Casual GF 因果格林函数

多体理论

---

定义 Definition

$$i G(1,2) = \left\langle \mathscr{T} \psi(1) \psi^\dagger(2) \right\rangle = \text{Tr} \left( \rho \mathscr{T} \psi(1) \psi^\dagger(2) \right)$$

Grand canonical ensemble 巨正则系综,

$$\rho = e^{\beta \Omega} e^{- \beta K}$$

$$K = H - \mu N$$

$$K \left| n \right\rangle = E_n K \left| n \right\rangle$$

$$p \left| n \right\rangle = P_n \left| n \right\rangle$$

calculate $iG(xt, x't')$, $x' \to 0$, $t' \to 0$, $x \to x$, $t \to t$,

$$i G(xt,x't') = i G(xt) = \left\langle \mathscr{T} \psi(xt) \psi^\dagger (0) \right\rangle$$

$$iG(xt) = \left\langle \psi(xt) \psi^\dagger(0) \right\rangle \theta(t) \mp \left\langle \psi^\dagger(0) \psi(xt) \right\rangle \theta(-t)$$

1st term,

$$e^{\beta \Omega} \sum\limits_{mn} \left\langle m \right| e^{-\beta K} e^{i Ht} e^{-i px} \psi e^{i px} e^{- i Ht} \left| n \right\rangle \left\langle n \right| \psi^\dagger \left| m \right\rangle \theta(t)$$

$$= e^{\beta \Omega} \sum\limits_{mn} e^{- \beta E_m}e^{iE_m t}e^{-i E_n t} e^{-i P_m x} e^{i P_n x} \left| \left\langle m \right| \psi \left| n \right\rangle \right|^2 \theta(t)$$

$$= e^{\beta \Omega} \sum\limits_{mn} e^{- \beta E_m} e^{i \omega_{mn}t } e^{-i P_{mn}x } \left| \left\langle m \right| \psi \left| n \right\rangle \right|^2 \theta(t)$$

likewise, 2nd term,

$$\mp e^{\beta \Omega} \sum\limits_{mn} \left\langle m \right| e^{-\beta K} \psi^\dagger \left| n \right\rangle \left\langle n \right| e^{i H t} e^{- i px} \psi e^{i px} e^{- i Ht} \left| m \right\rangle \theta(-t)$$

$$= \mp e^{\beta \Omega} \sum\limits_{mn} e^{- \beta E_m} e^{i E_n t} e^{-i E_m t} e^{-i P_n x} e^{i P_m x} \left| \left\langle n \right| \psi \left| m \right\rangle \right|^2 \theta(-t)$$

$$= \mp e^{\beta \Omega} \sum\limits_{mn} e^{- \beta E_m} e^{ - i \omega_{mn} t} e^{i P_{mn}x } \left| \left\langle n \right| \psi \left| m \right\rangle \right|^2 \theta(-t)$$

1st term ($n \leftrightarrow m$) + 2nd term,

$$e^{\beta \Omega} \sum\limits_{mn} \left| \left\langle n \right| \psi \left| m \right\rangle \right|^2 \left\{ e^{- \beta E_n} e^{- i \omega_{mn}t } e^{i P_{mn}x } \theta(t) \mp e^{- \beta E_m} e^{ - i \omega_{mn} t} e^{i P_{mn}x } \theta(-t) \right\}$$

FT,

$$G(p, \omega) = \int d^3 x dt e^{-i px} e^{i \omega t } G(x,t)$$

use,

$$\int d^3 x e^{i P_{mn} x} e^{-ipx} = (2\pi)^3 \delta(p-P_{mn})$$

$$\int_0^\infty dt e^{i \omega t} e^{-i \omega_{mn}t} e^{- \eta t} = - \frac{1}{ i (\omega - \omega_{mn} + i \eta )}$$

$$\int_{-\infty}^0 dt e^{i \omega t } e^{-i \omega_{mn}t} e^{\eta t} = \frac{1}{i (\omega - \omega_{mn} - i \eta)}$$

Then,

$$G(p, \omega) = \frac{(2\pi)^3}{2S+1} e^{\beta \Omega}\sum\limits_{mn} \left| \left\langle n \right| \psi \left| m \right\rangle \right|^2 \delta(p-P_{mn}) \left\{ \frac{e^{- \beta E_n}}{\omega - \omega_{mn} + i \eta } \pm \frac{e^{-\beta E_m}}{\omega - \omega_{mn} - i \eta } \right\}$$

or,

$$G(p, \omega) = \frac{(2\pi)^3}{2S+1} e^{\beta \Omega}\sum\limits_{mn} \left| \left\langle n \right| \psi \left| m \right\rangle \right|^2 e^{- \beta E_n} \delta(p-P_{mn}) \left\{ \frac{ 1 }{\omega - \omega_{mn} + i \eta } \pm \frac{e^{-\beta \omega_{mn}}}{\omega - \omega_{mn} - i \eta } \right\}$$

推迟和超前格林函数 Retarded and Advanced GF

$$iG^R(xt)= \left\langle [ \psi(xt), \psi^\dagger(0) ]_{\pm} \right\rangle \theta(t)$$

$$i G^A(xt) = - \left\langle [ \psi(xt), \psi^\dagger(0) ]_{\pm} \right\rangle \theta(-t)$$

Retarded GF

$$i G^R(xt) = \left\langle \psi(xt)\psi^\dagger(0) \pm \psi^\dagger(0) \psi(xt) \right\rangle \theta(t)$$

$$= e^{\beta \Omega} \sum\limits_{mn} \left| \psi_{nm} \right|^2 e^{-i \omega_{mn} t } e^{i P_{mn} x} \theta(t) \left\{ e^{- \beta E_n} \pm e^{-\beta E_m} \right\}$$

$$= e^{\beta \Omega} \sum\limits_{mn} e^{- \beta E_n} \left| \psi_{nm} \right|^2 e^{-i \omega_{mn} t } e^{i P_{mn} x} \theta(t) \left\{ 1 \pm e^{-\beta \omega_{mn}} \right\}$$

FT,

$$G^R(p, \omega) = \frac{(2\pi)^3}{2S+1} \sum\limits_{mn} \rho_n \left|\psi_{mn} \right|^2 \delta(p-P_{mn}) \left\{ \frac{ 1 \pm e^{- \beta \omega_{mn}} }{ \omega - \omega_{mn} + i \eta } \right\}$$

Likewise, for advanced GF,

$$i G^A(xt) = - \left\langle \psi(xt)\psi^\dagger(0) \pm \psi^\dagger(0) \psi(xt) \right\rangle \theta(-t)$$

$$= e^{\beta \Omega} \sum\limits_{mn} \left| \psi_{nm} \right|^2 e^{-i \omega_{mn} t }e^{i P_{mn} x} \theta(-t ) \left\{ - e^{- \beta E_n} \mp e^{- \beta E_m} \right\}$$

$$e^{\beta \Omega} \sum\limits_{mn} e^{- \beta E_n} \left| \psi_{nm} \right|^2 e^{-i \omega_{mn} t }e^{i P_{mn} x} \theta(-t) \left\{ -1 \mp e^{- \beta \omega_{mn} } \right\}$$

FT,

$$G^A(p, \omega) = \frac{(2\pi)^3}{2S+1} \sum\limits_{mn} \rho_n \left| \psi_{nm} \right|^2 \delta(p-P_{mn}) \left\{ \frac{1 \pm e^{-\beta \omega_{mn}}}{\omega - \omega_{mn} - i \eta } \right\}$$

in continuum limit, $\sum\limits_{mn} ...(P_{mn}, \omega_{mn})... \to \int d \omega' ...\rho(p, \omega')...$,

$$G^{R,A}(p, \omega) = \int_{-\infty}^{\infty} \frac{d \omega'}{2 \pi} \frac{\rho(p, \omega') }{ \omega - \omega' \pm i \eta}$$

where,

$$\rho(p, \omega) = \frac{(2\pi)^4}{2S+1}\sum\limits_{mn} \rho_n \left| \psi_{nm} \right|^2 \delta(p-P_{mn}) \delta(\omega -\omega_{mn}) \left( 1 \pm e^{-\beta \omega_{mn}} \right)$$

and

$$\rho(p, \omega) = -2 \Im G^R(p, \omega)$$

求和规则 The sum rule

$\rho(p, \omega)$ is the spectral density function, fulfills:

$$\int_{-\infty}^\infty \frac{d \omega}{2 \pi} \rho(p, \omega ) = 1$$

From expression of $G(p,\omega)$, $G^R(p,\omega)$ and $G^A(p,\omega)$, we see that:

$$\Re G^{R,A}(p, \omega) = \Re G(p, \omega)$$

$$\Im G^{R,A}(p, \omega) = \pm \Im G(p, \omega) \coth^{\pm} \frac{\beta \omega}{2}$$

$\coth \frac{\omega}{2}$ for fermions, and $\coth^{-1} \frac{\omega}{2}$ for bosons.

Then,

$$G^{R,A}(p, \omega) = \Re G(p, \omega) \pm i \coth^{\pm} \frac{\beta \omega}{2} \times \Im G(p, \omega)$$

“克莱莫斯-克罗尼格”关系 Kramers-Kronig relations

$$\Re G^R(p, \omega)= \frac{1}{\pi} \mathscr{P} \int_{-\infty}^{\infty} d \omega' \frac{ \Im G^R(p, \omega)}{\omega' - \omega}$$

证明 Prove

$$\oint d\omega' \frac{ G^R(p, \omega') }{\omega' - \omega} = 0$$

$\omega \to \infty$, $G^R(p, \omega) \to \frac{1}{\omega}$.

$$\mathscr{P} \int_{- \infty}^\infty d\omega' \frac{G^R(p, \omega')}{\omega'- \omega } - i \pi G^R(p, \omega) = 0$$

$$G^R(p, \omega) = \frac{1}{i \pi} \mathscr{P} \int_{-\infty}^{\infty} d\omega' \frac{G^R(p, \omega')}{\omega'- \omega }$$

i.e.,

$$\Re G^R(p, \omega) = \frac{1}{\pi} \mathscr{P} \int_{-\infty}^{\infty} d\omega' \frac{ \Im G^R(p, \omega') }{\omega' - \omega}$$

$$\Im G^R(p, \omega) = - \frac{1}{\pi} \mathscr{P} \int_{-\infty}^{\infty} d\omega' \frac{\Re G^R(p, \omega')}{\omega' - \omega}$$

Q.E.D.

---

• Wiki: Kramers-Kronig relations

$$\Re G(p, \omega) = \Re G^R (p, \omega) = \frac{1}{\pi} \mathscr{ P }\int_{-\infty}^{\infty} d\omega' \frac{\Im G^R (p, \omega) }{\omega' - \omega}$$

use,

$$\Im G^R (p, \omega) = \Im G(p, \omega) \coth^{\pm} \frac{\beta \omega}{2}$$

then,

$$\Re G(p, \omega) = \frac{1}{\pi} \mathscr{ P }\int_{-\infty}^{\infty} d\omega' \coth^{\pm} \frac{\beta \omega'}{2} \frac{\Im G (p, \omega') }{\omega' - \omega}$$

习题 Problems

The sum rule

$$\int_{-\infty}^{\infty} \frac{d \omega}{2 \pi} \rho( \bf{p},\omega) =1$$

Mahan, p151

Particle density

$$n_{\bf{p}} = \int_{- \infty}^{\infty} \frac{d \omega}{2 \pi} n_F(\omega) \rho(\bf{p}, \omega)$$

Mahan, p154

@季燕江