# 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

1. $n_v$ is the number of interactions.
2. $n_F$ 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

1. $n_v$ is the number of interactions.
2. $n_F$ 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,

$$$\chi^{-1} = P^{-1} + V,$$$