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.
Investigating the roots of the nonlinear Luttinger liquid phenomenology