Interaction
In triqs the interaction Hamiltonian is represented as a sum of monomials of quartic operators
(1)\[H_{int} =
\sum_{ \{\bar{a}\bar{b}cd \}_s} V(\bar{a}\bar{b}cd) \,\, \bar{a} \bar{b} c d\]
where the sum runs over all unique sets of \(\bar{a}\bar{b}cd\) of normal ordered and lexicographically ordered operators.
Note
This is a unique representation of the Hamiltonian and a unique representation of the prefactor \(V(\bar{a}\bar{b}cd)\), in contrast to representations where we allow any permutation of \(\bar{a}\bar{b}\) and \(cd\).
RPA tensor
In RPA we approximate the vertex \(\Gamma\) in the Bethe-Salpeter equation
(2)\[\chi^{(PH)}(\bar{a}b\bar{c}d) =
\chi_0(\bar{a}b\bar{c}d)
+ \chi_0(\bar{a}b\bar{p}q) \,
\Gamma^{(PH)}(q\bar{p}s\bar{r}) \,
\chi^{(PH)}(\bar{r}s\bar{c}d)\]
by a constant rank 4 tensor \(U(\bar{a}b\bar{c}d)\)
(3)\[\Gamma^{(PH)}(q\bar{p}s\bar{r}) \approx U(q\bar{p}s\bar{r})\]
To determine the relation between \(U(\bar{a}b\bar{c}d)\) and \(V(\bar{a}\bar{b}cd)\) we expand \(\chi\) to first order in \(V\)
The generalized susceptibility is defined as
(4)\[\chi(\bar{a}b\bar{c}d) =
\langle \bar{a}b\bar{c}d \rangle
- \langle b \bar{a} \rangle \langle d \bar{c} \rangle\]
to zeroth order in \(V\) we get the bare susceptibility
(5)\[[\chi]_0 = \chi_0 = - \langle d\bar{a} \rangle \langle b \bar{c} \rangle\]
the first order is given by
(6)\[\begin{split}[\chi]_1 =
- \langle \bar{a}b\bar{c}d H_{int} \rangle
+ \langle b \bar{a} H_{int} \rangle \langle d \bar{c} \rangle
+ \langle b \bar{a} \rangle \langle d \bar{c} H_{int} \rangle
\\ =
\sum_{ \{\bar{A}\bar{B}CD \}_s }
V(\bar{A}\bar{B}CD)
\langle \bar{a}\bar{c} CD \rangle \langle bd \bar{A}\bar{B} \rangle\end{split}\]
where we in the last step perform the restricted summation over unique interaction terms, as defined above, and use the fact that all contractions of \(d\bar{c}\) and \(b\bar{a}\) in the first term are canceled by the two last terms.
Performing the Wick contraction of the result and pairing the quadratic expectation values into \(\chi_0\) terms gives
(7)\[\begin{split}[\chi]_1 =
\sum_{ \{ \bar{A}\bar{B}CD \}_s}
V(\bar{A}\bar{B}CD)
\Big[
\chi_0(\bar{a}b \bar{A}C) \chi_0(\bar{B}D\bar{c}d)
- \chi_0(\bar{a}b \bar{A}D) \chi_0(\bar{B}C\bar{c}d) \\
- \chi_0(\bar{a}b \bar{B}C) \chi_0(\bar{A}D\bar{c}d)
+ \chi_0(\bar{a}b \bar{B}D) \chi_0(\bar{A}C\bar{c}d)
\Big]\end{split}\]
by defining the tensor \(U\) as
(8)\[\begin{split}U(\bar{A}C\bar{B}D) = + V(\bar{A}\bar{B}CD)\\
U(\bar{A}D\bar{B}C) = - V(\bar{A}\bar{B}CD)\\
U(\bar{B}C\bar{A}D) = - V(\bar{A}\bar{B}CD)\\
U(\bar{B}D\bar{A}C) = + V(\bar{A}\bar{B}CD)\end{split}\]
we can rewrite the above equation as an unrestricted sum over \(U(\bar{A}B\bar{C}D)\)
(9)\[[\chi]_1 =
\sum_{ \bar{A}B\bar{C}D }
\chi_0(\bar{a}b\bar{A}B)
U(\bar{A}B\bar{C}D)
\chi_0(\bar{C}D\bar{c}d)\]
which determines that the RPA \(U(\bar{A}C\bar{B}D)\) tensor transforms as the prefactor of
(10)\[-V(\bar{A}\bar{B}CD) \bar{A}\bar{B}CD\]
under permutations of the indices.