Feynman Rules
In general, we follow the convention in the textbook "Quantum Many-particle Systems" by J. Negele and H. Orland, Page 95,
Fourier Transform
\[G(\tau) = \frac{1}{\beta} \sum_n G(i\omega_n) \text{e}^{-i\omega_n \tau}\]
\[G(i\omega_n) = \int_0^\beta G(\tau) \text{e}^{i\omega_n \tau} d\tau\]
where the Matsubara-frequency $\omega_n=2\pi n/\beta$ for boson and $\omega_n = 2\pi (n+1)/\beta$ for fermion.
Action and Partition Sum
The partition sum associates with a generic action,
\[Z = \int \mathcal{D}\bar{\psi}\mathcal{D}\psi \exp\left(-S\right),\]
where the action takes a generic form,
\[S = \bar{\psi}_1\left(\frac{\partial}{\partial \tau} +\epsilon_k \right)\psi_1 + V_{1234}\bar{\psi}_1\bar{\psi}_2\psi_3\psi_4,\]
where $\bar{\psi}$ and $\psi$ are Grassman fields.
In the Matsubara-frequency domain, the action is,
\[S = \bar{\psi}_1\left(-i\omega_n +\epsilon_k \right)\psi_1 + V_{1234}\bar{\psi}_1\bar{\psi}_2\psi_3\psi_4,\]
Bare Propagator
- Imaginary time
\[g(\tau, k) = \left<\mathcal{T} \psi(k, \tau) \bar{\psi}(k, 0) \right>_0= \frac{e^{-\epsilon_k \tau}}{1+e^{-\epsilon_k \beta}}\theta(\tau)+\xi \frac{e^{-\epsilon_k (\beta+\tau)}}{1+e^{-\epsilon_k \beta}}\theta(-\tau),\]
where $\xi$ is $+1$ for boson and $-1$ for fermion.
- Matusbara frequency
\[g(i\omega_n, k) = -\frac{1}{i\omega_n-\epsilon_k},\]
Then the action takes a simple form,
\[S = \bar{\psi}_1g_{12}^{-1}\psi_2 + V_{1234}\bar{\psi}_1\bar{\psi}_2\psi_3\psi_4,\]
Dressed Propagator and Self-energy
The dressed propagator is given by,
\[G(\tau, k) = \left<\mathcal{T} \psi(k, \tau) \bar{\psi}(k, 0) \right>,\]
and we define the self-energy $\Sigma$ as the one-particle irreducible vertex function,
\[G^{-1} = g^{-1} + \Sigma,\]
so that
\[G = g - g\Sigma g + g\Sigma g \Sigma g - ...\]
Perturbative Expansion of the Green's Function
The sign of a Green's function diagram is given by $(-1)^{n_v} \xi^{n_F}$, where
- $n_v$ is the number of interactions.
- $n_F$ is the number of the fermionic loops.
Feynman Rules for the Self-energy
From the Green's function diagrams, one can derive the negative self-energy diagram,
\[\begin{aligned} -\Sigma = & (-1) \xi V_{34} g_{44}+(-1) V_{34} g_{34} \\ +&(-1)^2 \xi V_{34} V_{56} g_{46} g_{64} g_{43}+(-1)^2 V_{34} V_{56} g_{35} g_{54} g_{42}+\cdots \end{aligned}\]
The sign of a negative self-energy $-\Sigma$ diagram is given by $(-1)^{n_v} \xi^{n_F}$, where
- $n_v$ is the number of interactions.
- $n_F$ is the number of the fermionic loops.
Feynman Rules for the 3-point Vertex Function
The self-energy is related to the 3-point vertex function through an equation,
\[-\left(\Sigma_{3, x} -\Sigma^{Hartree}_{3, x}\right) = G_{3,y} \cdot \left(-V_{3, 4}\right) \cdot \Gamma^3_{4,y,x},\]
where the indices $x, y$ could be different from diagrams to diagrams, and $\Gamma_3$ is the inproper three-vertex function. Eliminate the additional sign, one derives,
\[\Sigma_{3, x} -\Sigma^{Hartree}_{3, x} = G_{3,y} \cdot V_{3, 4} \cdot \Gamma^3_{4,y,x},\]
The diagram weights are given by,
\[\begin{aligned} \Gamma^{(3)}= & 1 + (-1) \xi V_{56} g_{46} g_{64} + (-1) V_{56} g_{54} g_{46}\\ +&(-1)^2 \xi^2 V_{56} V_{78} g_{46} g_{64} g_{58} g_{85}+(-1)^2\xi V_{56} V_{78} g_{74} g_{46}+\cdots \end{aligned}\]
The sign of $\Gamma^{(3)}$ diagram is given by $(-1)^{n_v} \xi^{n_F}$.
Feynman Rules for the 4-point Vertex Function
The 4-point vertex function is related to the 3-point vertex function through an equation,
\[\Gamma^{(3)}_{4,y,x} = \xi \cdot G_{4,s} \cdot G_{t, 4} \cdot \Gamma^{(4)}_{s, t, y, x},\]
where the indices $x, y, s, t$ could be different from diagrams to diagrams.
The diagram weights are given by,
\[\begin{aligned} \Gamma^{(4)}= & (-1) V_{56}^{\text{direct}} + (-1)\xi V_{56}^{exchange}\\ +&(-1)^2 \xi V_{56} V_{78} g_{58} g_{85}+(-1)^2 V_{56} V_{78}+\cdots, \end{aligned}\]
where we used the identity $\xi^2 = 1$.
The sign of $\Gamma^{(4)}$ diagram is given by $(-1)^{n_v} \xi^{n_F}$ multiplied with a sign from the permutation of the external legs.
Feynman Rules for the Susceptibility
The susceptibility can be derived from $\Gamma^{(4)}$.
\[\chi_{1,2} \equiv \left<\mathcal{T} n_1 n_2\right>_{\text{connected}} = \xi G_{1,2} G_{2, 1} + \xi G_{1,s} G_{t, 1} \Gamma^{(4)}_{s, t, y, x} G_{2,y} G_{x, 2}\]
We define the polarization $P$ as the one-interaction irreducible (or proper) vertex function,
\[\chi^{-1} = P^{-1} + V,\]