SciPost Phys. 7, 047 (2019) ·
published 9 October 2019

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The nonlinear Luttinger liquid phenomenology of onedimensional correlated Fermi systems is an attempt to describe the effect of the band curvature beyond the TomonagaLuttinger 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 twoparticle 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 singleparticle 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) twoparticle potential. This dependence was recently shown to spoil power laws in the singleparticle spectral function which previously were believed to be part of the TomonagaLuttinger liquid universality. Although our second order results for the structure factor are consistent with powerlaw 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.
Mrs Markhof: "It is rather difficult for us ..."
in Report on Investigating the roots of the nonlinear Luttinger liquid phenomenology