相变与临界现象
$\quad\\$
伊辛模型
$\quad\\$
  $N$个自旋处于点阵格点位置，每个自旋只能取$+1$与$-1$两个态，这样的自旋系统称为伊辛模型，为简单起见，仅考虑相邻自旋的相互作用，以$\sum\limits_{\langle ij \rangle}$表示对相邻对求和，系统哈密顿量为
$$H=-J\sum_{\langle ij \rangle}s_is_j-\mu\mathscr{H}\sum_{i=1}^Ns_i$$
  正则系综配分函数
$$Z_N=\sum_{s_1}\sum_{s_2}\cdots\sum_{s_N}e^{-H/kT}=\sum_{\{s_i\}}e^{-H/kT}$$
  如可求得配分函数，则可由自由能求得诸热力学量
$$\quad\\
F=-kT\ln{Z_N}\\
\quad\\
\overline{E}=-T^2\left(\frac{\partial}{\partial T}\left(\frac{F}{T}\right)\right)_{\mathscr{H}}\\
\quad\\
C_{\mathscr{H}}=\left(\frac{\partial \overline{E}}{\partial T}\right)_{\mathscr{H}}=-T\left(\frac{\partial^2 F}{\partial T^2}\right)_{\mathscr{H}}\\
\quad\\
\overline{\mathscr{M}}=\overline{\mu\sum_i{s_i}}=N\mu\overline{s}=-\left(\frac{\partial F}{\partial \mathscr{H}}\right)_{T}$$
$\quad\\$
平均场近似
$\quad\\$
  伊辛模型哈密顿量中有双重指标，较难计算，以下采用平均场近似，设格点配位数为$z$，则
$$H=-\sum_i\mu s_i\left(\mathscr{H}+\frac{J}{\mu}\sum_{\langle ij\rangle}s_j\right)\\
\approx -\sum_i\mu s_i\left(\mathscr{H}+\frac{zJ}{\mu}\overline{s}\right)\\
\quad\\
Z_N=\sum_{\{s_i\}}\exp\left[{\sum_i\mu s_i\left(\mathscr{H}+\frac{zJ}{\mu}\overline{s}\right)/kT}\right]\\
=\sum_{\{s_i\}}\prod_i\exp\left[{\mu s_i\left(\mathscr{H}+\frac{zJ}{\mu}\overline{s}\right)/kT}\right]\\
=\prod_i\sum_{s_i=\pm 1}\exp\left[{\mu s_i\left(\mathscr{H}+\frac{zJ}{\mu}\overline{s}\right)/kT}\right]\\
=\left[2\cosh\left(\frac{\mu\mathscr{H}}{kT}+\frac{zJ}{kT}\overline{s}\right)\right]^N\\
\quad\\
F=-NkT\left[\ln{2}+\ln{\cosh}\left(\frac{\mu\mathscr{H}}{kT}+\frac{zJ}{kT}\overline{s}\right)\right]\\
\quad\\
\overline{\mathscr{M}}=N\mu\overline{s}=N\mu\tanh\left(\frac{\mu\mathscr{H}}{kT}+\frac{zJ}{kT}\overline{s}\right)$$
  由此得到未知量$\overline{s}$的自洽方程
$$\overline{s}=\tanh\left(\frac{\mu\mathscr{H}}{kT}+\frac{zJ}{kT}\overline{s}\right)$$
  令$T_c=\displaystyle\frac{zJ}{k}$，称为临界温度，自洽方程变为$\displaystyle\overline{s}=\tanh\left(\frac{\mu\mathscr{H}}{kT}+\frac{T_c}{T}\overline{s}\right)$
  当外界磁场$\mathscr{H}=0$时，方程化为$\displaystyle\overline{s}=\tanh\left(\frac{T_c}{T}\overline{s}\right)$，$T>T_c$时，方程只有零解，$T<T_c$时，方程对应于自由能极小的解不为零，即产生自发磁化的现象
  观察系统微观哈密顿量，$H_{\{-s_i\}}=H_{\{s_i\}}$，其关于自旋对称，当$T>T_c$时，$\overline{\mathscr{M}}=0$，系统处于顺磁相，宏观状态保持与微观哈密顿量同样的对称性；当$T<T_c$时，$\overline{\mathscr{M}}\neq 0$，系统处于铁磁相，宏观状态发生对称性自发破缺，这是连续相变的普遍特征
  观察序参量$\overline{M}$在$T\to T_c^-$时的变化，此时$\overline{s}\to 0$
$$\displaystyle\tanh\left(\frac{T_c}{T}\overline{s}\right)\approx \frac{T_c}{T}\overline{s}-\frac{1}{3}\left(\frac{T_c}{T}\overline{s}\right)^3\\
\quad\\
\overline{s}\approx \sqrt{3}\left(1-\frac{T}{T_c}\right)^{\frac{1}{2}}\\
\quad\\
\overline{\mathscr{M}}\sim (T_c-T)^{\frac{1}{2}}$$
  序参量$\overline{\mathscr{M}}$以幂律形式趋于零，临界指数为$\displaystyle\frac{1}{2}$
  得到$\overline{s}$与$T$的关系，则可由$F$求得$\overline{E}$及$C_{\mathscr{H}}$的表达式
$$\quad\\
\overline{E}=Nk\tanh\left(\frac{T_c}{T}\overline{s}\right)\left[-T_c\overline{s}+T_cT\frac{\partial \overline{s}}{\partial T}\right]\\
\quad\\
C_{\mathscr{H}}=Nk\left[1-\tanh^2\left(\frac{T_c}{T}\overline{s}\right)\right]\left[\frac{T_c^2}{T^2}\overline{s}^2-2\frac{T_c^2}{T}\overline{s}\frac{\partial \overline{s}}{\partial T}+T_c^2\left(\frac{\partial \overline{s}}{\partial T}\right)^2\right]\\+Nk\tanh\left(\frac{T_c}{T}\overline{s}\right)T_cT\frac{\partial^2\overline{s}}{\partial T^2}
$$
  得到热容在临界点附近的极限为
$$C_{\mathscr{H}}=\left\lbrace\begin{array}{l}0&T\to T_c^+\\\displaystyle\frac{3Nk}{2}&T\to T_c^-\end{array}\right.$$
  在顺磁相$T>T_c$，当外界磁场$\mathscr{H}\neq 0$但很弱时，$\overline{s}\approx 0$，自洽方程变为
$$\quad\\
\overline{s}=\tanh\left(\frac{\mu\mathscr{H}}{kT}+\frac{T_c}{T}\overline{s}\right)\approx \frac{\mu\mathscr{H}}{kT}+\frac{T_c}{T}\overline{s}\\
\quad\\
\overline{s}\approx \frac{\mu}{k}\frac{1}{T-T_c}\mathscr{H}\\
\quad\\
\overline{\mathscr{M}}\sim (T-T_c)^{-1}$$