时间反演对称和空间反演对称性
哈密顿量:
\[H(r)=\sum_ke^{ikr}H(k)e^{-ikr}
\]
一，时间反演对称性 \(\hat{T}\):
\([\hat{T},H(r)]=\hat{T}H(r)-H(r)\hat{T}=0\) 得到: \(\hat{T}H(r)\hat{T}^{-1}=H(r)\)
\[\hat{T}\sum_{k}e^{ikr}H(k)e^{-ikr}\hat{T}^{-1} = H(r) \\
=\sum_{k}e^{-ikr}\hat{T}H(k)\hat{T}^{-1}e^{ikr} =\sum_ke^{ikr}H(k)e^{-ikr} \\
\hat{T}H(k)\hat{T}^{-1} = H(-k)
\]
时间反演算符 \(T=UK\).
\[\hat{U}\hat{K}H(k)\hat{K}^{-1}\hat{U}^{-1} = H(-k)
\]
时间反演算符为 \(T=-i\sigma_y K\)，作用到哈密顿量上，
\[H(-k)=TH(k)T^{-1}=
\begin{pmatrix}
0&-I\\
I&0
\end{pmatrix}K
\begin{pmatrix}
h_{\uparrow}(k)&0\\
0&h_{\downarrow}(k)
\end{pmatrix}
K^{-1}
\begin{pmatrix}
0&I\\
-I&0
\end{pmatrix}\\
=\begin{pmatrix}
h_{\downarrow}(k)^{*}&0\\
0&h_{\uparrow}(k)^{*}
\end{pmatrix}
\]
二，空间反演对称性 \(\hat{P}\):
\([\hat{P},H(r)]=\hat{P}H(r)-H(r)\hat{P}=0\) 得到: \(\hat{P}H(r)\hat{P}^{-1}=H(r)\)
\[\hat{P}\sum_{k}e^{ikr}H(k)e^{-ikr}\hat{P}^{-1} = H(r) \\
=\sum_{k}e^{-ikr}\hat{P}H(k)\hat{P}^{-1}e^{ikr} =\sum_ke^{ikr}H(k)e^{-ikr} \\
\hat{P}H(k)\hat{P}^{-1} = H(-k)
\]
三，旋转对称性 \(\hat{R}\):
\([\hat{R},H(r)]=\hat{R}H(r)-H(r)\hat{R}=0\) 得到: \(\hat{R}H(r)\hat{R}^{-1}=H(r)\)
\[\hat{R}\sum_{k}e^{ikr}H(k)e^{-ikr}\hat{R}^{-1} = H(r) \\
=\sum_{k}e^{ik\hat{R}r}\hat{R}H(k)\hat{R}^{-1}e^{-ik\hat{R}r} =\sum_ke^{ikr}H(k)e^{-ikr} \\
\hat{R}H(k)\hat{R}^{-1} = H(\hat{R}k)
\]
