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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: 2019-04-15
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 argued to form the basis of the nonlinear Luttinger liquid phenomenology. Surprisingly, the second order contribution was so far not computed. We close this gap and show that it is consistent with the conjectured power law. We take this as a motivation to critically assess the steps leading to the mobile impurity Hamiltonian. We, in particular, highlight 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 which so far were widely believed to be part of the Tomonaga-Luttinger liquid universality. This result raises doubts that the conjectured nonlinear Luttinger liquid phenomenology can be considered as universal.

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Reports on this Submission

Anonymous Report 1 on 2019-4-21 Invited Report


Brute-force calculation of the second-order in interaction contribution to the spectral function of 1D Fermi gas presented in Appendix.


1. No new results.
2. Vague and misleading conclusions based on the lack of understanding of a textbook-level theory of Fermi edge singularity.


This work addresses the derivation of the power-law singularity of a spectral function of interacting one-dimensional Fermi gas associated with the threshold of the spectral continuum. The Authors take the perturbative approach to the problem, and successfully reproduce the well-known lowest-order in interaction result, obtaining the $\ln\omega$ term in the asymptote (energy $\omega$ is measured from the edge). Then they proceed to the second order in the interaction potential, and -- predictably -- find the $\ln^2\omega$ contribution in the leading logarithmic approximation series. The sub-leading terms are relegated to the Appendix. The Authors state that two first terms of the leading-logarithmic expansion support the power-law form of the edge singularity. There is nothing new in this statement. Since the 1969 work of Schotte and Schotte, there are better ways to derive the asymptotic power law. Later on, Don Hamann (Phys. Rev. Lett. v. 26, 1030, 1971) offered a workable way of finding the sub-leading corrections. The Authors may want to familiarize themselves with an exposition of this problem, e.g., in the monograph by Gogolin et al, "Bosonization and Strongly Correlated Systems".

Probably the Authors could assess the corrections to the power-law asymptote using the material of their Appendix. Being sub-leading, these corrections may become important at larger $\omega$, but should not affect the validity of the power-law asymptote. Instead, the Authors added Section 4-6 which have unclear meaning and no real conclusion. The vague sense of these sections of their work is condensed into a misleading statement occupying three last sentences of the abstract to their manuscript.

Requested changes


  • validity: poor
  • significance: poor
  • originality: poor
  • clarity: poor
  • formatting: perfect
  • grammar: good

Author Lisa Markhof on 2019-04-29 (in reply to Report 1 on 2019-04-21)
reply to objection

There seems to be a misunderstanding about the purpose of our work. We agree with the Referee that if it concerned the Fermi edge singularity, it would indeed be textbook material. However, we consider the problem of 1d correlated fermions with parabolic dispersion. The calculation of the second order perturbative correction to the dynamical structure factor (DSF) is therefore a difficult task and it was previously not clear - and not at all obvious - that the leading terms of this correction are indeed consistent with a power law. We stress that this is an important and new result.

In previous works (see references in our paper), an analogy between the problem at hand (the calculation of correlation functions of the 1d interacting Fermi gas) to the Fermi edge singularity problem was drawn. But in contrast to the latter, where an exact solution is known, no such exact solution confirms power laws in the DSF of the interacting Fermi gas. As we point out in our paper, the mobile-impurity model constructed to exhibit this power law can be derived only heuristically. A thorough investigation of this construction is the motivation for our Sects. 4-5. We disagree with the Referee that Sects. 4-6 are vague. We clearly state that the momentum dependence of the interaction potential, which is known to be non-negligible in the Tomonaga-Luttinger model beyond the scaling limit (see the Introduction of our paper), has to be neglected at least partially in the construction of an exactly solvable mobile-impurity model. This connection is also highlighted in the last three sentences of the abstract.

Concerning the sub-leading corrections, we wish to underline that we are only interested in the leading behavior close to the thresholds (as also clearly stated in our work). We note that we are familiar with the references the Referee points out, and we emphasize that those are about the Fermi edge singularity problem rather than about the 1d interacting electron gas. Even for the mobile-impurity model, a more complicated calculation is needed as a mobile impurity has to be considered rather than an immobile core hole.

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