We consider critical XXZ model with anisotropy $\Delta =(q+q^{-1})/2$, $|q|=1$. For this model we study the space of quasi-local operators $q^{2\alpha S(0)}\mathcal{O}$ with $S(0)=\textstyle {\frac 1 2}\sum _{j=-\infty}^0\sigma ^3 _j$ and $\mathcal{O}$ localized on finite number of cites. We explain that this space is created from the primary field $q^{2\alpha S(0)}$ by two fermions. Similarly to Baxter's $Q$-operator, these fermions are constructed using $q$-oscillators representations of quantum affine algebra. The Vacuum Expectation Values in the fermionic basis are given by determinants, like in the free theory.