# Relevant IMSRG equations

Normal-ordered operators:

<!-- Equation labels as ordinary links -->
<div id="_auto1"></div>

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

<!-- Equation labels as ordinary links -->
<div id="_auto2"></div>

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

<!-- Equation labels as ordinary links -->
<div id="_auto3"></div>

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

<!-- Equation labels as ordinary links -->
<div id="_auto4"></div>

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

IMSRG flow equations:

<!-- Equation labels as ordinary links -->
<div id="_auto5"></div>

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

<!-- Equation labels as ordinary links -->
<div id="_auto6"></div>

$$
\begin{equation}
\begin{aligned}
\frac{d}{ds} f_{ij} = \sum_{a} \left(\eta_{ia}f_{aj}-f_{ia}\eta_{aj}\right)
&+ \sum_{ab}(n_a\bar{n}_b-\bar{n}_a n_b) \left(\eta_{ab}\Gamma_{biaj}-f_{ab}\eta_{biaj} \right)
\\
&+ \frac{1}{2} \sum_{abc}(n_an_b\bar{n}_c + \bar{n}_a\bar{n}_bn_c) \left( \eta_{ciab}\Gamma_{abcj} - \Gamma_{ciab}\eta_{abcj} \right)
\end{aligned}
\label{_auto6} \tag{6}
\end{equation}
$$

<!-- Equation labels as ordinary links -->
<div id="_auto7"></div>

$$
\begin{equation}
\begin{aligned}
\frac{d}{ds}\Gamma_{ijkl} = &\sum_{a}\left[ (1-P_{ij}\left( \eta_{ia}\Gamma_{ajkl}-f_{ia}\eta_{ajkl} \right) - ( 1-P_{kl}) \left( \eta_{ak}\Gamma_{kjal} - f_{ak}\eta_{ijal} \right) \right]
\\
&+ \frac{1}{2} \sum_{ab} (\bar{n}_a\bar{n}_b-n_a n_b)\left( \eta_{ijab}\Gamma_{abkl} - \Gamma_{ijab}\eta_{abkl} \right) \\
&+ (1-P_{ij})(1-P_{kl}) \sum_{ab}(n_a\bar{n}_b-\bar{n}_an_b) \eta_{aibk}\Gamma_{bjal}
\end{aligned}
\label{_auto7} \tag{7}
\end{equation}
$$

The White generator:

<!-- Equation labels as ordinary links -->
<div id="_auto8"></div>

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

<!-- Equation labels as ordinary links -->
<div id="_auto9"></div>

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

Energy denominators:

<!-- Equation labels as ordinary links -->
<div id="_auto10"></div>

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

<!-- Equation labels as ordinary links -->
<div id="_auto11"></div>

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

Simplification of flow equation for nuclear matter:

<!-- Equation labels as ordinary links -->
<div id="_auto12"></div>

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

<!-- Equation labels as ordinary links -->
<div id="_auto13"></div>

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

<!-- Equation labels as ordinary links -->
<div id="_auto14"></div>

$$
\begin{equation}
\begin{aligned}
\frac{d}{ds}\Gamma_{ijkl} &= (A_{ii} + A_{jj} - A_{kk} - A_{ll})B_{ijkl} - (B_{ii} + B_{jj} - B_{kk} - B_{ll})A_{ijkl}
\\
&+ \frac{1}{2} \sum_{ab} (\bar{n}_a\bar{n}_b-n_a n_b)\left( \eta_{ijab}\Gamma_{abkl} - \Gamma_{ijab}\eta_{abkl} \right) \\
&+ (1-P_{ij})(1-P_{kl}) \sum_{ab}(n_a\bar{n}_b-\bar{n}_an_b) \eta_{aibk}\Gamma_{bjal}
\end{aligned}
\label{_auto14} \tag{14}
\end{equation}
$$

# 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 $[\eta(s), H(s)]$.

<table border="1">
<thead>
<tr><th align="center">$s$</th> <th align="center"> $E_0$ </th> <th align="center">  EMBPT2  </th> <th align="center"> $dE/ds$  </th> </tr>
</thead>
<tbody>
<tr><td align="center">   0.0    </td> <td align="center">   1.50000    </td> <td align="center">   -0.0623932    </td> <td align="center">   0.0000000     </td> </tr>
<tr><td align="center">   0.1    </td> <td align="center">   1.48752    </td> <td align="center">   -0.0531358    </td> <td align="center">   -0.1247860    </td> </tr>
<tr><td align="center">   0.2    </td> <td align="center">   1.47689    </td> <td align="center">   -0.0453987    </td> <td align="center">   -0.1062720    </td> </tr>
<tr><td align="center">   0.3    </td> <td align="center">   1.46781    </td> <td align="center">   -0.0388940    </td> <td align="center">   -0.0907975    </td> </tr>
<tr><td align="center">   0.4    </td> <td align="center">   1.46004    </td> <td align="center">   -0.0333983    </td> <td align="center">   -0.0777880    </td> </tr>
<tr><td align="center">   0.5    </td> <td align="center">   1.45336    </td> <td align="center">   -0.0287359    </td> <td align="center">   -0.0667967    </td> </tr>
</tbody>
</table>

The same numbers as before, but now the $[\eta_2, H_2]_2$
particle-hole commutator term is omitted.

<table border="1">
<thead>
<tr><th align="center">$s$</th> <th align="center"> $E_0$ </th> <th align="center">  EMBPT2  </th> <th align="center"> $dE/ds$  </th> </tr>
</thead>
<tbody>
<tr><td align="center">   0      </td> <td align="center">   1.5        </td> <td align="center">   -0.0623932    </td> <td align="center">   0             </td> </tr>
<tr><td align="center">   0.1    </td> <td align="center">   1.48752    </td> <td align="center">   -0.0531358    </td> <td align="center">   -0.124786     </td> </tr>
<tr><td align="center">   0.2    </td> <td align="center">   1.47689    </td> <td align="center">   -0.0453551    </td> <td align="center">   -0.106272     </td> </tr>
<tr><td align="center">   0.3    </td> <td align="center">   1.46782    </td> <td align="center">   -0.0387878    </td> <td align="center">   -0.0907102    </td> </tr>
<tr><td align="center">   0.4    </td> <td align="center">   1.46007    </td> <td align="center">   -0.0332251    </td> <td align="center">   -0.0775756    </td> </tr>
<tr><td align="center">   0.5    </td> <td align="center">   1.45342    </td> <td align="center">   -0.0284994    </td> <td align="center">   -0.0664503    </td> </tr>
<tr><td align="center">   0.6    </td> <td align="center">   1.44772    </td> <td align="center">   -0.0244746    </td> <td align="center">   -0.0569988    </td> </tr>
<tr><td align="center">   0.7    </td> <td align="center">   1.44283    </td> <td align="center">   -0.0210395    </td> <td align="center">   -0.0489492    </td> </tr>
</tbody>
</table>

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

<table border="1">
<thead>
<tr><th align="center">$s$</th> <th align="center">  $E0$ </th> <th align="center">  EMBPT2  </th> <th align="center"> $dE/ds$  </th> </tr>
</thead>
<tbody>
<tr><td align="center">   0.0    </td> <td align="center">   1.50000    </td> <td align="center">   -0.0623932    </td> <td align="center">   0             </td> </tr>
<tr><td align="center">   0.1    </td> <td align="center">   1.48752    </td> <td align="center">   -0.0531358    </td> <td align="center">   -0.124786     </td> </tr>
<tr><td align="center">   0.2    </td> <td align="center">   1.47689    </td> <td align="center">   -0.0453551    </td> <td align="center">   -0.106272     </td> </tr>
<tr><td align="center">   0.3    </td> <td align="center">   1.46782    </td> <td align="center">   -0.0387878    </td> <td align="center">   -0.0907102    </td> </tr>
<tr><td align="center">   0.4    </td> <td align="center">   1.46007    </td> <td align="center">   -0.0332251    </td> <td align="center">   -0.0775756    </td> </tr>
<tr><td align="center">   0.5    </td> <td align="center">   1.45342    </td> <td align="center">   -0.0284994    </td> <td align="center">   -0.0664503    </td> </tr>
</tbody>
</table>

d8930af56b34148b1c944d2f03708319b7154a7b
ipynb
Jupyter Notebook
doc/pub/IMSRG/ipynb/IMSRG.ipynb
NuclearTalent/ManyBody2018
2339ed834777fa10f6156344f17494b9a7c0bf91
CC0-1.0
2018-07-17T01:09:17.000Z
2021-04-08T02:34:02.000Z
doc/pub/IMSRG/ipynb/IMSRG.ipynb
NuclearTalent/ManyBody2018
2339ed834777fa10f6156344f17494b9a7c0bf91
CC0-1.0
doc/pub/IMSRG/ipynb/IMSRG.ipynb
NuclearTalent/ManyBody2018
2339ed834777fa10f6156344f17494b9a7c0bf91
CC0-1.0
2018-07-16T06:31:54.000Z
2021-05-01T07:53:38.000Z
Qwen/Qwen-72B
1. YES
2. YES