11.15主要内容

路径积分补遗

1. Grassmanian number: (1)\(\eta \neq x+iy\); (2)\(\int d\eta \eta\)积分上下限.
2. 路径积分与拓扑

   

   Path integral and topology
   1. Adiabatic invariant
      \(I=\oint_{S^{1}} p dq, \ dI=0\)
   2. Geometry phase
      \(\gamma=i\oint\left \langle \varphi(\vec{R}) \right | \frac{\partial}{\partial \vec{R}} \left | \varphi(\vec{R}) \right \rangle \cdot d \vec{R}\) —————— Berry phase
      在spin \(s=\frac{1}{2}\)情况下：
      
      \(\left | \psi(\phi) \right \rangle=\bigl( \begin{smallmatrix} u(\theta) \\ v(\theta)e^{-i\phi} \end{smallmatrix} \bigr)\), \(\gamma=i\oint\left \langle \varphi(\vec{R}) \right | \frac{\partial}{\partial \vec{R}} \left | \varphi(\vec{R}) \right \rangle \cdot d \vec{R}=2\pi v^2\ (v=\cos \frac{\theta}{2})\),
      \(\int_{\partial \Sigma} \omega = \int_{\Sigma} d\omega = \frac{1}{2}\int d\Omega\)(差一个常数不重要) ，这与Haldane model等价
   3. Wess-Zumino(WZ) term
      In a shpere, \(\int_{0}^{T} dt \left \langle \vec{n}) | \dot{\vec{n}} \right \rangle \)
      \(d\Omega = \vec{n}\cdot (\vec{n_x} \times \vec{n_y}) \) —————— Wess-Zumino(WZ) term
      \(\frac{1}{4\pi}\oint \vec{n} \cdot (\vec{n_x} \times \vec{n_y}) = \frac{1}{4\pi} \oint d\Omega = \mathbb{Z}\)(整数) —————— skyrmion number
3. 路径积分在统计物理中的应用
   1. Boson：
      
      统计物理方法：\(H=\omega a^{\dagger} a\), \(Z = {\rm Tr}(e^{-\beta H}) = \sum_{n=0}^{\infty} e^{-\beta \omega_n} = \frac{1}{1-e^{-\beta \omega}} = e^{-{\rm ln}(1-e^{-\beta \omega})}\)
      路径积分方法：\(Z=\int_{-\infty}^{+\infty} D \alpha^{*} \alpha e^{-\int_{0}^{\beta} (\alpha^{*} \frac{\partial}{\partial t} \alpha + \omega \alpha^{*} \alpha)dt} = e^{-\sum_n {\rm ln}(i\omega_n+\omega)} \)
   2. Fermion：
      
      统计物理方法：\(H=\omega C^{\dagger} C\), \(Z = {\rm Tr}(e^{-\beta H}) = \sum_{n=0}^{1} e^{-\beta \omega_n} = 1+e^{-\beta \omega} \)
      路径积分方法：\(Z=\int D \overline{\eta} D \eta e^{-\int_{0}^{\beta} (\overline{\eta} \frac{\partial}{\partial t} \eta + \omega \overline{\eta} \eta)dt} = e^{\sum_n {\rm ln}(i\omega_n+\omega)} \)
      可以check两种方法所得解的奇点完全相同

自发对称性破缺(Spontaneous symmetry breaking)

1. Background
   1. from Condensed matter physics
      
      Magnetism：\(\left | \uparrow \right \rangle + \left | \downarrow \right \rangle \)
      Superconductivity：(1) 序参量：\(\left \langle \phi_{\uparrow} \phi_{\downarrow} \right \rangle \neq 0:\ \phi \rightarrow e^{i\theta} \phi, \ \left \langle \phi_{\uparrow} \phi_{\downarrow} \right \rangle \rightarrow e^{i 2 \theta} \left \langle \phi_{\uparrow} \phi_{\downarrow} \right \rangle \) (Cooper pair) break U(1) symmetry
             (2) Meissner effect
   2. from particle physics
      
      mass Generation
2. Summary of different models

   

   Summary of different models
3. \(\phi^4\) theory: \(\mathscr{L}=\frac{1}{2}\partial_{\mu}\phi \partial^{\mu}\phi-\frac{m^2}{2} \phi^2 - \frac{m^2}{2}\phi^2 - \frac{\lambda}{4!} \phi^4\), 而\(\left \langle \phi \right \rangle = \sqrt{-\frac{6m^2}{\lambda}} \neq 0\)，\(\phi^4\)会发生SSB，且破缺后\(\left \langle \phi \right \rangle \neq 0\)
4. L\(\sigma\)M：\(\mathscr{L}=\frac{1}{2}\partial_{\mu}\phi^{\dagger} \partial^{\mu}\phi - \frac{m^2}{2} \phi^{\dagger} \phi - \frac{\lambda}{4!} (\phi^{\dagger} \phi)^2\)

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