# 别列津斯基-科斯特利茨-索利斯相变

## XY模型

XY模型哈密頓

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

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

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

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

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

## 動力學

### 単独渦旋的能量

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

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

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

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

### 一對渦旋的能量

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

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