Normal-ordered operators: $$ \begin{equation} E_0 = \sum_{i} n_i t_{ii} + \frac{1}{2}\sum_{ij}n_in_j V_{ijij} + \frac{1}{6} n_i n_j n_k V_{ijkijk}^{(3)} \label{_auto1} \end{equation} $$ $$ \begin{equation} f_{pq} = t_{pq} + \sum_{i}n_i V_{piqi} + \frac{1}{2} \sum_{ij}n_i n_j V_{pijqij}^{(3)} \label{_auto2} \end{equation} $$ $$ \begin{equation} \Gamma_{pqrs} = V_{pqrs} + \sum_{i}n_i V_{pqirsi}^{(3)} \label{_auto3} \end{equation} $$ $$ \begin{equation} W_{pqrstu} = V_{pqrstu}^{(3)} \label{_auto4} \end{equation} $$
IMSRG flow equations. Note that here we do not follow the CC/MBPT convetion that \( ijk \) are hole indices and \( abc \) are particle indices. Instead, \( abc \) are used for interal (contracted) indices which are summed over, and \( ijk \) are external indices. $$ \begin{equation} \frac{d}{ds}E_0 = \sum_{ab} n_a\bar{n}_b \left(\eta_{ab}f_{ba}-f_{ab}\eta_{ba} \right) +\frac{1}{4} \sum_{abcd} n_a n_b \bar{n}_c\bar{n}_d \left( \eta_{abcd}\Gamma_{cdab} - \Gamma_{abcd}\eta_{cdab} \right) \label{_auto5} \end{equation} $$ $$ \begin{equation} \begin{aligned} \frac{d}{ds} f_{ij} = \sum_{a} \left(\eta_{ia}f_{aj}-f_{ia}\eta_{aj}\right) &+ \sum_{ab}(n_a\bar{n}_b-\bar{n}_a n_b) \left(\eta_{ab}\Gamma_{biaj}-f_{ab}\eta_{biaj} \right) \\ &+ \frac{1}{2} \sum_{abc}(n_an_b\bar{n}_c + \bar{n}_a\bar{n}_bn_c) \left( \eta_{ciab}\Gamma_{abcj} - \Gamma_{ciab}\eta_{abcj} \right) \end{aligned} \label{_auto6} \end{equation} $$ $$ \begin{equation} \begin{aligned} \frac{d}{ds}\Gamma_{ijkl} = &\sum_{a}\left[ (1-P_{ij}\left( \eta_{ia}\Gamma_{ajkl}-f_{ia}\eta_{ajkl} \right) - ( 1-P_{kl}) \left( \eta_{ak}\Gamma_{kjal} - f_{ak}\eta_{ijal} \right) \right] \\ &+ \frac{1}{2} \sum_{ab} (\bar{n}_a\bar{n}_b-n_a n_b)\left( \eta_{ijab}\Gamma_{abkl} - \Gamma_{ijab}\eta_{abkl} \right) \\ &+ (1-P_{ij})(1-P_{kl}) \sum_{ab}(n_a\bar{n}_b-\bar{n}_an_b) \eta_{aibk}\Gamma_{bjal} \end{aligned} \label{_auto7} \end{equation} $$
The White generator: $$ \begin{equation} \eta^{\text{Wh}}_{ai} = \frac{f_{ai}}{\Delta_{ai}} - h.c. \label{_auto8} \end{equation} $$ $$ \begin{equation} \eta^{\text{Wh}}_{abij} = \frac{\Gamma_{abij}}{\Delta_{abij}} - h.c. \label{_auto9} \end{equation} $$
Here we use the Moller-Plesset energy denominators. Other choices are possible, but these will suffice for our purposes. $$ \begin{equation} \Delta_{ai} = f_{aa} - f_{ii} \label{_auto10} \end{equation} $$ $$ \begin{equation} \Delta_{abij} = f_{aa} + f_{bb} - f_{ii} - f_{jj} \label{_auto11} \end{equation} $$
For nuclear matter, momentum conservation restricts one-body operators to be diagonal. This allows for some simplification of flow equations: $$ \begin{equation} \frac{d}{ds}E_0 = \frac{1}{4} \sum_{abcd} n_a n_b \bar{n}_c\bar{n}_d \left( \eta_{abcd}\Gamma_{cdab} - \Gamma_{abcd}\eta_{cdab} \right) \label{_auto12} \end{equation} $$ $$ \begin{equation} \frac{d}{ds} f_{ij} = \frac{1}{2} \sum_{abc}(n_an_b\bar{n}_c + \bar{n}_a\bar{n}_bn_c) \left( \eta_{ciab}\Gamma_{abcj} - \Gamma_{ciab}\eta_{abcj} \right) \label{_auto13} \end{equation} $$ $$ \begin{equation} \begin{aligned} \frac{d}{ds}\Gamma_{ijkl} &= (A_{ii} + A_{jj} - A_{kk} - A_{ll})B_{ijkl} - (B_{ii} + B_{jj} - B_{kk} - B_{ll})A_{ijkl} \\ &+ \frac{1}{2} \sum_{ab} (\bar{n}_a\bar{n}_b-n_a n_b)\left( \eta_{ijab}\Gamma_{abkl} - \Gamma_{ijab}\eta_{abkl} \right) \\ &+ (1-P_{ij})(1-P_{kl}) \sum_{ab}(n_a\bar{n}_b-\bar{n}_an_b) \eta_{aibk}\Gamma_{bjal} \end{aligned} \label{_auto14} \end{equation} $$
Solving the IMSRG flow equation with a simple Euler step method with step size \( ds = 0.1 \). \( E_0 \) is the zero-body piece of the flowing Hamiltonian \( H(s) \). EMBPT2 is the second order MBPT energy using \( H(s) \), and \( dE/ds \) is the zero body part of \( [\eta(s), H(s)] \).
| \( s \) | \( E_0 \) | EMBPT2 | \( dE/ds \) |
|---|---|---|---|
| 0.0 | 1.50000 | -0.0623932 | 0.0000000 |
| 0.1 | 1.48752 | -0.0531358 | -0.1247860 |
| 0.2 | 1.47689 | -0.0453987 | -0.1062720 |
| 0.3 | 1.46781 | -0.0388940 | -0.0907975 |
| 0.4 | 1.46004 | -0.0333983 | -0.0777880 |
| 0.5 | 1.45336 | -0.0287359 | -0.0667967 |
The same numbers as before, but now the \( [\eta_{2b}, H_{2b}]_{2b} \) particle-hole commutator term is omitted.
| \( s \) | \( E_0 \) | EMBPT2 | \( dE/ds \) |
|---|---|---|---|
| 0 | 1.5 | -0.0623932 | 0 |
| 0.1 | 1.48752 | -0.0531358 | -0.124786 |
| 0.2 | 1.47689 | -0.0453551 | -0.106272 |
| 0.3 | 1.46782 | -0.0387878 | -0.0907102 |
| 0.4 | 1.46007 | -0.0332251 | -0.0775756 |
| 0.5 | 1.45342 | -0.0284994 | -0.0664503 |
| 0.6 | 1.44772 | -0.0244746 | -0.0569988 |
| 0.7 | 1.44283 | -0.0210395 | -0.0489492 |