科斯特里茲-托勒斯相變

**別列津斯基 - 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 模型中的一種相變。它是指超過某一臨界溫度時，系統中的渦旋 - 反渦旋束縛態融化成為不成對的渦旋和反渦旋的相變。這種相變是以凝聚態物理學家瓦季姆・別列津斯基、約翰・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導中的這一相變與 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

### 単獨渦旋的能量

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

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

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

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

### 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 （俄語） . 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 [2016-10-04], Bibcode:1971JETP...32..493B, （原始內容存檔 (PDF) 於 2019-07-13）
* Березинский, В. Л., Разрушение дальнего порядка в одномерных и двумерных системах с непрерывной группой симметрии II. Квантовые системы, ЖЭТФ, 1971, **61** (3): 1144–1156 （俄語） . 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 [2016-10-04], Bibcode:1972JETP...34..610B, （原始內容存檔 (PDF) 於 2019-07-13）
* 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（頁面存檔備份，存於網際網路檔案館）. Read pp. 618–688);_
* H. Kleinert, _Multivalued Fields in Condensed Matter, Electrodynamics, and Gravitation_, World Scientific (Singapore, 2008) (also available online: here （頁面存檔備份，存於網際網路檔案館）)