# Investigating the roots of the nonlinear Luttinger liquid phenomenology

## TL;DR

This paper examines the nonlinear Luttinger liquid phenomenology in one-dimensional correlated Fermi systems, analyzing second-order perturbations and critiquing the mobile impurity model's assumptions, questioning its universality.

## Contribution

It provides the first second-order perturbative analysis of the structure factor and critically evaluates the mobile impurity model's limitations regarding momentum dependence.

## Key findings

- Second-order perturbation results are consistent with power-law behavior.
- The mobile impurity model cannot incorporate momentum dependence of the two-particle potential.
- Power laws in spectral functions may not be universal in nonlinear Luttinger liquids.

## Abstract

The nonlinear Luttinger liquid phenomenology of one-dimensional correlated Fermi systems is an attempt to describe the effect of the band curvature beyond the Tomonaga-Luttinger liquid paradigm. It relies on the observation that the dynamical structure factor of the interacting electron gas shows a logarithmic threshold singularity when evaluated to first order perturbation theory in the two-particle interaction. This term was interpreted as the linear one in an expansion which was conjectured to resum to a power law. A field theory, the mobile impurity model, which is constructed such that it provides the power law in the structure factor, was suggested to be the proper effective model and used to compute the single-particle spectral function. This forms the basis of the nonlinear Luttinger liquid phenomenology. Surprisingly, the second order perturbative contribution to the structure factor was so far not studied. We first close this gap and show that it is consistent with the conjectured power law. Secondly, we critically assess the steps leading to the mobile impurity Hamiltonian. We show that the model does not allow to include the effect of the momentum dependence of the (bulk) two-particle potential. This dependence was recently shown to spoil power laws in the single-particle spectral function which previously were believed to be part of the Tomonaga-Luttinger liquid universality. Although our second order results for the structure factor are consistent with power-law scaling, this raises doubts that the conjectured nonlinear Luttinger liquid phenomenology can be considered as universal. We conclude that more work is required to clarify this.