别列津斯基-科斯特利茨-索利斯相变

别列津斯基-科斯特利茨-索利斯相變（英語：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瓦季姆·別列津斯基、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模型的哈密頓是

$H=-J\sum _{\langle i,j\rangle }\mathbf {S} _{i}\cdot \mathbf {S} _{j}=-J\sum _{\langle i,j\rangle }\cos(\theta _{i}-\theta _{j})$

${\boldsymbol {S}}_{i}=(\cos {\theta _{i}},\sin {\theta _{i}})$

格林函數（傳播子）是

$G({\boldsymbol {r}}_{i}-{\boldsymbol {r}}_{j})=\langle {\boldsymbol {S}}_{i}\cdot {\boldsymbol {S}}_{j}\rangle =\langle \cos(\theta _{i}-\theta _{j})\rangle =\langle e^{i(\theta _{i}-\theta _{j})}\rangle$

$G({\boldsymbol {r}}_{i}-{\boldsymbol {r}}_{j})\sim \exp \left(-r\log {\frac {2}{\beta J}}\right)$

$G({\boldsymbol {r}}_{i}-{\boldsymbol {r}}_{j})\sim \left({\frac {1}{r}}\right)^{1/2\pi \beta J}$

動力學

単独渦旋的能量

$E\sim {\frac {J}{2}}\int {\mathrm {d} }^{2}r(\nabla \theta )^{2}={\frac {J}{2}}\int _{a}^{L}d^{2}r2\pi {\mathrm {d} }r{\frac {1}{r^{2}}}=\pi J\log \left({\frac {L}{a}}\right)$

$S=\log \left({\frac {L}{a}}\right)^{2}$

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

$T_{BKT}={\frac {\pi J}{2}}$

一對渦旋的能量

$E\sim -\pi J\sum _{i\neq j}n_{i}n_{j}\log \left|{\frac {{\boldsymbol {r}}_{i}-{\boldsymbol {r}}_{j}}{a}}\right|+\pi J\left(\sum _{i}n_{i}\right)^{2}\log \left({\frac {L}{a}}\right)$

$E\sim 2\pi J\log \left|{\frac {{\boldsymbol {r}}_{i}-{\boldsymbol {r}}_{j}}{a}}\right|$

參考資料

• Березинский, В. Л., Разрушение дальнего порядка в одномерных и двумерных системах с непрерывной группой симметрии I. Классические системы, ЖЭТФ, 1970, 59 (3): 907–920 （俄语） . Translation available: Berezinskii, V. L., Destruction of long-range order in one-dimensional and two-dimensional systems having a continuous symmetry group I. Classical systems (pdf), Sov. Phys. JETP, 1971, 32 (3): 493–500, Bibcode:1971JETP...32..493B 
• Березинский, В. Л., Разрушение дальнего порядка в одномерных и двумерных системах с непрерывной группой симметрии II. Квантовые системы, ЖЭТФ, 1971, 61 (3): 1144–1156 （俄语） . Translation available: Berezinskii, V. L., Destruction of long-range order in one-dimensional and two-dimensional systems having a continuous symmetry group II. Quantum systems (pdf), Sov. Phys. JETP, 1972, 34 (3): 610–616, Bibcode:1972JETP...34..610B 
• Kosterlitz, J. M.; Thouless, D. J., Ordering, metastability and phase transitions in two-dimensional systems, Journal of Physics C: Solid State Physics, 1973, 6: 1181–1203, Bibcode:1973JPhC....6.1181K, doi:10.1088/0022-3719/6/7/010 
• McBryan, O.; Spencer, T., On the decay of correlations in SO(n)-symmetric ferromagnets, Commun. Math. Phys., 1977, 53: 299, Bibcode:1977CMaPh..53..299M, doi:10.1007/BF01609854 
• B. I. Halperin, D. R. Nelson, Phys. Rev. Lett. 41, 121 (1978)
• A. P. Young, Phys. Rev. B 19, 1855 (1979)
• Resnick, D.J.; Garland, J.C.; Boyd, J.T.; Shoemaker, S.; Newrock, R.S., Kosterlitz Thouless Transition in Proximity Coupled Superconducting Arrays, Phys. Rev. Lett., 1981, 47: 1542, Bibcode:1981PhRvL..47.1542R, doi:10.1103/PhysRevLett.47.1542 
• Fröhlich, Jürg; Spencer, Thomas, The Kosterlitz–Thouless transition in two-dimensional abelian spin systems and the Coulomb gas, Comm. Math. Phys., 1981, 81 (4): 527–602, Bibcode:1981CMaPh..81..527F, doi:10.1007/bf01208273 
• Z. Hadzibabic; et al, Berezinskii–Kosterlitz–Thouless crossover in a trapped atomic gas, Nature, 2006, 41: 1118, Bibcode:2006Natur.441.1118H, arXiv:cond-mat/0605291 , doi:10.1038/nature04851 

相關書籍

• J.V. Jose, 40 Years of Berezinskii–Kosterlitz–Thouless Theory, World Scientific, 2013, ISBN 978-981-4417-65-5
• H. Kleinert, Gauge Fields in Condensed Matter, Vol. I, " SUPERFLOW AND VORTEX LINES", pp. 1–742, World Scientific (Singapore, 1989); Paperback ISBN 9971-5-0210-0 (also available online: Vol. I （页面存档备份，存于互联网档案馆）. Read pp. 618–688);
• H. Kleinert, Multivalued Fields in Condensed Matter, Electrodynamics, and Gravitation, World Scientific (Singapore, 2008) (also available online: here)