Anderson Hamiltonian (product of number operators) in 1st quantization?

[AA1983]

In the Anderson model, it cost an energy Un_{\Uparrow}n_{\Downarrow} for a quantum dot level to be occupied by two electrons. Here n_{\Uparrow} is the second quantized number operator, counting the number of particles with spin \Uparrow. I need the term Un_{\Uparrow}n_{\Downarrow} in first quantization. Here is what I know:

Un_{\Uparrow}n_{\Downarrow} =
Ud_{\Uparrow}^{\dagger}d_{\Uparrow}d_{\Downarrow}^ {\dagger}d_{\Downarrow}
=
-Ud_{\Uparrow}^{\dagger}d_{\Downarrow}^{\dagger}d_{ \Uparrow}d_{\Downarrow}
=
\frac{1}{2}\sum_{\eta_{1}\eta_{2}\eta_{3}\eta_{4}} V_{\eta_{1}\eta_{2}\eta_{3}\eta_{4}}d_{\eta_{1}}^{ \dagger}d_{\eta_{2}}^{\dagger}d_{\eta_{3}}d_{\eta_ {4}}

where


V_{\eta_{1}\eta_{2}\eta_{3}\eta_{4}}=\Big\{
\begin{array}{c}
-2U \qquad \text{for} \qquad \eta_{1}=\eta_{2}=\Uparrow,\: \eta_{2}=\eta_{4}=\Downarrow\\
0 \qquad \text{elsewhere}
\end{array}.

V is also given by


V_{\eta_{1}\eta_{2}\eta_{3}\eta_{4}}=\int dx_{j} dx_{k} \psi_{\eta_{1}}^{\ast}(x_{j})\psi_{\eta_{2}}^{\ast }(x_{k})V(x_{j}-x_{k})
\psi_{\eta_{3}}(x_{j})\psi_{\eta_{4}}(x_{k})

Now, what is V(x_{j}-x_{k}) ?

[AA1983]

Is it V=-2U\delta(x_{j}-x_{k})\delta_{\eta_{1}\Uparrow}\delta_{\eta_{2}\Do wnarrow}\delta_{\eta_{3}\Uparrow}\delta_{\eta_{4}\ Downarrow} ?

[Lonewolf]

I agree in principle, but shouldn't V_{\eta_{1}\eta_{2}\eta_{3}\eta_{4}} be non-zero for other combinations of indices? For example \eta_{1}=\eta_{4}=\uparrow, \eta_{2}=\eta_{3}=\downarrow should probably be allowed, since you're not creating or annihilating two of the same type of spin. Also, since flipping spins means swapping two pairs of fermionic operators in -Ud_{\Uparrow}^{\dagger}d_{\Downarrow}^{\dagger}d_{ \Uparrow}d_{\Downarrow}, you won't pick up a minus sign, so it should probably be


V=-2U\delta(x_{j}-x_{k})\left(\delta_{\eta_{1}\eta_{3}}\delta_{\eta_ {2}\eta_{4}} - \delta_{\eta_{1}\eta4}\delta_{\eta_{2}\eta_{3}}\ri ght)


so \eta_{1} is either the same as \eta_{3} or \eta_{4}, and it picks up a minus sign in the latter. Similarly for \eta_{2}.