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Investigating the roots of the nonlinear Luttinger liquid phenomenology

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

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Submission summary

Authors (as Contributors): Lisa Markhof · Volker Meden
Submission information
Arxiv Link: (pdf)
Date submitted: 2019-06-07 02:00
Submitted by: Markhof, Lisa
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
Approach: Theoretical


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:
Has been resubmitted

Reports on this Submission

Anonymous Report 2 on 2019-7-5 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1904.06220v2, delivered 2019-07-05, doi: 10.21468/SciPost.Report.1053


Second order perturbative calculation of the structure factor


A (too) large part of the paper has a review character and does not really present original results


In this revised version the authors have improved some aspects of the presentation of the results. The results themselves and the message of the paper are essentially unchanged.

As mentioned in the previous report the main, and in my opinion, only new result of the paper is the calculation of the perturbative expansion (up to second order) of the structure factor. This calculation is sound and constitute a new result that would be worthy of publication by itself, at the level of a "brief report" (the editor of Scipost should fix the threshold level they want for the papers published in Scipost).

However, the authors want to put this result in the more general context of discussing the validity and assumptions behind the non-linear Luttinger liquid model/derivation that exists in the literature. This is done in several stages:

- the introduction (Section 1) contains a reminder (recalling the results of papers [7,8]) that for systems with momentum dependent interactions one does not find the TLL powerlaw behavior of one is at a finite energy scale (e.g. by having $k-k_F \neq 0$. It then mentions the impurity model that was used to address this finite energy scale question.

- the Sections 2 and 3 contain the explicit perturbative calculation of the structure factor. The result is found consistent with the expansion of the powerlaw behavior predicted by the impurity model (up to second order of course).

- Section 4 is reminder/rederivation of the impurity model (essentially introduced in e.g. Ref [6]). This section reproduces essentially the steps of Ref.[6] pointing out explicitly (for example just below equ. (26)) where approximations were made in the initial derivation. A summary of these is given at the end of the section (points (i)-(iv)). In my opinion this part does not contain new results but is simply a guided derivation of the model of Ref[6].

- Section 5 contains a discussion of the various results that exist in the literature concerning the impurity model with the conclusion that none of them is conclusive to justify the universal result claimed by the model. Again in this section there is no original contribution by the authors, except a critical reading of the literature.

The final conclusion of these two sections is largely inconclusive, with the fact that "more research is needed" to decide if the impurity model has indeed a universal behavior or not.

I am not contesting the warnings that are made in these two last sections, but I reiterate my opinion that Sections 4 and 5 are not presenting original results but more a critical reading and derivation of existing material in the literature. I would be perfectly happy to see such chapters in e.g. notes of a school, but I am uneasy to see this as a sizable part of an original research article, moreover in a way that is essentially decoupled from the part that contains the original result of the paper.

I realize that the latter is subjective and different people might have different thresholds for what is considered as the content of an original research paper. For my part I think that the paper would benefit from seriously compressing Sections 4 (for example one could keep mostly the points (i)-(iv)) and Section 5, to provide a single section of discussion that would follow the results of the second order expansion (consistent with the predictions of the impurity model) with some comments of caution on the model.

Requested changes

- strong compression of the Sections 4 and 5

  • validity: good
  • significance: ok
  • originality: good
  • clarity: high
  • formatting: excellent
  • grammar: perfect

Anonymous Report 1 on 2019-6-15 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1904.06220v2, delivered 2019-06-15, doi: 10.21468/SciPost.Report.1020


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  [id 541]

(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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