Bosonization for a Wigner-Jordan-like transformation: Backscattering and umklapp processes on a fictitious lattice

We analyze the asymptotic behavior of the exponential form in the fermionic density operators as a function of ruling parameter Q. In the particular case Q=π this exponential is associated with the Wigner-Jordan transformation in XY spin chain model. We compare the bosonization approach and the evaluation using the Toeplitz determinant. In the latter method, the Szeg\"o-Kac theorem suggests that at Q>π/3 a bosonized solution is accompanied by a faster decaying one. The second solution is revealed as an umklapp process on the fictitious lattice, and originates from backscattering terms in the bosonized density. Our finding remains true in a wide range of fermion filling ratios.