Spin degrees of freedom in the presence of SU(2) symmetry
This section concerns electronic systems with SU(2) spin symmetry. For systems without SU(2) symmetry (e.g., in the presence of spin-orbit coupling or magnetic fields), the spin structure of two-particle quantities is more complicated and not covered here.
SU(2) symmetry entails spin conservation for every correlation function, meaning the total spin of the incoming electrons must equal the total spin of the outgoing electrons. Furthermore, all quantities must be invariant under a global spin flip.
Two-point functions are, as a consequence of spin conservation, diagonal in their spin arguments, G σ 2 σ 1 ∼ δ σ 2 , σ 1 G_{\sigma_2 \sigma_1} \sim \delta_{\sigma_2, \sigma_1} G σ 2 σ 1 ∼ δ σ 2 , σ 1 σ i ∈ { ↑ , ↓ } \sigma_i \in \{\uparrow, \downarrow\} σ i ∈ { ↑ , ↓ }
For four-point functions, in particular vertices, spin conservation implies F σ 1 σ 2 σ 3 σ 4 ∼ δ σ 1 + σ 3 , σ 2 + σ 4 F_{\sigma_1\sigma_2\sigma_3\sigma_4} \sim \delta_{\sigma_1+\sigma_3, \sigma_2+\sigma_4} F σ 1 σ 2 σ 3 σ 4 ∼ δ σ 1 + σ 3 , σ 2 + σ 4
F σ σ ‾ σ ‾ σ ≡ F ↑ ↓ F σ σ σ ‾ σ ‾ ≡ F ↑ ↓ ‾ F σ σ σ σ ≡ F ↑ ↑ , \begin{align}
F_{\sigma \overline{\sigma} \overline{\sigma} \sigma} &\equiv F_{\uparrow \downarrow} \\
F_{\sigma \sigma \overline{\sigma} \overline{\sigma}} &\equiv F_{\overline{\uparrow\downarrow}} \\
F_{\sigma\sigma\sigma\sigma} &\equiv F_{\uparrow \uparrow}\, ,
\end{align} F σ σ σ σ F σσ σ σ F σσσσ ≡ F ↑↓ ≡ F ↑↓ ≡ F ↑↑ , where σ ‾ \overline{\sigma} σ σ \sigma σ
In some papers (especially from the Vienna community), the definitions of F ↑ ↓ F_{\uparrow \downarrow} F ↑↓ F ↑ ↓ ‾ F_{\overline{\uparrow\downarrow}} F ↑↓ p h ↔ p h ‾ ph \leftrightarrow \overline{ph} p h ↔ p h note
The three spin components above are furthermore related by the additionally identity
F ↑ ↑ = F ↑ ↓ + F ↑ ↓ ‾ , \begin{align}
F_{\uparrow \uparrow} = F_{\uparrow \downarrow} + F_{\overline{\uparrow\downarrow}}\, ,
\end{align} F ↑↑ = F ↑↓ + F ↑↓ , reducing the number of independent spin components to two.
The number of independent spin components can be further reduced to just one by using crossing symmetry, relating F ↑ ↓ F_{\uparrow \downarrow} F ↑↓ F ↑ ↓ ‾ F_{\overline{\uparrow\downarrow}} F ↑↓
Depending on the context, it is often useful to work with linear combinations of the spin components defined above. One common choice is the magnetic/density basis, defined as
F m ≡ F ↑ ↑ − F ↑ ↓ F d ≡ F ↑ ↑ + F ↑ ↓ . \begin{align}
F_m &\equiv F_{\uparrow \uparrow} - F_{\uparrow \downarrow} \\
F_d &\equiv F_{\uparrow \uparrow} + F_{\uparrow \downarrow}\, .
\end{align} F m F d ≡ F ↑↑ − F ↑↓ ≡ F ↑↑ + F ↑↓ . Another common choice is the singlet/triplet basis, defined as
F s ≡ F ↑ ↓ − F ↑ ↓ ‾ = 2 F ↑ ↓ − F ↑ ↑ F t ≡ F ↑ ↓ + F ↑ ↓ ‾ = F ↑ ↑ . \begin{align}
F_s &\equiv F_{\uparrow \downarrow} - F_{\overline{\uparrow \downarrow}} = 2F_{\uparrow \downarrow} - F_{\uparrow \uparrow} \\
F_t &\equiv F_{\uparrow \downarrow} + F_{\overline{\uparrow \downarrow}} = F_{\uparrow \uparrow}\, .
\end{align} F s F t ≡ F ↑↓ − F ↑↓ = 2 F ↑↓ − F ↑↑ ≡ F ↑↓ + F ↑↓ = F ↑↑ . The main reason for introducing these linear combinations is that connections of two four-point vertices via bubbles in the p h ph p h p p pp pp
[ A ∘ χ 0 p h ∘ B ] m / d = A m / d ∙ χ 0 p h ∙ B m / d [ A ∘ χ 0 p p ∘ B ] s / t = A s / t ∙ χ 0 p p ∙ B s / t , \begin{align}
[A \circ \chi_0^{ph} \circ B]_{m/d} &= A_{m/d} \bullet \chi_0^{ph} \bullet B_{m/d} \\
[A \circ \chi_0^{pp} \circ B]_{s/t} &= A_{s/t} \bullet \chi_0^{pp} \bullet B_{s/t}\, ,
\end{align} [ A ∘ χ 0 p h ∘ B ] m / d [ A ∘ χ 0 pp ∘ B ] s / t = A m / d ∙ χ 0 p h ∙ B m / d = A s / t ∙ χ 0 pp ∙ B s / t , where A A A B B B χ 0 p h / p p \chi_0^{ph/pp} χ 0 p h / pp two-particle channels ∙ \bullet ∙
In the p h ph p h [ χ 0 ] σ 4 σ 3 σ 2 σ 1 p h ∼ δ σ 2 , σ 1 δ σ 4 , σ 3 [\chi_0]^{ph}_{\sigma_4 \sigma_3 \sigma_2 \sigma_1} \sim \delta_{\sigma_2, \sigma_1} \delta_{\sigma_4, \sigma_3} [ χ 0 ] σ 4 σ 3 σ 2 σ 1 p h ∼ δ σ 2 , σ 1 δ σ 4 , σ 3 G σ 2 σ 1 ∼ δ σ 2 , σ 1 G_{\sigma_2 \sigma_1} \sim \delta_{\sigma_2, \sigma_1} G σ 2 σ 1 ∼ δ σ 2 , σ 1
[ A ∘ χ 0 p h ∘ B ] σ 1 σ 2 σ 3 σ 4 = ∑ σ 5 , σ 6 σ 7 , σ 8 A σ 5 σ 2 σ 3 σ 6 ∙ [ χ 0 ] σ 8 σ 5 σ 6 σ 7 p h ∙ B σ 1 σ 8 σ 7 σ 4 = ∑ σ 5 , σ 6 A σ 5 σ 2 σ 3 σ 6 ∙ χ 0 p h ∙ B σ 1 σ 5 σ 6 σ 4 . \begin{align}
[A \circ \chi_0^{ph} \circ B]_{\sigma_1 \sigma_2 \sigma_3 \sigma_4} &= \sum_{\substack{\sigma_5, \sigma_6 \\ \sigma_7, \sigma_8}} A_{\sigma_5 \sigma_2 \sigma_3 \sigma_6} \bullet [\chi_0]^{ph}_{\sigma_8 \sigma_5 \sigma_6 \sigma_7} \bullet B_{\sigma_1 \sigma_8 \sigma_7 \sigma_4} \\
&= \sum_{\sigma_5, \sigma_6} A_{\sigma_5 \sigma_2 \sigma_3 \sigma_6} \bullet \chi_0^{ph} \bullet B_{\sigma_1 \sigma_5 \sigma_6 \sigma_4}\, .
\end{align} [ A ∘ χ 0 p h ∘ B ] σ 1 σ 2 σ 3 σ 4 = σ 5 , σ 6 σ 7 , σ 8 ∑ A σ 5 σ 2 σ 3 σ 6 ∙ [ χ 0 ] σ 8 σ 5 σ 6 σ 7 p h ∙ B σ 1 σ 8 σ 7 σ 4 = σ 5 , σ 6 ∑ A σ 5 σ 2 σ 3 σ 6 ∙ χ 0 p h ∙ B σ 1 σ 5 σ 6 σ 4 . For the ↑ ↑ \uparrow \uparrow ↑↑ ↑ ↓ \uparrow\downarrow ↑↓
[ A ∘ χ 0 p h ∘ B ] ↑ ↑ = [ A ∘ χ 0 p h ∘ B ] σ σ σ σ = ∑ σ 5 , σ 6 A σ 5 σ σ σ 6 ∙ χ 0 p h ∙ B σ σ 5 σ 6 σ = A σ σ σ σ ∙ χ 0 p h ∙ B σ σ σ σ + A σ ‾ σ σ σ ‾ ∙ χ 0 p h ∙ B σ σ ‾ σ ‾ σ = A ↑ ↑ ∙ χ 0 p h ∙ B ↑ ↑ + A ↑ ↓ ∙ χ 0 p h ∙ B ↑ ↓ [ A ∘ χ 0 p h ∘ B ] ↑ ↓ = [ A ∘ χ 0 p h ∘ B ] σ σ ‾ σ ‾ σ = ∑ σ 5 , σ 6 A σ 5 σ ‾ σ ‾ σ 6 ∙ χ 0 p h ∙ B σ σ 5 σ 6 σ = A σ σ ‾ σ ‾ σ ∙ χ 0 p h ∙ B σ σ σ σ + A σ ‾ σ ‾ σ ‾ σ ‾ ∙ χ 0 p h ∙ B σ σ ‾ σ ‾ σ = A ↑ ↓ ∙ χ 0 p h ∙ B ↑ ↑ + A ↑ ↑ ∙ χ 0 p h ∙ B ↑ ↓ . \begin{align}
[A \circ \chi_0^{ph} \circ B]_{\uparrow \uparrow} &= [A \circ \chi_0^{ph} \circ B]_{\sigma\sigma\sigma\sigma} = \sum_{\sigma_5, \sigma_6} A_{\sigma_5 \sigma \sigma \sigma_6} \bullet \chi_0^{ph} \bullet B_{\sigma \sigma_5 \sigma_6 \sigma} \\
&= A_{\sigma\sigma\sigma\sigma} \bullet \chi_0^{ph} \bullet B_{\sigma\sigma\sigma\sigma} + A_{\overline{\sigma} \sigma \sigma \overline{\sigma}} \bullet \chi_0^{ph} \bullet B_{\sigma \overline{\sigma} \overline{\sigma} \sigma} \\
&= A_{\uparrow \uparrow} \bullet \chi_0^{ph} \bullet B_{\uparrow \uparrow} + A_{\uparrow \downarrow} \bullet \chi_0^{ph} \bullet B_{\uparrow \downarrow} \\ & \\
[A \circ \chi_0^{ph} \circ B]_{\uparrow \downarrow} &= [A \circ \chi_0^{ph} \circ B]_{\sigma \overline{\sigma} \overline{\sigma} \sigma} = \sum_{\sigma_5, \sigma_6} A_{\sigma_5 \overline{\sigma} \overline{\sigma} \sigma_6} \bullet \chi_0^{ph} \bullet B_{\sigma \sigma_5 \sigma_6 \sigma} \\
&= A_{\sigma \overline{\sigma} \overline{\sigma} \sigma} \bullet \chi_0^{ph} \bullet B_{\sigma \sigma \sigma \sigma} + A_{\overline{\sigma} \overline{\sigma} \overline{\sigma} \overline{\sigma}} \bullet \chi_0^{ph} \bullet B_{\sigma \overline{\sigma} \overline{\sigma} \sigma} \\
&= A_{\uparrow \downarrow} \bullet \chi_0^{ph} \bullet B_{\uparrow \uparrow} + A_{\uparrow \uparrow} \bullet \chi_0^{ph} \bullet B_{\uparrow \downarrow}\, .
\end{align} [ A ∘ χ 0 p h ∘ B ] ↑↑ [ A ∘ χ 0 p h ∘ B ] ↑↓ = [ A ∘ χ 0 p h ∘ B ] σσσσ = σ 5 , σ 6 ∑ A σ 5 σσ σ 6 ∙ χ 0 p h ∙ B σ σ 5 σ 6 σ = A σσσσ ∙ χ 0 p h ∙ B σσσσ + A σ σσ σ ∙ χ 0 p h ∙ B σ σ σ σ = A ↑↑ ∙ χ 0 p h ∙ B ↑↑ + A ↑↓ ∙ χ 0 p h ∙ B ↑↓ = [ A ∘ χ 0 p h ∘ B ] σ σ σ σ = σ 5 , σ 6 ∑ A σ 5 σ σ σ 6 ∙ χ 0 p h ∙ B σ σ 5 σ 6 σ = A σ σ σ σ ∙ χ 0 p h ∙ B σσσσ + A σ σ σ σ ∙ χ 0 p h ∙ B σ σ σ σ = A ↑↓ ∙ χ 0 p h ∙ B ↑↑ + A ↑↑ ∙ χ 0 p h ∙ B ↑↓ . We thus find for the magnetic and density components
[ A ∘ χ 0 p h ∘ B ] m / d = [ A ∘ χ 0 p h ∘ B ] ↑ ↑ ∓ [ A ∘ χ 0 p h ∘ B ] ↑ ↓ = A ↑ ↑ ∙ χ 0 p h ∙ B ↑ ↑ + A ↑ ↓ ∙ χ 0 p h ∙ B ↑ ↓ ∓ A ↑ ↓ ∙ χ 0 p h ∙ B ↑ ↑ ∓ A ↑ ↑ ∙ χ 0 p h ∙ B ↑ ↓ = ( A ↑ ↑ ∓ A ↑ ↓ ) ∙ χ 0 p h ∙ ( B ↑ ↑ ∓ B ↑ ↓ ) = A m / d ∙ χ 0 p h ∙ B m / d . ✓ \begin{align}
[A \circ \chi_0^{ph} \circ B]_{m/d} &= [A \circ \chi_0^{ph} \circ B]_{\uparrow \uparrow} \mp [A \circ \chi_0^{ph} \circ B]_{\uparrow \downarrow} \\
&= A_{\uparrow \uparrow} \bullet \chi_0^{ph} \bullet B_{\uparrow \uparrow} + A_{\uparrow \downarrow} \bullet \chi_0^{ph} \bullet B_{\uparrow \downarrow} \mp A_{\uparrow \downarrow} \bullet \chi_0^{ph} \bullet B_{\uparrow \uparrow} \mp A_{\uparrow \uparrow} \bullet \chi_0^{ph} \bullet B_{\uparrow \downarrow} \\
&= (A_{\uparrow \uparrow} \mp A_{\uparrow \downarrow}) \bullet \chi_0^{ph} \bullet (B_{\uparrow \uparrow} \mp B_{\uparrow \downarrow}) \\
&= A_{m/d} \bullet \chi_0^{ph} \bullet B_{m/d}\, . \ \checkmark
\end{align} [ A ∘ χ 0 p h ∘ B ] m / d = [ A ∘ χ 0 p h ∘ B ] ↑↑ ∓ [ A ∘ χ 0 p h ∘ B ] ↑↓ = A ↑↑ ∙ χ 0 p h ∙ B ↑↑ + A ↑↓ ∙ χ 0 p h ∙ B ↑↓ ∓ A ↑↓ ∙ χ 0 p h ∙ B ↑↑ ∓ A ↑↑ ∙ χ 0 p h ∙ B ↑↓ = ( A ↑↑ ∓ A ↑↓ ) ∙ χ 0 p h ∙ ( B ↑↑ ∓ B ↑↓ ) = A m / d ∙ χ 0 p h ∙ B m / d . ✓ In the p p pp pp [ χ 0 ] σ 4 σ 3 σ 2 σ 1 p p ∼ δ σ 4 , σ 1 δ σ 2 , σ 3 [\chi_0]^{pp}_{\sigma_4 \sigma_3 \sigma_2 \sigma_1} \sim \delta_{\sigma_4, \sigma_1} \delta_{\sigma_2, \sigma_3} [ χ 0 ] σ 4 σ 3 σ 2 σ 1 pp ∼ δ σ 4 , σ 1 δ σ 2 , σ 3 G σ 2 σ 1 ∼ δ σ 2 , σ 1 G_{\sigma_2 \sigma_1} \sim \delta_{\sigma_2, \sigma_1} G σ 2 σ 1 ∼ δ σ 2 , σ 1
[ A ∘ χ 0 p h ∘ B ] σ 1 σ 2 σ 3 σ 4 = ∑ σ 5 , σ 6 σ 7 , σ 8 A σ 1 σ 5 σ 3 σ 6 ∙ [ χ 0 ] σ 6 σ 8 σ 5 σ 7 p h ∙ B σ 7 σ 2 σ 8 σ 4 = ∑ σ 5 , σ 6 A σ 1 σ 5 σ 3 σ 6 ∙ χ 0 p h ∙ B σ 6 σ 2 σ 5 σ 4 . \begin{align}
[A \circ \chi_0^{ph} \circ B]_{\sigma_1 \sigma_2 \sigma_3 \sigma_4} &= \sum_{\substack{\sigma_5, \sigma_6 \\ \sigma_7, \sigma_8}} A_{\sigma_1 \sigma_5 \sigma_3 \sigma_6} \bullet [\chi_0]^{ph}_{\sigma_6 \sigma_8 \sigma_5 \sigma_7} \bullet B_{\sigma_7 \sigma_2 \sigma_8 \sigma_4} \\
&= \sum_{\sigma_5, \sigma_6} A_{\sigma_1 \sigma_5 \sigma_3 \sigma_6} \bullet \chi_0^{ph} \bullet B_{\sigma_6 \sigma_2 \sigma_5 \sigma_4}\, .
\end{align} [ A ∘ χ 0 p h ∘ B ] σ 1 σ 2 σ 3 σ 4 = σ 5 , σ 6 σ 7 , σ 8 ∑ A σ 1 σ 5 σ 3 σ 6 ∙ [ χ 0 ] σ 6 σ 8 σ 5 σ 7 p h ∙ B σ 7 σ 2 σ 8 σ 4 = σ 5 , σ 6 ∑ A σ 1 σ 5 σ 3 σ 6 ∙ χ 0 p h ∙ B σ 6 σ 2 σ 5 σ 4 . For the ↑ ↑ \uparrow \uparrow ↑↑
[ A ∘ χ 0 p p ∘ B ] t = [ A ∘ χ 0 p p ∘ B ] ↑ ↑ = [ A ∘ χ 0 p p ∘ B ] σ σ σ σ = ∑ σ 5 , σ 6 A σ σ 5 σ σ 6 ∙ χ 0 p p ∙ B σ 6 σ σ 5 σ = A σ σ σ σ ∙ χ 0 p p ∙ B σ σ σ σ = A ↑ ↑ ∙ χ 0 p p ∙ B ↑ ↑ = A t ∙ χ 0 p p ∙ B t . ✓ \begin{align}
[A \circ \chi_0^{pp} \circ B]_{t} &= [A \circ \chi_0^{pp} \circ B]_{\uparrow \uparrow} = [A \circ \chi_0^{pp} \circ B]_{\sigma \sigma \sigma \sigma} = \sum_{\sigma_5, \sigma_6} A_{\sigma \sigma_5 \sigma \sigma_6} \bullet \chi_0^{pp} \bullet B_{\sigma_6 \sigma \sigma_5 \sigma} \\
&= A_{\sigma \sigma \sigma \sigma} \bullet \chi_0^{pp} \bullet B_{\sigma \sigma \sigma \sigma} = A_{\uparrow \uparrow} \bullet \chi_0^{pp} \bullet B_{\uparrow \uparrow} = A_{t} \bullet \chi_0^{pp} \bullet B_{t}\, .\ \checkmark
\end{align} [ A ∘ χ 0 pp ∘ B ] t = [ A ∘ χ 0 pp ∘ B ] ↑↑ = [ A ∘ χ 0 pp ∘ B ] σσσσ = σ 5 , σ 6 ∑ A σ σ 5 σ σ 6 ∙ χ 0 pp ∙ B σ 6 σ σ 5 σ = A σσσσ ∙ χ 0 pp ∙ B σσσσ = A ↑↑ ∙ χ 0 pp ∙ B ↑↑ = A t ∙ χ 0 pp ∙ B t . ✓ For the ↑ ↓ \uparrow \downarrow ↑↓ ↑ ↓ ‾ \overline{\uparrow \downarrow} ↑↓
[ A ∘ χ 0 p p ∘ B ] ↑ ↓ = [ A ∘ χ 0 p p ∘ B ] σ σ ‾ σ ‾ σ = ∑ σ 5 , σ 6 A σ σ 5 σ ‾ σ 6 ∙ χ 0 p p ∙ B σ 6 σ ‾ σ 5 σ = A σ σ σ ‾ σ ‾ ∙ χ 0 p p ∙ B σ ‾ σ ‾ σ σ + A σ σ ‾ σ ‾ σ ∙ χ 0 p p ∙ B σ σ ‾ σ ‾ σ = A ↑ ↓ ‾ ∙ χ 0 p p ∙ B ↑ ↓ ‾ + A ↑ ↓ ∙ χ 0 p p ∙ B ↑ ↓ [ A ∘ χ 0 p p ∘ B ] ↑ ↓ ‾ = [ A ∘ χ 0 p p ∘ B ] σ σ σ ‾ σ ‾ = ∑ σ 5 , σ 6 A σ σ 5 σ ‾ σ 6 ∙ χ 0 p p ∙ B σ 6 σ σ 5 σ ‾ = A σ σ ‾ σ ‾ σ ∙ χ 0 p p ∙ B σ σ σ ‾ σ ‾ + A σ σ σ ‾ σ ‾ ∙ χ 0 p p ∙ B σ ‾ σ σ σ ‾ = A ↑ ↓ ∙ χ 0 p p ∙ B ↑ ↓ ‾ + A ↑ ↓ ‾ ∙ χ 0 p p ∙ B ↑ ↓ . \begin{align}
[A \circ \chi_0^{pp} \circ B]_{\uparrow \downarrow} &= [A \circ \chi_0^{pp} \circ B]_{\sigma \overline{\sigma} \overline{\sigma} \sigma} = \sum_{\sigma_5, \sigma_6} A_{\sigma \sigma_5 \overline{\sigma} \sigma_6} \bullet \chi_0^{pp} \bullet B_{\sigma_6 \overline{\sigma} \sigma_5 \sigma} \\
&= A_{\sigma \sigma \overline{\sigma} \overline{\sigma}} \bullet \chi_0^{pp} \bullet B_{\overline{\sigma} \overline{\sigma} \sigma \sigma} + A_{\sigma \overline{\sigma} \overline{\sigma} \sigma} \bullet \chi_0^{pp} \bullet B_{\sigma \overline{\sigma} \overline{\sigma} \sigma} \\
&= A_{\overline{\uparrow \downarrow}} \bullet \chi_0^{pp} \bullet B_{\overline{\uparrow \downarrow}} + A_{\uparrow \downarrow} \bullet \chi_0^{pp} \bullet B_{\uparrow \downarrow} \\ & \\
[A \circ \chi_0^{pp} \circ B]_{\overline{\uparrow \downarrow}} &= [A \circ \chi_0^{pp} \circ B]_{\sigma \sigma \overline{\sigma} \overline{\sigma}} = \sum_{\sigma_5, \sigma_6} A_{\sigma \sigma_5 \overline{\sigma} \sigma_6} \bullet \chi_0^{pp} \bullet B_{\sigma_6 \sigma \sigma_5 \overline{\sigma}} \\
&= A_{\sigma \overline{\sigma} \overline{\sigma} \sigma} \bullet \chi_0^{pp} \bullet B_{\sigma \sigma \overline{\sigma} \overline{\sigma}} + A_{\sigma \sigma \overline{\sigma} \overline{\sigma}} \bullet \chi_0^{pp} \bullet B_{\overline{\sigma} \sigma \sigma \overline{\sigma}} \\
&= A_{\uparrow \downarrow} \bullet \chi_0^{pp} \bullet B_{\overline{\uparrow \downarrow}} + A_{\overline{\uparrow \downarrow}} \bullet \chi_0^{pp} \bullet B_{\uparrow \downarrow}\, .
\end{align} [ A ∘ χ 0 pp ∘ B ] ↑↓ [ A ∘ χ 0 pp ∘ B ] ↑↓ = [ A ∘ χ 0 pp ∘ B ] σ σ σ σ = σ 5 , σ 6 ∑ A σ σ 5 σ σ 6 ∙ χ 0 pp ∙ B σ 6 σ σ 5 σ = A σσ σ σ ∙ χ 0 pp ∙ B σ σ σσ + A σ σ σ σ ∙ χ 0 pp ∙ B σ σ σ σ = A ↑↓ ∙ χ 0 pp ∙ B ↑↓ + A ↑↓ ∙ χ 0 pp ∙ B ↑↓ = [ A ∘ χ 0 pp ∘ B ] σσ σ σ = σ 5 , σ 6 ∑ A σ σ 5 σ σ 6 ∙ χ 0 pp ∙ B σ 6 σ σ 5 σ = A σ σ σ σ ∙ χ 0 pp ∙ B σσ σ σ + A σσ σ σ ∙ χ 0 pp ∙ B σ σσ σ = A ↑↓ ∙ χ 0 pp ∙ B ↑↓ + A ↑↓ ∙ χ 0 pp ∙ B ↑↓ . For the singlet component, we thus find
[ A ∘ χ 0 p p ∘ B ] s = [ A ∘ χ 0 p p ∘ B ] ↑ ↓ − [ A ∘ χ 0 p p ∘ B ] ↑ ↓ ‾ = A ↑ ↓ ‾ ∙ χ 0 p p ∙ B ↑ ↓ ‾ + A ↑ ↓ ∙ χ 0 p p ∙ B ↑ ↓ − A ↑ ↓ ∙ χ 0 p p ∙ B ↑ ↓ ‾ − A ↑ ↓ ‾ ∙ χ 0 p p ∙ B ↑ ↓ = ( A ↑ ↓ − A ↑ ↓ ‾ ) ∙ χ 0 p p ∙ ( B ↑ ↓ − B ↑ ↓ ‾ ) = A s ∙ χ 0 p p ∙ B s . ✓ \begin{align}
[A \circ \chi_0^{pp} \circ B]_{s} &= [A \circ \chi_0^{pp} \circ B]_{\uparrow \downarrow} - [A \circ \chi_0^{pp} \circ B]_{\overline{\uparrow \downarrow}} \\
&= A_{\overline{\uparrow \downarrow}} \bullet \chi_0^{pp} \bullet B_{\overline{\uparrow \downarrow}} + A_{\uparrow \downarrow} \bullet \chi_0^{pp} \bullet B_{\uparrow \downarrow} - A_{\uparrow \downarrow} \bullet \chi_0^{pp} \bullet B_{\overline{\uparrow \downarrow}} - A_{\overline{\uparrow \downarrow}} \bullet \chi_0^{pp} \bullet B_{\uparrow \downarrow} \\
&= (A_{\uparrow \downarrow} - A_{\overline{\uparrow \downarrow}}) \bullet \chi_0^{pp} \bullet (B_{\uparrow \downarrow} - B_{\overline{\uparrow \downarrow}}) \\
&= A_{s} \bullet \chi_0^{pp} \bullet B_{s}\, . \ \checkmark
\end{align} [ A ∘ χ 0 pp ∘ B ] s = [ A ∘ χ 0 pp ∘ B ] ↑↓ − [ A ∘ χ 0 pp ∘ B ] ↑↓ = A ↑↓ ∙ χ 0 pp ∙ B ↑↓ + A ↑↓ ∙ χ 0 pp ∙ B ↑↓ − A ↑↓ ∙ χ 0 pp ∙ B ↑↓ − A ↑↓ ∙ χ 0 pp ∙ B ↑↓ = ( A ↑↓ − A ↑↓ ) ∙ χ 0 pp ∙ ( B ↑↓ − B ↑↓ ) = A s ∙ χ 0 pp ∙ B s . ✓