# Relevant IMSRG equations

Normal-ordered operators:

$$\begin{equation} E_0 = \sum_{i} n_i t_{ii} + \frac{1}{2}\sum_{ij}n_in_j V_{ijij} + \frac{1}{6} n_i n_j n_k V_{ijkijk}^{(3)} \label{_auto1} \tag{1} \end{equation}$$

$$\begin{equation} f_{pq} = t_{pq} + \sum_{i}n_i V_{piqi} + \frac{1}{2} \sum_{ij}n_i n_j V_{pijqij}^{(3)} \label{_auto2} \tag{2} \end{equation}$$

$$\begin{equation} \Gamma_{pqrs} = V_{pqrs} + \sum_{i}n_i V_{pqirsi}^{(3)} \label{_auto3} \tag{3} \end{equation}$$

$$\begin{equation} W_{pqrstu} = V_{pqrstu}^{(3)} \label{_auto4} \tag{4} \end{equation}$$

IMSRG flow equations:

$$\begin{equation} \frac{d}{ds}E_0 = \sum_{ab} n_a\bar{n}_b \left(\eta_{ab}f_{ba}-f_{ab}\eta_{ba} \right) +\frac{1}{4} \sum_{abcd} n_a n_b \bar{n}_c\bar{n}_d \left( \eta_{abcd}\Gamma_{cdab} - \Gamma_{abcd}\eta_{cdab} \right) \label{_auto5} \tag{5} \end{equation}$$

\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} \tag{6} \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} \tag{7} \end{equation}

The White generator:

$$\begin{equation} \eta^{\text{Wh}}_{ai} = \frac{f_{ai}}{\Delta_{ai}} \label{_auto8} \tag{8} \end{equation}$$

$$\begin{equation} \eta^{\text{Wh}}_{abij} = \frac{\Gamma_{abij}}{\Delta_{abij}} \label{_auto9} \tag{9} \end{equation}$$

Energy denominators:

$$\begin{equation} \Delta_{ai} = f_{aa} - f_{ii} \label{_auto10} \tag{10} \end{equation}$$

$$\begin{equation} \Delta_{abij} = f_{aa} + f_{bb} - f_{ii} - f_{jj} \label{_auto11} \tag{11} \end{equation}$$

Simplification of flow equation for nuclear matter:

$$\begin{equation} \frac{d}{ds}E_0 = \frac{1}{4} \sum_{abcd} n_a n_b \bar{n}_c\bar{n}_d \left( \eta_{abcd}\Gamma_{cdab} - \Gamma_{abcd}\eta_{cdab} \right) \label{_auto12} \tag{12} \end{equation}$$

$$\begin{equation} \frac{d}{ds} f_{ij} = \frac{1}{2} \sum_{abc}(n_an_b\bar{n}_c + \bar{n}_a\bar{n}_bn_c) \left( \eta_{ciab}\Gamma_{abcj} - \Gamma_{ciab}\eta_{abcj} \right) \label{_auto13} \tag{13} \end{equation}$$

\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} \tag{14} \end{equation}

# Benchmarking the pairing model

Pairing model with 4 particles, in 4 doubly degenerate levels, for $\delta=1$ and $g=+0.5$

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 $[\eta_2, H_2]_2$ 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

These are the same as before, but with the painful $[\eta_{2b}, H_{2b}]_{2b}$ particle-hole term omitted.

$s$ $E0$ EMBPT2 $dE/ds$
0.0 1.50000 -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