Applying the Jordan-Wigner String Operator in the Kondo Problem

The string operator F ensures that fermionic operators at different sites anticommute correctly. My question is whether the string operator should be applied even to the Hilbert space of the Kondo spin. Specifically, the Kondo spin is essentially a single-occupied site, which introduces a prefactor of -1.
For example, consider the following expression:
\left\langle\hat{s}_{\mathrm{imp}} \cdot c_{+1, \alpha}^{\dagger} \boldsymbol{\sigma}_{\alpha \beta} c_{-1, \beta}\right\rangle-\text { H.c. }
Here, the Kondo spin \hat{s}_{imp} is located at site 0, interacting with nearest fermions from the left (-1) and right (+1) lead. Should the string operator, assuming it starts from the left side, be applied to the site -1 and the Kondo spin site 0 or just the site -1?

There is no need to apply the prefactor -1 for the Kondo spin site, the Kondo spin represents the degree of freedom of f electron in Heavy fermion systems. It is a boson operator, s_{imp} is quadratic for f electron operators, move the conduction electron across a spin site always pass the f electron operator twice, the prefactor
will not occur in practice and we just enforce the anticommutation relation for the conduction electron and treat the spins of impurities as bosons.

Thank you for your detailed explanation regarding the application of the Jordan-Wigner string operator in the Kondo problem. I understand your point that the Kondo spin itself does not introduce a string operator. However, I partially disagree regarding the conduction electron creation operator c_{+1,\alpha}^\dag at site +1. If the string operator is assumed to start from the leftmost, it should apply up to the nearest left site of the conduction electron, which I am inclined to believe is site -1 and not site 0, although I am not certain.

It turns out that the application of the string operator at site 0 is inconsequential. This is because it effectively contributes an extra -1, which can be absorbed, for example, into the right lead as a phase factor. Numerical tests confirm this as well.
Furthermore, it’s the single-channel problem(without the mixed term \hat{c}_{-1,\alpha}^{\dagger} \hat{\vec{S}}_{imp}\cdot\vec{\sigma}_{\alpha,\beta}\hat{c}_{1,\beta} + h.c., it’s a two-channel problem), so it’s natural to introduce the new modes as \hat{c}_{-1,\alpha}^{\dagger}+\hat{c}_{1,\alpha}^{\dagger} and \hat{c}_{-1,\alpha}^{\dagger}-\hat{c}_{1,\alpha}^{\dagger} to replace \hat{c}_{-1,\alpha}^{\dagger} and \hat{c}_{1,\alpha}^{\dagger}.

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