Bosonization for Wigner-Jordan-like Transformation : Backscattering and Umklapp-processes on Fictitious Lattice

TL;DR

This paper investigates the bosonization of fermionic density operators related to the Wigner-Jordan transformation, revealing how backscattering and umklapp processes manifest on a fictitious lattice and their dependence on the parameter Q.

Contribution

It introduces a novel analysis of bosonization for the Wigner-Jordan transformation, highlighting the role of umklapp processes and divergent series in the asymptotic behavior.

Findings

Bosonization approach aligns with Toeplitz determinant evaluation.

Umklapp processes emerge from backscattering terms.

Divergent series indicate a bosonized solution for Q > π/3.

Abstract

We analyze the asymptotic behavior of the exponential form in the fermionic density operators as the function of ruling parameter Q. In the particular case Q=\pi this exponential associates with the Wigner-Jordan transformation for XY spin chain model. We compare the bosonization approach and the evaluation via the Toeplitz determinant. The use of Szego-Kac theorem suggests that at Q>\pi/3 the divergent series for intrinsic logarithm provides a bosonized solution and faster decaying one, found as the logarithm's value on another sheet of the complex plane. The second solution is revealed as umklapp-process on the fictitious lattice while originates from backscattering terms in bosonized density. Our finding preserves in a wide range of fermion filling ratios.