查看“︁别列津斯基-科斯特利茨-索利斯相变”︁的源代码

'''别列津斯基-科斯特利茨-索利斯相變'''（{{lang-en|Berezinskii–Kosterlitz–Thouless transition}}，又稱'''BKT相變；'''科斯特利茨-索利斯相變及KT相變）是二維[[XY模型]]中的一种[[相變]]。它是指超过某一臨界溫度时，系统中的渦旋-反渦旋束縛態融化成为不成對的涡旋和反涡旋的相變。這種相變是以[[凝聚態物理學]]家{{le|瓦季姆·別列津斯基|Vadim Berezinskii}}、[[约翰·科斯特利茨]]和[[戴維·索利斯]]命名的。BKT相變在凝聚態物理學中多個可用XY模型作近似的系統中出現，例如[[约瑟夫森效应|約瑟夫森接面]]陣列和薄無序超導顆粒膜。這個詞最近還被研究二維超導絕緣體相變的社群應用，用於把[[庫珀對]]釘在絕緣區，能夠這樣做是因為超导中的这一相变与BKT相變有相似的地方。

對這種相變的研究使得索利斯和科斯特利茨於2016年與[[鄧肯·霍爾丹]]一同獲授[[諾貝爾物理學獎]]。

== XY模型 ==
[[XY模型]]的[[哈密頓函數|哈密頓]]是

<math>H = -J \sum_{\langle i,j\rangle}\mathbf{S}_i \cdot \mathbf{S}_j = -J \sum_{\langle i,j\rangle} \cos(\theta_i-\theta_j)</math>

<math>\boldsymbol{S}_i = (\cos{\theta_i}, \sin{\theta_i})</math>

[[格林函數]]（[[傳播子]]）是

<math>G(\boldsymbol{r}_i - \boldsymbol{r}_j) = \langle \boldsymbol{S}_i \cdot \boldsymbol{S}_j \rangle = \langle \cos(\theta_i - \theta_j) \rangle= \langle e^{i(\theta_i - \theta_j)} \rangle</math>

<math>G(\boldsymbol{r}_i - \boldsymbol{r}_j) \sim \exp \left( -r \log \frac{2}{\beta J} \right) </math>

<math>G(\boldsymbol{r}_i - \boldsymbol{r}_j) \sim \left(\frac{1}{r} \right)^{1/2\pi \beta J}</math>

== 動力學 ==

=== 単独渦旋的能量 ===
<math>E \sim \frac{J}{2}\int {\mathrm d}^2 r (\nabla \theta)^2 = \frac{J}{2}\int_a^L d^2 r 2\pi {\mathrm d} r \frac{1}{r^2} = \pi J \log \left( \frac{L}{a} \right)</math>

<math>S = \log\left( \frac{L}{a} \right)^2</math>

<math>F = E - TS = (\pi J - 2T) \log \left( \frac{L}{a} \right)</math>

<math>T_{BKT} = \frac{\pi J}{2}</math>

=== 一對渦旋的能量 ===
<math>E \sim - \pi J \sum_{i \neq j}  n_i n_j \log \left| \frac{ \boldsymbol{r}_i - \boldsymbol{r}_j }{a} \right| + \pi J \left( \sum_{i} n_i \right)^2 \log \left( \frac{L}{a} \right)</math>

<math>E \sim 2\pi J \log \left| \frac{ \boldsymbol{r}_i - \boldsymbol{r}_j }{a} \right| </math>

== 參考資料 ==
* {{citation|first=В. Л.|last=Березинский|title=Разрушение дальнего порядка в одномерных и двумерных системах с непрерывной группой симметрии I. Классические системы|journal=ЖЭТФ|volume=59|issue=3|year=1970|pages=907–920|language=ru}}. Translation available: {{citation|first=V. L.|last=Berezinskii|title=Destruction of long-range order in one-dimensional and two-dimensional systems having a continuous symmetry group I. Classical systems|journal=Sov. Phys. JETP|volume=32|issue=3|pages=493–500|year=1971|url=http://www.jetp.ac.ru/cgi-bin/dn/e_032_03_0493.pdf|format=pdf|bibcode=1971JETP...32..493B|accessdate=2016-10-04|archive-date=2019-07-13|archive-url=https://web.archive.org/web/20190713084858/http://www.jetp.ac.ru/cgi-bin/dn/e_032_03_0493.pdf|dead-url=no}}
* {{citation|first=В. Л.|last=Березинский|title=Разрушение дальнего порядка в одномерных и двумерных системах с непрерывной группой симметрии II. Квантовые системы|journal=ЖЭТФ|volume=61|issue=3|year=1971|pages=1144–1156|language=ru}}. Translation available: {{citation|first=V. L.|last=Berezinskii|title=Destruction of long-range order in one-dimensional and two-dimensional systems having a continuous symmetry group II. Quantum systems|journal=Sov. Phys. JETP|volume=34|issue=3|pages=610–616|year=1972|url=http://www.jetp.ac.ru/cgi-bin/dn/e_034_03_0610.pdf|format=pdf|bibcode=1972JETP...34..610B|accessdate=2016-10-04|archive-date=2019-07-13|archive-url=https://web.archive.org/web/20190713064739/http://www.jetp.ac.ru/cgi-bin/dn/e_034_03_0610.pdf|dead-url=no}}
* {{citation|first1=J. M.|last1=Kosterlitz|first2=D. J.|last2=Thouless|title=Ordering, metastability and phase transitions in two-dimensional systems|journal=Journal of Physics C: Solid State Physics|volume=6|pages=1181–1203|year=1973|doi=10.1088/0022-3719/6/7/010|bibcode = 1973JPhC....6.1181K }}
* {{citation|first1=O.|last1=McBryan|first2=T.|last2=Spencer|journal=Commun. Math. Phys.|volume=53|page=299|year=1977|bibcode = 1977CMaPh..53..299M |doi = 10.1007/BF01609854 |title=On the decay of correlations in SO(n)-symmetric ferromagnets}}
* B. I. Halperin, D. R. Nelson, Phys. Rev. Lett. 41, 121 (1978)
* A. P. Young, Phys. Rev. B 19, 1855 (1979)
* {{citation | first1=D.J.|last1=Resnick|first2=J.C.| last2=Garland |first3=J.T.| last3=Boyd|first4=S.| last4=Shoemaker|first5=R.S.|last5=Newrock|title=Kosterlitz Thouless Transition in Proximity Coupled Superconducting Arrays|journal=Phys. Rev. Lett.|volume=47|page=1542|year=1981|bibcode = 1981PhRvL..47.1542R |doi = 10.1103/PhysRevLett.47.1542 }}
* {{citation|first1=Jürg|last1=Fröhlich|first2=Thomas|last2=Spencer|title=The Kosterlitz–Thouless transition in two-dimensional abelian spin systems and the Coulomb gas|journal=Comm. Math. Phys.|volume=81|year=1981|issue=4|pages=527–602|doi=10.1007/bf01208273|bibcode = 1981CMaPh..81..527F }}
* {{citation|author= Z. Hadzibabic|title=Berezinskii–Kosterlitz–Thouless crossover in a trapped atomic gas|journal=Nature|volume=41|page=1118|year=2006|doi=10.1038/nature04851|arxiv = cond-mat/0605291 |bibcode = 2006Natur.441.1118H |display-authors=etal}}

== 相關書籍 ==
* J.V. Jose, ''40 Years of Berezinskii–Kosterlitz–Thouless Theory'', World Scientific, 2013, ISBN 978-981-4417-65-5
* [[哈根·克莱纳特|H. Kleinert]], ''Gauge Fields in Condensed Matter'', Vol. I, " SUPERFLOW AND VORTEX LINES", pp.&nbsp;1–742, [https://archive.today/20060514143926/http://www.worldscibooks.com/physics/0356.htm 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]{{Wayback|url=http://www.physik.fu-berlin.de/~kleinert/kleiner_reb1/contents1.html |date=20080527174206 }}. Read pp.&nbsp;618–688);
* [[哈根·克莱纳特|H. Kleinert]], ''Multivalued Fields in Condensed Matter, Electrodynamics, and Gravitation'', [https://web.archive.org/web/20080315225354/http://www.worldscibooks.com/physics/6742.html World Scientific (Singapore, 2008)] (also available online: [http://www.physik.fu-berlin.de/~kleinert/re.html#B9 here] {{Wayback|url=http://www.physik.fu-berlin.de/~kleinert/re.html#B9 |date=20080510224427 }})

{{量子場論}}
{{統計力學}}
[[Category:理論物理]]
[[Category:統計力學]]
[[Category:量子場論]]
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