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Investigating the roots of the nonlinear Luttinger liquid phenomenology

by L. Markhof, M. Pletyukhov, V. Meden

Submission summary

As Contributors: Lisa Markhof · Volker Meden
Arxiv Link:
Date submitted: 2019-06-07
Submitted by: Markhof, Lisa
Submitted to: SciPost Physics
Domain(s): Theoretical
Subject area: Condensed Matter Physics - Theory


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.

Current status:
Editor-in-charge assigned

Reports on this Submission

Anonymous Report 1 on 2019-6-15 Invited Report


Explicit calculation of the higher-order correction to an observable in the Appendix to the main text.


Misleading conclusions based on the erroneous previously-published works of the same authors and vague inconclusive discussion presented in the current manuscript.


The response to the previous Report solidifies my opinion that the Authors harbor misconceptions about basics of one-dimensional quantum fluids theory. The conventional Luttinger liquid theory does provide the correct phenomenology of a generic 1D quantum fluid (with a nonlinear spectrum of constituent particles) for certain observables and in a certain limit. The validity of Luttinger liquid theory is illustrated by rather than based on an exactly solvable model.

The nonlinear Luttinger liquid specifies the limit in which the linear theory works. It makes rather minimal extension of the theory to account for the effect of curvature of the single-particle spectrum on the observables close to the threshold singularities (which presence is protected by kinematics of particles in 1D). The preservation of the power-law singularities is warranted by the same general principles which warrant the existence of commonly-known Fermi edge singularities beyond the theory of free fermions.

A correct detailed microscopic theory for the spectral function would be of value. However the Authors' papers preceding the current submission are flawed in a pretty basic part, as the linear dispersion is unstable with respect to the momentum-dependent interactions. One has to account, e.g., for the Harteree-Fock corrections which would curve the dispersion relation and break the spurious degeneracy of the many-body spectrum inherent for the linear dispersion. The Authors seem to be oblivious to that, coming to entirely wrong conclusions in their previous work. The main text (not the Appendix) of the submitted one is no better. In my view, its publication would do further damage to the field.

  • validity: poor
  • significance: poor
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Author Lisa Markhof on 2019-06-15 (in reply to Report 1 on 2019-06-15)
reply to objection

It is rather difficult for us to reply to this report. The referee mainly, actually almost exclusively, criticizes (partly in the first paragraph but mainly the third one of the report) an earlier paper of two of us (LM and VM). Needless to say that we do not agree with the very vague arguments against this earlier work. In any case this earlier work is not the topic of the present reviewing process.

However, the first paragraph shows that it is not us, but the referee who suffers from misconceptions of Tomonaga-Luttinger theory. The crucial argument to show universality is the RG irrelevance of the band curvature and the momentum dependence of the two-particle interactions. In an RG procedure the Tomonaga-Luttinger model is the fixed point. It stands at the heart of Tomonaga-Luttinger liquid universality and not only illustrates this concept. In contrast to the referee, in our manuscript we clearly spell out the observables which are part of the Tomonaga-Luttinger liquid universality. Most crucially it only applies if RG arguments can be used, that is if all energy scales vanish.

In the second paragraph the referee just repeats the standard arguments used to support the mapping to the mobile impurity model. The referee does not even try to argue against our detailed analysis of the weaknesses of the mapping.

We believe that the last sentence of the report shows that the referee is highly biased and not open to scientific arguments which indicate weaknesses in the basis of the nonlinear Luttinger liquid phenomenology. Just take the fact that the referee ignores that any search for this phenomenology in microscopic models was so far not successful (for details see our manuscript).

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