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

TL;DR

This paper investigates the orthogonality catastrophe in a one-dimensional lattice of spinless fermions with impurities, emphasizing the roles of impurity scaling, finite size corrections, and umklapp scattering, using DMRG to analyze the effects.

Contribution

It demonstrates the importance of including impurity scaling, finite size corrections, and umklapp flow in understanding the orthogonality catastrophe, providing evidence for the asymptotic impurity exponent of 1/16.

Findings

Finite size corrections of the form ln(L)/L are significant.

The impurity backscattering contribution approaches 1/16 in certain regimes.

Umklapp scattering becomes relevant near the charge-density wave transition.

Abstract

We show that to understand the orthogonality catastrophe in the half-filled lattice model of spinless fermions with repulsive nearest neighbor interaction and 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 umklapp scattering. The latter defines a length scale $L_u$ which becomes exceedingly large the closer the system is to its transition into the charge-density wave phase. Beyond this transition umklapp scattering is relevant in the renormalization group sense. Field theory can only be employed for length scales 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 the finite size corrections resulting from (i) and (ii) we provide strong evidence that the impurity backscattering contribution to the orthogonality exponent is asymptotically given by $1/16$. While further increasing the two-particle interaction leads to a faster renormalization group flow of the impurity towards the cut chain fixed point, the increased bare amplitude of the umklapp scattering renders it virtually impossible to confirm the expected asymptotic value of $1/16$ given the accessible system sizes. We employ the density matrix renormalization group.