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We show that to understand the orthogonality catastrophe in the half-filledlattice model of spinless fermions with repulsive nearest neighbor interactionand a local impurity in its Luttinger liquid phase one has to take into account(i) the impurity scaling, (ii) unusual finite size $L$ corrections of the form$\ln(L)/L$, as well as (iii) the renormalization group flow of the umklappscattering. The latter defines a length scale $L_u$ which becomes exceedinglylarge the closer the system is to its transition into the charge-density wavephase. Beyond this transition umklapp scattering is relevant in therenormalization group sense. Field theory can only be employed for lengthscales larger than $L_u$. For small to intermediate two-particle interactions,for which the regime $L > L_u$ can be accessed, and taking into account thefinite size corrections resulting from (i) and (ii) we provide strong evidencethat the impurity backscattering contribution to the orthogonality exponent isasymptotically given by $1/16$. While further increasing the two-particleinteraction leads to a faster renormalization group flow of the impuritytowards the cut chain fixed point, the increased bare amplitude of the umklappscattering renders it virtually impossible to confirm the expected asymptoticvalue of $1/16$ given the accessible system sizes. We employ the density matrixrenormalization group.

Renormalization group flows in one-dimensional lattice models: impurity scaling, umklapp scattering and the orthogonality catastrophe