~#[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)]		*[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/10.1088/0022-3719/6/7/010 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)]		*[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/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.19.2457 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/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/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.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/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/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)]
	#[http://dx.doi.org/10.1103/RevModPhys.60.161 Katherine J. Strandburg "Two-dimensional melting", Reviews of Modern Physics '''60''' pp. 161-207 (1988)]		*[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)]
	#[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)]
	[[category: phase transitions]]		[[category: phase transitions]]