We analyze the asymptotic behavior of the exponential form in the fermionic density operators as a function of ruling parameter (Formula presented). In the particular case (Formula presented) this exponential is associated with the Wigner-Jordan transformation in (Formula presented) spin chain model. We compare the bosonization approach and the evaluation using the Toeplitz determinant. In the latter method, the Szegö-Kac theorem suggests that at (Formula presented) 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.