diff options
Diffstat (limited to 'content/know/concept/self-energy')
| -rw-r--r-- | content/know/concept/self-energy/dyson.png | bin | 14904 -> 15660 bytes | |||
| -rw-r--r-- | content/know/concept/self-energy/fullgf.png | bin | 16781 -> 16012 bytes | |||
| -rw-r--r-- | content/know/concept/self-energy/index.pdc | 14 | ||||
| -rw-r--r-- | content/know/concept/self-energy/selfenergy.png | bin | 27003 -> 26221 bytes |
4 files changed, 7 insertions, 7 deletions
diff --git a/content/know/concept/self-energy/dyson.png b/content/know/concept/self-energy/dyson.png Binary files differindex f576632..168505c 100644 --- a/content/know/concept/self-energy/dyson.png +++ b/content/know/concept/self-energy/dyson.png diff --git a/content/know/concept/self-energy/fullgf.png b/content/know/concept/self-energy/fullgf.png Binary files differindex 5767dba..0ea6958 100644 --- a/content/know/concept/self-energy/fullgf.png +++ b/content/know/concept/self-energy/fullgf.png diff --git a/content/know/concept/self-energy/index.pdc b/content/know/concept/self-energy/index.pdc index 7e67143..935cca8 100644 --- a/content/know/concept/self-energy/index.pdc +++ b/content/know/concept/self-energy/index.pdc @@ -27,7 +27,7 @@ and $\beta = 1 / (k_B T)$: $$\begin{aligned} G_{s_b s_a}(\vb{r}_b, \tau_b; \vb{r}_a, \tau_a) - = - \frac{\expval{\mathcal{T}\Big\{ \hat{K}(\hbar \beta, 0) \hat{\Psi}_{s_b}(\vb{r}_b, \tau_b) \hat{\Psi}_{s_a}^\dagger(\vb{r}_a, \tau_a) \Big\}}} + = \frac{\expval{\mathcal{T}\Big\{ \hat{K}(\hbar \beta, 0) \hat{\Psi}_{s_b}(\vb{r}_b, \tau_b) \hat{\Psi}_{s_a}^\dagger(\vb{r}_a, \tau_a) \Big\}}} {\hbar \expval{\hat{K}(\hbar \beta, 0)}} \end{aligned}$$ @@ -50,7 +50,7 @@ and $\hat{\Psi}_a \equiv \hat{\Psi}_{s_a}(\vb{r}_a, \tau_a)$: $$\begin{aligned} G_{ba} - &= - \frac{\displaystyle\sum_{n = 0}^\infty \frac{1}{n!} \Big( \!-\!\frac{1}{\hbar} \Big)^n \idotsint_0^{\hbar \beta} + &= \frac{\displaystyle\sum_{n = 0}^\infty \frac{1}{n!} \Big( \!-\!\frac{1}{\hbar} \Big)^n \idotsint_0^{\hbar \beta} \expval{\mathcal{T}\Big\{ \hat{W}(\tau_1) \cdots \hat{W}(\tau_n) \hat{\Psi}_b \hat{\Psi}_a^\dagger \Big\}} \dd{\tau_1} \cdots \dd{\tau_n}} {\hbar \displaystyle\sum_{n = 0}^\infty \frac{1}{n!} \Big( \!-\!\frac{1}{\hbar} \Big)^n \idotsint_0^{\hbar \beta} \expval{\mathcal{T}\Big\{ \hat{W}(\tau_1) \cdots \hat{W}(\tau_n) \Big\}} \dd{\tau_1} \cdots \dd{\tau_n}} @@ -84,7 +84,7 @@ The full $G_{ba}$ thus becomes: $$\begin{aligned} G_{ba} - &= - \frac{\displaystyle\sum_{n = 0}^\infty \frac{1}{n!} \Big( \!-\! \frac{1}{2 \hbar} \Big)^n (-\hbar)^{2n+1} + &= \frac{\displaystyle\sum_{n = 0}^\infty \frac{1}{n!} \Big( \!-\! \frac{1}{2 \hbar} \Big)^n (-\hbar)^{2n+1} \idotsint W_{1'1} \cdots W_{n'n} \: \Big( G^0_\mathrm{num} \Big) \dd{1'} \dd{1} \cdots \dd{n'} \dd{n}} {\hbar \displaystyle\sum_{n = 0}^\infty \frac{1}{n!} \Big( \!-\! \frac{1}{2 \hbar} \Big)^n (-\hbar)^{2n} \idotsint W_{1'1} \cdots W_{n'n} \: \Big( G^0_\mathrm{den} \Big) \dd{1'} \dd{1} \cdots \dd{n'} \dd{n}} @@ -131,7 +131,7 @@ times $(-1)^p$ to account for swaps of fermionic operators: $$\begin{aligned} G_{ba} - &= \frac{\displaystyle\sum_{n = 0}^\infty \frac{1}{n!} \Big( \!-\! \frac{\hbar}{2} \Big)^n + &= -\frac{\displaystyle\sum_{n = 0}^\infty \frac{1}{n!} \Big( \!-\! \frac{\hbar}{2} \Big)^n \idotsint W_{1'1} \cdots W_{n'n} \: \Big( \sum_{p} (-1)^p \prod_{m = 1}^{2 n + 1} G^0_{(p,m)} \Big) \dd{1}' \dd{1} \cdots \dd{n'} \dd{n}} {\displaystyle\sum_{n = 0}^\infty \frac{1}{n!} \Big( \!-\! \frac{\hbar}{2} \Big)^n \idotsint W_{1'1} \cdots W_{n'n} \: \Big( \sum_{p} (-1)^p \prod_{m = 1}^{2 n} G^0_{(p,m)} \Big) \dd{1'} \dd{1} \cdots \dd{n'} \dd{n}} @@ -172,7 +172,7 @@ $$\begin{aligned} &= \frac{\displaystyle\sum_{n = 0}^\infty \frac{1}{2^n n!} \bigg[ \sum_{m = 0}^{n} \frac{n!}{m! (n \!-\! m)!} \binom{1 \; \mathrm{external}}{\mathrm{order} \; m}_{\!\Sigma\mathrm{all}} \binom{\mathrm{0\;or\;more\;internal}}{\mathrm{total\;order} \; (n \!-\! m)}_{\!\Sigma\mathrm{all}} \bigg]} - {-\hbar \displaystyle\sum_{n = 0}^\infty \frac{1}{2^n n!} \binom{\mathrm{0\;or\;more\;internal}}{\mathrm{total\;order} \; n}_{\!\Sigma\mathrm{all}}} + {\hbar \displaystyle\sum_{n = 0}^\infty \frac{1}{2^n n!} \binom{\mathrm{0\;or\;more\;internal}}{\mathrm{total\;order} \; n}_{\!\Sigma\mathrm{all}}} \end{aligned}$$ Where the total order is the sum of the orders of all considered diagrams, @@ -186,7 +186,7 @@ $$\begin{aligned} &= \frac{\displaystyle\sum_{m = 0}^{\infty} \frac{1}{2^m m!} \binom{1 \; \mathrm{external}}{\mathrm{order} \; m}_{\!\Sigma\mathrm{all}} \bigg[ \sum_{n = 0}^\infty \frac{1}{2^{n-m} (n \!-\! m)!} \binom{\mathrm{0\;or\;more\;internal}}{\mathrm{total\;order} \; (n \!-\! m)}_{\!\Sigma\mathrm{all}} \bigg]} - {-\hbar \displaystyle\sum_{n = 0}^\infty \frac{1}{2^n n!} \binom{\mathrm{0\;or\;more\;internal}}{\mathrm{total\;order} \; n}_{\!\Sigma\mathrm{all}}} + {\hbar \displaystyle\sum_{n = 0}^\infty \frac{1}{2^n n!} \binom{\mathrm{0\;or\;more\;internal}}{\mathrm{total\;order} \; n}_{\!\Sigma\mathrm{all}}} \end{aligned}$$ Since both $n$ and $m$ start at zero, @@ -194,7 +194,7 @@ and the sums include all possible diagrams, we see that the second sum in the numerator does not actually depend on $m$: $$\begin{aligned} - -\hbar G_{ba} + \hbar G_{ba} &= \frac{\displaystyle\sum_{m = 0}^{\infty} \frac{1}{2^m m!} \binom{1 \; \mathrm{external}}{\mathrm{order} \; m}_{\!\Sigma\mathrm{all}} \bigg[ \sum_{n = 0}^\infty \frac{1}{2^n n!} \binom{\mathrm{0\;or\;more\;internal}}{\mathrm{total\;order} \; n}_{\!\Sigma\mathrm{all}} \bigg]} {\displaystyle\sum_{n = 0}^\infty \frac{1}{2^n n!} \binom{\mathrm{0\;or\;more\;internal}}{\mathrm{total\;order} \; n}_{\!\Sigma\mathrm{all}}} diff --git a/content/know/concept/self-energy/selfenergy.png b/content/know/concept/self-energy/selfenergy.png Binary files differindex 55e182e..84d3651 100644 --- a/content/know/concept/self-energy/selfenergy.png +++ b/content/know/concept/self-energy/selfenergy.png |
