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Hagen Kleinert

2012-07-28T10:07:43Z

130.133.8.114:

[[Image:Hagmich.jpg|thumb|Hagen Kleinert, Photo taken in 2006]]
'''Hagen Kleinert''' (born 15 June 1941 in Festenberg, Germany, now [[Twardogóra]], Poland) is Professor of [[Theoretical Physics]] at the [[Free University of Berlin]], [[Germany]] (since 1968),
[http://users.physik.fu-berlin.de/~kleinert/BITMAPS/Kleinert_with_Academic_Senate_of_West_University_of_Timisoara.jpg Honorary Doctor] at the
[[West University of Timişoara|West University]] of [[Timişoara]], [http://users.physik.fu-berlin.de/~kleinert/news/drhc1a.jpg and]
at the
[http://www.4icu.org/reviews/3122.htm Kyrgyz-Russian Slavic University]
in [[Bishkek]]. He is also
[http://users.physik.fu-berlin.de/~kleinert/BITMAPS/akademieru.jpg Honorary Member] of the
[http://www.physik.fu-berlin.de/~kleinert/news/academycreative.html Russian Academy of Creative Endeavors]. For his contributions to particle and solid state physics he was [http://www.iop.org/activity/awards/International_Bilateral_Awards/The_Max_Born_Medal_and_Prize/Born_medal_winners/page_29137.html awarded] the [[Max Born prize]] 2008 with [http://www.dpg-physik.de/dpg/preise/preistraeger/max-born-preis.pdf Medal].
His contribution<ref>
{{cite journal
| title = From Landau's Order Parameter to Modern Disorder Fields
| first3 = Gregory
| last3 = Vereshchagin
| first2 = Remo
| last2 = Ruffini
| author = Kleinert H.
| journal = In "Lev Davidovich Landau and his Impact on Contemporary Theoretical Physics", publ. in
"Horizons in World Physics")
| volume = 264
| pages = 103
| year = 2009
| doi =
10.1063/1.3382313| url = http://users.physik.fu-berlin.de/~kleinert/373/373.pdf
| bibcode = 2010AIPC.1205..103K}}</ref> to the [https://www.novapublishers.com/catalog/product_info.php?products_id=9382 memorial volume] celebrating the 100th birthday of [[Lev Davidovich Landau]] earned him the [[Majorana Prize]] 2008 with Medal.
==Publications==
Kleinert has written ~400 papers on [[mathematical physics]] and the [[physics]] of [[elementary particle]]s, [[Atomic nucleus|nuclei]], [[solid|solid state]] systems, [[liquid crystal]]s, [[biomembrane]]s, [[microemulsion]]s, [[polymer]]s, and the theory of [[financial markets]]<ref>[http://users.physik.fu-berlin.de/~kleinert/kleinert/?p=articles&page=all His Papers].</ref>. He has written several books on theoretical physics<ref>[http://users.physik.fu-berlin.de/~kleinert/kleinert/?p=booklist His books].</ref>, the most notable of which, ''Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets,'' has been published in five editions since 1990 and has received enthusiastic reviews.<ref>
{{cite journal
| title = Book Reviews
| author = Henry B.I.
| journal = Australian Physics
| volume = 44
| issue = 3
| pages = 110
| year = 2007
| doi =
| url = http://www.physik.fu-berlin.de/~kleinert/b5/reviews/picrithenry.doc}}</ref>
==Education==
After his first years as a student at the [[University of Hanover]] and [[Georgia Institute of Technology]],<ref>{{cite web|url=http://www.icranet.org/misc/Scientific_Report_2008/CV/Kleinert.pdf|title=Kleinert Hagen|publisher=International Center for Relativistic Astrophysics Network|accessdate=2011-07-09}}</ref> he learned [[general relativity]] as a graduate student from [[George Gamov]], one of the fathers of the [[Big Bang]] theory.
==Career==
As a young professor in 1972, Kleinert visited [[California Institute of Technology|Caltech]] and was impressed by noted US physicist [[Richard Feynman]]. Later, Kleinert was to collaborate with Feynman<ref>
{{cite journal
| title = Travailler avec Feynman
| author = Kleinert H.
| journal = Science (French edition)
| volume = 19
| pages = 89–95
| year = 2004
| doi =
| url = http://users.physik.fu-berlin.de/~kleinert/feynman/feynmanw-Dateien/FEYNMAN_CH11_30avr.pdf}}</ref> in some of the latter's last work.<ref>
{{cite journal
| title = Effective classical partition functions
| author = [[Richard P. Feynman|Feynman R.P.]], Kleinert H.
| journal = [[Physical Review]] A
| volume = 34
| issue =
6| pages = 5080–5084
| year = 1986
| doi = 10.1103/PhysRevA.34.5080
| url = http://www.physik.fu-berlin.de/~kleinert/159/159.pdf
| pmid=9897894|bibcode = 1986PhRvA..34.5080F }}</ref>
This collaboration led to a mathematical method for converting [[divergent]] weak-coupling [[power series]] into [[wikt:convergent|convergent]] strong-coupling ones. This so-called [[variational perturbation theory]] yields at present the most accurate theory of critical exponents<ref>
Kleinert, H., [http://www.physik.fu-berlin.de/~kleinert/279/279.pdf "Critical exponents from seven-loop strong-coupling φ4 theory in three dimensions". [[Physical Review]] D 60, 085001 (1999)]</ref>
observable close to second-order [[phase transition]]s, as confirmed for [[superfluid]] helium in satellite experiments.<ref>
{{cite journal
| title = Specific heat of liquid helium in zero gravity very near the lambda point
| author = Lipa J.A.
| journal = [[Physical Review]] B
| volume = 68
| issue =
17| pages = 174518
| year = 2003
| doi = 10.1103/PhysRevB.68.174518
| url = http://www.physik.fu-berlin.de/~kleinert/papers/lipa.pdf
| last2 = Nissen
| first2 = J.
| last3 = Stricker
| first3 = D.
| last4 = Swanson
| first4 = D.
| last5 = Chui
| first5 = T.|arxiv = cond-mat/0310163 |bibcode = 2003PhRvB..68q4518L }}</ref> He also discovered an alternative to Feynman's time-sliced path integral construction which can be used to solve the [[Path integral formulation|path integral]] formulations of the [[hydrogen]] atom and the centrifugal barrier, i.e. to calculate their energy levels and eigenstates, as special cases of a general strategy for treating systems with [[Mathematical singularity | singular]] potentials using path integrals. <ref>
{{cite journal
| title = Solution of the path integral for the H-atom
| author = Duru I.H., Kleinert H.
| journal = Physics Letters B
| volume = 84
| issue = 2
| pages = 185–188
| year = 1979
| doi = 10.1016/0370-2693(79)90280-6
| url = http://www.physik.fu-berlin.de/~kleinert/kleiner_re65/65.pdf|bibcode = 1979PhLB...84..185D }}</ref><ref>
{{cite journal
| title = Quantum Mechanics of H-Atom from Path Integrals
| author = Duru I.H., Kleinert H.
| journal = Fortschr. Phys
| volume = 30
| issue = 2
| pages = 401–435
| year = 1982
| doi = 10.1002/prop.19820300802
| url = http://www.physik.fu-berlin.de/~kleinert/kleiner_re83/83.pdf}}</ref>

Within the quantum field theories of quarks
he found the origin<ref>
{{cite journal
| title = Bilocal Form Factors and Regge Couplings
| author = Kleinert H.
| journal = Nucl. Physics
| volume = B65
| pages = 77–111
| year = 1973
| doi = 10.1016/0550-3213(73)90276-9
| url = http://www.physik.fu-berlin.de/~kleinert/45/45.pdf|bibcode = 1973NuPhB..65...77K }}</ref> of the [http://www.physik.fu-berlin.de/~kleinert/papers/chn.pdf algebra of Regge residues] conjectured by [[Nicola Cabibbo|N. Cabibbo]], L. Horwitz, and
[[Yuval Ne'eman|Y. Ne'eman]] (see [http://www.physik.fu-berlin.de/~kleinert/papers/neeman1.pdf p.232] in Ref.<ref>
{{cite journal
| title = Universality in the Algebra of Vertex Strengths as Generated by Bilocal Currents
| author = Ne'eman Y, Reddy V.T.N.
| journal = Nucl. Phys. B
| volume = 84
| pages = 221–233
| year = 1981
| doi = 10.1016/0550-3213(75)90547-7
| url = http://www.physik.fu-berlin.de/~kleinert/papers/neemredd.pdf|bibcode = 1975NuPhB..84..221N }}
</ref>).

==Discoveries==
For [[superconductor]]s he predicted in 1982 a tricritical point in the [[phase diagram]] between type-I and type-II superconductors where the order of the transition changes from second to first.<ref>
{{cite journal
| title = Disorder Version of the Abelian Higgs Model and the Order of the Superconductive Phase Transition
| author = Kleinert H.
| journal = Lett. Nuovo Cimento
| volume = 35
| issue = 13
| pages = 405–412
| year = 1982
| doi = 10.1007/BF02754760
| url = http://www.physik.fu-berlin.de/~kleinert/97/97.pdf}}
</ref> The predictions were confirmed in 2002 by [[Monte Carlo method | Monte Carlo]] [[Computer simulation| computer simulations]].<ref>
{{cite journal
| title = Vortex interactions and thermally induced crossover from type-I to type-II superconductivity
| author = Hove J., Mo S., Sudbo A.
| journal = Phys. Rev. B
| volume = 66
| issue = 6
| pages = 064524
| year = 2002
| doi = 10.1103/PhysRevB.66.064524
| url = http://www.physik.fu-berlin.de/~kleinert/papers/sudbotre064524.pdf|arxiv = cond-mat/0202215 |bibcode = 2002PhRvB..66f4524H }}
</ref>

The theory is based on a [[disorder field|disorder field theory]] [[Duality (mathematics) | dual]] to the [[order parameter|order field theory]] of [[Lev Landau|L.D. Landau]] for [[phase transition]]s which Kleinert developed in the books on ''Gauge Fields in Condensed Matter''. In this theory, the statistical properties of fluctuating [[vortex]] or [[Crystallographic defect|defect lines]] are described as elementary excitations with the help of fields, whose [[Feynman diagram]]s are the pictures of the lines.

At the 1978 summer school in [[Erice]] he proposed the existence of broken [[supersymmetry]] in atomic nuclei,<ref>
{{cite journal
| title = The New Aspects of Subnuclear Physics
| author = Ferrara S., Discussion Section of 1978 Erice Lecture publ. in
| journal = Plenum Press, N.Y., Zichichi A. Ed.
| volume =
| pages = 40
| year = 1980
| doi =
| url = http://www.physik.fu-berlin.de/~kleinert/55/1978/supersnuc.pdf}}
</ref> which has since been observed experimentally.<ref>
{{cite journal
| title = Evidence for the Existence of Supersymmetry in Atomic Nuclei
| author = Metz A., Jolie J., Graw G., Hertenberger R., Gröger J., Günther C., Warr N., Eisermann Y.
| year = 1999
| journal = [[Physical Review Letters]]
| volume = 83
| issue = 8
| page = 1542
| doi = 10.1103/PhysRevLett.83.1542
| bibcode=1999PhRvL..83.1542M}}
</ref>

His theory of collective quantum fields<ref>
{{cite journal
| title = Collective Quantum Fields
| author = Kleinert H.
| journal = Fortschritte der Physik
| volume = 36
| issue = 11-12
| pages = 565–671
| year = 1978
| doi = 10.1002/prop.19780261102
| url = http://www.physik.fu-berlin.de/~kleinert/55/55.pdf}}
</ref> and the Hadronization of Quark Theories<ref>
{{cite journal
| title = On the Hadronization of Quark Theories
| author = Kleinert H., Lectures presented at the Erice Summer Institute 1976
| journal = Understanding the Fundamental Constituents of Matter, Plenum Press, New York, 1978 (A. Zichichi ed.)
| volume =
62
| issue = 4| pages = pp. 289–390
| year = 1978
| doi =
10.1016/0370-2693(76)90676-6| url = http://www.physik.fu-berlin.de/~kleinert/53/53.pdf
| bibcode = 1976PhLB...62..429K}}
</ref> are prototypes for numerous developments in the theory of [[condensed matter]], [[nuclear physics|nuclear]] and [[elementary particle physics]].

Together with K. Maki he proposed and clarified in 1981 a possible [[icosahedral phase]] of [[quasicrystal]]s.<ref>
{{cite journal
| title = Lattice Textures in Cholesteric Liquid Crystals
| author = Kleinert H., Maki K.
| journal = Fortschritte der Physik
| volume = 29
| issue = 5
| pages = 219–259
| year = 1981
| doi = 10.1002/prop.19810290503
| url = http://www.physik.fu-berlin.de/~kleinert/75/75.pdf}}
</ref>
This structure was discovered three years later in aluminum transition metal alloys by [[Dan Shechtman]], which earned him the Nobel Prize 2011.
See [http://users.physik.fu-berlin.de/~kleinert/icosahedral/index.html historical notes].

==String theory contributions==
In 1986 he introduced<ref>
{{cite journal
| title = The Membrane Properties of Condensing Strings
| author = Kleinert H.
| journal = Phys. Lett. B
| volume = 174
| issue =
3| pages = 335
| year = 1989
| doi = 10.1016/0370-2693(86)91111-1
| url = http://www.physik.fu-berlin.de/~kleinert/149/149.pdf|bibcode = 1986PhLB..174..335K }}</ref> stiffness into the [[string theory|theory of strings]],
which had formerly been characterized by tension alone. This greatly improved the description of the physical properties of strings. The Russian physicist [[Alexander Markovich Polyakov|A. Polyakov]]
simultaneously proposed a similar extension, and so the model is now known as the [http://prola.aps.org/abstract/PRD/v41/i8/p2634_1 Polyakov-Kleinert string].
==Theory of distributions==
Together with A. Chervyakov, Kleinert developed an extension of the theory of [[Distribution (mathematics)|distributions]] from linear spaces to [[semigroup]]s by defining their products uniquely (in the mathematical theory, only linear combinations are defined). The extension is motivated by the physical requirement that the corresponding [[path integral formulation|path integrals]] must be invariant under coordinate transformations,<ref>
{{cite journal
| title =Rules for integrals over products of distributions from coordinate independence of path integrals
| author = Kleinert H., Chervyakov A.
| journal = Europ. Phys. J. C
| volume = 19
| issue =
4| pages = 743–747
| year = 2001
| doi = 10.1007/s100520100600
| url = http://www.physik.fu-berlin.de/~kleinert/kleiner_re303/wardepl.pdf|arxiv = quant-ph/0002067 |bibcode = 2001EPJC...19..743K }}</ref> which is necessary for the equivalence of the [[path integral formulation]] to [[Schrödinger equation | Schrödinger theory]].

==String theory alternative==
As an alternative to [[string theory]], Kleinert used the complete analogy between [[non-Euclidean geometry]] and the geometry of crystals with [[Crystallographic defect|defects]] to construct a model of the universe called the [[World Crystal]] or [http://www.physik.fu-berlin.de/~kleinert/papers/planckklcZN.pdf Planck-Kleinert crystal]. In this model, matter creates defects in spacetime which generate curvature. This curvature reproduces all the effects of [[general relativity]], but leads to different physics than string theory at the scale of the [[Planck length]]. This theory inspired Italian artist [http://www.laurapesce.it/ Laura Pesce] to create glass sculptures entitled [http://www.laurapesce.it/main/exhibitions/DICE2006/DICE2006.html "world crystal"] (see also lower left on [http://www.laurapesce.it/main/exhibitions/ArteScienza/ArteScienza.html this page]).
==Current==
Kleinert is a senior member of the faculty for the International Relativistic Astrophysics Ph.D. [http://www.icra.it/IRAPPhD/2007/Welcome.htm (IRAP)] Project, which forms part of the international network for astrophysics ([[ICRANet]]). He was also involved in the [[European Science Foundation]]'s project [http://www.esf.org/activities/research-networking-programmes/physical-and-engineering-sciences-pesc/current-research-networking-programmes/cosmology-in-the-laboratory-coslab.html Cosmology in the Laboratory].

Kleinert's 60. birthday was honored by a [[Festschrift]] and a [http://users.physik.fu-berlin.de/~kleinert/books.html Festcolloquium] with 65 contributions by international colleagues (for instance [[Yuval Ne'eman|Y. Ne'eman]], [[Roman Jackiw|R. Jackiw]], [[Harald Fritzsch|H. Fritzsch]], [[Remo Ruffini|R. Ruffini]], [[Cécile DeWitt|C. DeWitt]], [[Louis H. Kauffman|L. Kauffman]], [[Jozef T. Devreese|J. Devreese]], and [[Kazumi Maki|K. Maki]]).

== References ==
{{Reflist}}

== Books ==

* ''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])''

* ''Critical Properties of φ<sup>4</sup>-Theories'', [http://www.worldscibooks.com/physics/4733.html World Scientific (Singapore, 2001)]; Paperback ISBN 981-02-4658-7'' (also available [http://users.physik.fu-berlin.de/~kleinert/kleinert/?p=booklist&details=6 online]) (together with V. Schulte-Frohlinde)''

* ''Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets'', 5th edition, [http://www.worldscibooks.com/physics/7305.html World Scientific (Singapore, 2009)] (also available [http://users.physik.fu-berlin.de/~kleinert/kleinert/?p=booklist&details=11 online])

* ''Multivalued Fields in in Condensed Matter, Electrodynamics, and Gravitation'', [http://www.worldscibooks.com/physics/6742.html World Scientific (Singapore, 2008)] (also available [http://users.physik.fu-berlin.de/~kleinert/kleinert/?p=booklist&details=9 online])

* ''Proceedings of the [http://www.icra.it/MG/mg11 Eleventh Marcel Grossmann Meeting] on [[General Relativity]]'', [http://www.worldscibooks.com/physics/6997.html World Scientific (Singapore, 2008)] (together with R.T. Jantzen)''

== External links ==
*[http://www.physik.fu-berlin.de/~kleinert/ Personal homepage]

Phase transitions

2012-07-28T10:05:03Z

130.133.8.114:

A '''phase transition''' occurs at points on the [[Phase diagrams | phase diagram]] where the [[Helmholtz energy function]] is a singular function of the thermodynamic variables within it
<ref>[http://dx.doi.org/10.1007/s10955-009-9814-1 Leo P. Kadanoff "More is the Same; Phase Transitions and Mean Field Theories", Journal of Statistical Physics '''137''' pp. 777-797 (2009)]</ref>.
The following is (an incomplete) list of phase transitions. The change from one state to another is usually monitored by some sort of [[order parameters | order parameter]].
*[[1-dimensional phase transitions]]
*[[First-order transitions]]
*[[Gas-liquid phase transitions]]
*[[Glass transition]]
*[[Isotropic-nematic phase transition]]
*[[Kosterlitz-Thouless transition]]
*[[Landau theory of second-order phase transitions]]
*[[Liquid-liquid phase transitions]]
*[[Martensitic phase transitions]]
*[[Nematic-smectic phase transition]]
*[[Nonequilibrium phase transitions]]
*[[Order-disorder transitions]]
*[[Quantum phase transitions]]
*[[Second-order transitions]]
*[[Solid-liquid phase transitions]]
*[[Topological phase transitions]]
*[[Gas-liquid phase transitions | Vapour-liquid phase transitions]]
'''Other transitions'''
*[[Coil-globule transition]]
*[[Helix-coil transition]]
==See also==
*[[Computation of phase equilibria]]
==References==
<references/>
'''Recommended reading'''
*Paul Ehrenfest "Phase conversions in a general and enhanced sense, classified according to the specific singularities of the thermodynamic potential", Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen '''36''' pp. 153-157 (1933)
*[http://www.oup.com/uk/catalogue/?ci=9780195053166 H. Eugene Stanley "Introduction to Phase Transitions and Critical Phenomena", Oxford University Press (1988)]
*[http://www.oup.com/uk/catalogue/?ci=9780198517306 J. M. Yeomans "Statistical Mechanics of Phase Transitions", Oxford University Press (1992)]
*[http://dx.doi.org/10.1021/jp9536460 Benjamin Widom "Theory of Phase Equilibrium", Journal of Physical Chemistry '''100''' pp 13190-13199 (1996)]
*[http://dx.doi.org/10.1070/PU1999v042n06ABEH000543 Georgii A. Martynov "The problem of phase transitions in statistical mechanics", Physics-Uspekhi '''42''' pp. 517-543 (1999)]
*[http://www.nature.com/nature/journal/v402/n6761supp/full/402c73a0.html Philip Ball "Transitions still to be made", Nature '''402''' C73-C76 (1999)]
* [[Hagen Kleinert|H. Kleinert]] and V. Schulte-Frohlinde, ''Critical Properties of φ<sup>4</sup>-Theories'', [http://www.worldscibooks.com/physics/4733.html World Scientific (Singapore, 2001)]; Paperback ISBN 981-02-4659-5 (Read online at [http://www.physik.fu-berlin.de/~kleinert/b8])''
[[category: phase transitions]]

Kosterlitz-Thouless transition

2012-07-28T10:01:52Z

130.133.8.114:

{{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>
<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]].
==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)]

[[category: phase transitions]]

Kosterlitz-Thouless transition

2012-07-28T10:00:23Z

130.133.8.114:

{{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>
<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]].
==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 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)]

[[category: phase transitions]]

Hagen Kleinert

2012-07-28T08:59:31Z

130.133.8.114: Created page with "Hagen Kleinert, Photo taken in 2006 '''Hagen Kleinert''' (born 15 June 1941 in Festenberg, Germany, now Twardogóra, Poland) is Professor of [[..."