  Recently, in the Luttinger liquid phase of the one-dimensional generalized
antiferromagnetic Kitaev spin-1/2 model, it has been found that the abelian
bosonization formulas of the local spin operators only respect the exact
discrete nonsymmorphic symmetry group of the model, not the emergent U(1)
symmetry. In this work, we perform a renormalization group (RG) study to
provide explanations for the origin of the U(1) breaking terms in the
bosonization formulas. We find that the lack of U(1) symmetry originates from
the wavefunction renormalization effects in the spin operators along the RG
flow induced by the U(1) breaking interactions in the microscopic Hamiltonian.
In addition, the RG analysis can give predictions to the signs and order of
magnitudes of the coefficients in the bosonization formulas. Our work is
helpful to understand the rich nonsymmorphic physics in one-dimensional Kitaev
spin models.
