I was wondering if someone could explain how to derive the discrete version of Gauss law in 1+1 QED using staggered Fermions.
The result I am trying to reproduce is found in multiple references [see for instance [this paper][1], Appendix F, paragraph below Eq. (F2)]. Gauss' law in the staggered representation takes the form
$$E_{j, j+1} - E_{j-1, j} = \Phi^\dagger_j \Phi_j - \frac{1 - (-1)^j}{2} \ .$$
The LHS seems clear; one simply approximates the derivative with a central finite difference, $\partial E(x) \rightarrow \frac{1}{a} (E_{j, j+1} - E_{j-1, j})$.

If I take the staggered fields defined on even and odd lattice sites such that $\Phi_{2j} := \sqrt{a}\psi_{e^-}(x_{2j})$ and $\Phi_{2j+1} := \sqrt{a}\psi^\dagger_{e^+}(x_{2j+1})$, respectively, then, using anticommutation relations $\{\psi_i, \psi_j^\dagger\}=\delta_{ij}$, I get for the RHS of Gauss' law
$$\psi^\dagger_j \psi_j = 
    \begin{cases}
    \Phi^\dagger_j \Phi_j & j \ \text{even} \\
    \Phi_j \Phi^\dagger_j & j \ \text{odd}
    \end{cases}
    =
    \begin{cases}
    \Phi^\dagger_j \Phi_j \\
    - \Phi^\dagger_j \Phi_j + 1
    \end{cases}
    = (-1)^j \Phi^\dagger_j \Phi_j + \frac{1 - (-1)^j}{2} \ .$$

Can anyone tell me what I do wrong?


  [1]: https://arxiv.org/abs/1810.03421