# ISING模型的重正化

### 实空间的重正化

$\displaystyle{ p(\{s_i\})=\frac{1}{Z}\exp(-\frac{E_{\{s_i\}}}{kT}) }$

$\displaystyle{ Z(H,T)=\sum_{\{s_i\}}\exp(-\frac{E_{\{s_i\}}}{kT}) }$

$\displaystyle{ \frac{\partial{\ln{Z(T,H)}}}{\partial{H}}=\frac{1}{Z}\sum_{\{s_i\}}{\frac{\sum_{i=1}^N {s_i}}{kT}}\exp{(-\frac{\sum_{\{s_i\}}E_{\{s_i\}}}{kT})}=\frac{\langle M\rangle}{kT} }$

$\displaystyle{ \chi=kT\frac{\partial {^2\ln{Z(T,H)}}}{\partial{H^2}} }$

$\displaystyle{ c=-T\frac{\partial^2{\ln{Z}}}{\partial{T^2}} }$

### 一维ISING模型的重正化

$\displaystyle{ Z(T)=\sum_{\{s_i\}}{\exp(\frac{1}{kT}\sum_{i=1}^{N}s_is_{i+1})}=\sum_{odd}\sum_{even}{\exp(\frac{1}{kT}\sum_{i=1}^{N}s_is_{i+1})} }$

\displaystyle{ \begin{align}&Z(T)=\sum_{odd}\sum_{even}{\exp(\frac{1}{kT}\sum_{i=1}^{N}s_is_{i+1})}\\ &=\sum_{odd}\sum_{even}{\exp(\frac{1}{kT}((s_1s_2+s_2s_3)+(s_2s_3+s_3s_4)+\cdot\cdot\cdot+(s_{N-2}s_{N-1}+s_{N-1}s_{N}))}\\ &=\sum_{odd}\sum_{even}{\exp(\frac{1}{kT}(s_1s_2+s_2s_3))\cdot\cdot\cdot\exp(\frac{1}{kT}(s_{N-2}s_{N-1}+s_{N-1}s_{N})} \end{align} }

$\displaystyle{ \sum_{s_i=\pm 1}\exp(\frac{1}{kT}s_i(s_{i-1}+s_{i+1}))=2 \cosh (\frac{1}{kT}(s_{i-1}+s_{i+1})) }$

$\displaystyle{ Z'(T)=\sum_{odd}2 \cosh(\frac{1}{kT}(s_1+s_3))\cdot\cdot\cdot 2 \cosh(\frac{1}{kT}(s_{N-1}+s_N))=\sum_{\{s_I\}}2 \cosh(\frac{1}{kT}(s_1+s_3))\cdot\cdot\cdot 2 \cosh(\frac{1}{kT}(s_{N-1}+s_N)) }$

$\displaystyle{ 2 \cosh(\frac{1}{kT}(s_{i-1}+s_{i+1})=\exp(k_0+\frac{1}{kT'}s_{i-1}s_{i+1}) }$

$\displaystyle{ \left\{\begin{array}{ll} 2\cosh 2 \frac{1}{kT}=\exp(k_0+\frac{1}{kT'}) & \mbox {when }s_{i-1}=s_{i+1}, \\ 2=\exp(k_0-\frac{1}{kT'}) & \mbox {when }s_{i-1}\neq s_{i+1}.\end{array}\right. }$

$\displaystyle{ Z(T)=\sum_{\{s_I\}}{\exp(\sum_{I=1}^{N/2}{k_0})}\exp(\frac{1}{kT'}\sum_{I=1}^{N/2}{s_{I}s_{I+1}})=\exp(Nk_0/2)\sum_{\{s_I\}}\exp(\frac{1}{kT'}\sum_{I=1}^{N/2}{s_{I}s_{I+1}})=\exp(Nk_0/2)Z(T') }$

$\displaystyle{ Z(T)\sim Z(T') }$

$\displaystyle{ \frac{1}{kT'}=\frac{1}{2}\log (\cosh (2\frac{1}{kT})) }$

$\displaystyle{ \frac{1}{kT^{(s+1)}}=\frac{1}{2}\log (\cosh (2\frac{1}{kT^{(s)}})) }$

 $\displaystyle{ \frac{1}{kT^{*}}=\frac{1}{2}\log (\cosh (\frac{2}{kT^{*}})) }$


$\displaystyle{ T^*=0 }$

### 二维ISING模型的重正化

$\displaystyle{ Z(K_1,N)=\sum_{white}\sum_{black}\exp(K_1\sum_{\lt ij\gt }s_is_j) }$

$\displaystyle{ Z(K_1,N)=\sum_{white}\cdot\cdot\cdot 2\cosh(K_1(s_{i,1}+s_{i,2}+s_{i,3}+s_{i,4}))\cdot\cdot\cdot }$

$\displaystyle{ \cosh(K_1(s_{i,1}+s_{i,2}+s_{i,3}+s_{i,4}))=\exp(K_0'+\frac{K_1'}{2}(s_{i,1}s_{i,2}+s_{i,1}s_{i,4}+s_{i,2}s_{i,3}+s_{i,3}s_{i,4})+K_2'(s_{i,1}s_{i,3}+s_{i,2}s_{i,4})+K_3'(s_{i,1}s_{i,2}s_{i,3}s_{i,4})) }$

$\displaystyle{ K_1^{(s+1)}=\frac{3}{8}\log(\cosh 4 K_1^{(s)}) }$

### Wilson的重正化群理论

Wilson重正化方法的核心观点是，首先，针对广义的Ising模型，我们用一个参数向量$\displaystyle{ (K_0,K_1,K_2,\cdot\cdot\cdot) }$来刻画，这是最广义的Ising模型，它包含了近邻、次近邻、三体、四体等等各种相互作用。其次，Wilson讨论一般的重正化算符R，把它反复应用在Ising模型上，就得到了广义参数空间$\displaystyle{ (K_0,K_1,K_2,\cdot\cdot\cdot) }$中的一个重正化流。而这个重正化流的最终吸引子就是临界点（对应临界参数K*）。最后，将重正化算子R在临界点K*附近泰勒展开，就可以把R进行线性化，而线性化算子的特征根的对数就对应了相关的临界指数，而特征向量就是重正化操作中重要的参数。

#### 广义的ISING模型

$\displaystyle{ E_{\{s_i\}}=-kT (K_0+K_1 \sum_{neighbors}s_i s_j+K_2 \sum_{nnn}s_i s_j + \cdot\cdot\cdot+ K_3 \sum_{three}s_is_js_k+\cdot\cdot\cdot ) }$

$\displaystyle{ K=\{K_0,K_1,K_2,\cdot\cdot\cdot\} }$

#### 重正化群算符

$\displaystyle{ K'=R_{b_1}(K) }$

$\displaystyle{ K''=R_{b_2}(K')=R_{b_2b_1}(K), }$ $\displaystyle{ K'''=R_{b_3}(K'')=R_{b_3b_2}(K')=R_{b_3b_2b_1}(K) }$

$\displaystyle{ R_{b_2 b_3}=R_{b_3}R_{b_2} }$

#### 重正化流

$\displaystyle{ K\rightarrow K'\rightarrow K''\rightarrow K'''\cdot\cdot\cdot }$

$\displaystyle{ K^*=R(K^*) }$

#### 临界指数

$\displaystyle{ R(K)=R(K^*)+\frac{\partial{R}}{\partial{K}}\delta K+O(\delta K^2) }$

$\displaystyle{ R(K)=K'=K^*+\delta K' }$

$\displaystyle{ R(K^*)=K^* }$

$\displaystyle{ \delta K'=L_R\delta K+O(\delta K^2) }$

$\displaystyle{ \delta K'=L_R\delta K }$

$\displaystyle{ (L_R)_{ij}=\frac{\partial_i K_i'}{\partial_j K_j} }$

$\displaystyle{ \begin{bmatrix} \lambda_1 & & & \\ & \lambda_2 & & \\ & & \cdot & \\ & & & \lambda_l\\ \end{bmatrix} }$

$\displaystyle{ \begin{bmatrix} \Delta K_1'\\\Delta K_2'\\\cdot\cdot\cdot\\\Delta K_l' \end{bmatrix}_{\mathbf{v}} = \begin{bmatrix} \lambda_1 & & & \\ & \lambda_2 & & \\ & & \cdot & \\ & & & \lambda_l\\ \end{bmatrix}_{\mathbf{v}} \begin{bmatrix} \Delta K_1\\\Delta K_2\\\cdot\cdot\cdot\\\Delta K_l \end{bmatrix}_{\mathbf{v}} }$

$\displaystyle{ \Delta K_i'=\lambda_i\Delta K_i }$

$\displaystyle{ \lambda_i(b_1)\lambda_i(b_2)=\lambda_i(b_1b_2) }$

$\displaystyle{ \lambda_i(b)=b^{y_i} }$

$\displaystyle{ \Delta K_i'=b^{y_i}\Delta K_i }$

• $\displaystyle{ b^{y_i}\gt 1 }$，则最终得到的$\displaystyle{ \Delta K_i' }$会大于0，这样的参数$\displaystyle{ K_i }$我们叫做相关的(relevant)；
• $\displaystyle{ b^{y_i}\lt 1 }$，则经过s次乘方后$\displaystyle{ \Delta K_i' }$会趋近于0，这样的参数$\displaystyle{ K_i }$叫做不相关的(Irrelevant)；
• $\displaystyle{ b^{y_i}=1 }$，则经过s次乘方后$\displaystyle{ \Delta K_i' }$会=1，这样的参数$\displaystyle{ K_i }$叫做边界的（Marginal）。

$\displaystyle{ T'-T_C=b^{y_t} (T-T_C), H'-H_C=b^{y_h} (H-H_C) }$

$\displaystyle{ \log Z(T',H')=b^{-d}\log Z(b^{y_t}(T-T_C),b^{y_h}(H-H_C)) }$

 $\displaystyle{ \alpha=2-\frac{d}{y_t}, \beta=\frac{d-y_h}{y_t}, \gamma=\frac{2y_h-d}{y_t}, \eta=d+2-2y_h, \nu=\frac{1}{y_t} }$


### 波数空间的重正化

#### Ginzburg Landau方程

$\displaystyle{ Z=\int D\phi \exp(-\frac{1}{kT}\int dx L(\phi(x))) }$

#### 傅立叶变换

$\displaystyle{ \phi(k)=\int_{-\infty}^{+\infty}\phi(x)\exp(-ixk)dk }$

$\displaystyle{ Z=\int D\phi \exp(-\frac{1}{kT}\int_0^{+\infty} dk L(\phi(k))) }$

$\displaystyle{ L(\phi(k)) }$为相应的傅立叶变换结果。现在积分区间上限到无穷大，我们将对k的积分进行截断到$\displaystyle{ \Lambda }$

$\displaystyle{ \int_0^{\Lambda} dk L(\phi(k))\doteq S }$

#### 波数空间的缩放

$\displaystyle{ S=\int_0^{+\Lambda} dk L(\phi(k))=\int_0^{\Lambda/b} dk L(\phi(k))+\int_{\Lambda/b}^{\Lambda} dk L(\phi(k)) }$

#### 粗粒化

$\displaystyle{ S=\int_0^{\Lambda/b} dk L(\phi(k))+\int_{\Lambda/b}^{\Lambda} L(\phi(k)) dk=\int_0^{\Lambda/b} L(\phi(k))dk+g(L(\phi))=\int_0^{\Lambda/b} L'(\phi)dk }$

#### 重正化方程

$\displaystyle{ K'=R(K) }$

## 参考文献

Kopietz, P.; Bartosch, L.; Schutz, F. (2010). Introduction to the Functional Renormalization Group. Springer. ISBN 978-3-642-05093-0.