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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: 2019-08-02
Submitted by: Meden, Volker
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

Author comments upon resubmission

Dear Editor,

Thank you very much for reconsidering our submission and for your

We have uploaded a revised version of our manuscript to arXiv.
We followed your and the former editor's advise to significantly
shorten sections 4 and 5. In fact, section 4 now only has less
than half the length it used to have and more or less exclusively
describes our observation that crucial parts of the momentum dependence
of the two-particle interaction cannot be kept in the "derivation" of
the mobile impurity model from the 1d interacting electron gas. This is
our second new result (the first one being the second order perturbation
theory for the DSF described in section 3 and the appendix). We only kept
a small part of section 5 and merged it with the old section 6 to a new
section 5.

Sections 1 to 3 of the revised submission only contain minor changes.
They were required for consistency reasons (with respect to the changes
in the other sections).

Concerning your more specific remarks (1) to (4).

(1) We do not completely agree with your statement that everybody in the
field is fully aware of the phenomenological nature and the weaknesses
of the nonlinear Luttinger liquid approach. In fact, when discussing
our new results with colleagues we frequently encountered exactly the
opposite. Many colleagues believe that with the advent of the nonlinear
Luttinger liquid phenomenology the issue of the nonlinearity of the
single-particle dispersion is settled. However, in our revised version
we express that certain parts of the community are aware that many steps
in the "derivation" of the mobile impurity model from the interacting
electron gas are at most phenomenological. In this respect we also

(2) included the reference you mentioned and frequently refer to it.

(3) We did not intend to indicate that it is not settled why the
Calogero-Sutherland model does not fall into the (alleged) nonlinear
Luttinger liquid universality class. With the revisions of section
5 this is no longer an issue.

(4) Also your remark concerning the old Ref. [25] (new Ref. [28]) is no
longer an issue.

As emphasized in an earlier conversation with the former editor of our
submission we do not doubt that the first referee has worked on 1d
correlated systems. However, this referee is highly biased and was
unwilling to engage in any scientific discussion. Instead, the second
report of this referee contains impudent statements. We would very much
appreciate if this would be acknowledged from the editorial side. On
general grounds we believe that reports of this type should simply be

We hope that the revised version can now be accepted for publication
in SciPost.

Yours sincerely,

Lisa Markhof
Mikhail Pletyukhov
Volker Meden

Reports on this Submission

Anonymous Report 1 on 2019-8-20 Invited Report


1-The second-order calculation of the dynamical structure factor is technically challenging and the result nicely confirms the expectation of a threshold singularity.


1-The criticism of nonlinear Luttinger liquid phenomenology is vague and based on a misunderstanding of the validity of the approach.


The point of this manuscript is to put the nonlinear Luttinger liquid theory to the test. First, the authors tested the prediction of a threshold singularity in the dynamical structure factor (DSF) calculated within an effective mobile impurity model by Pustilnik et al. in Ref. [17]. They find that the perturbative calculation of the DSF to second order in the fermionic is consistent with the threshold singularity. This is the only result of the manuscript.

The authors claim to have a second result which is actually a remark about their own difficulty in handling the momentum dependence of the interaction within the mobile impurity model. This is not a serious objection to the nonlinear Luttinger liquid theory because it is based on a misunderstanding of the regime of validity of the mobile impurity model. As clearly stated in the original paper by Pustilnik et al., the definition of the impurity subband centered at momentum kF-q requires a momentum cutoff which is much smaller than q. This is important to distinguish it from the low-energy subband of right movers centered at momentum kF. Such procedure is devised to compute the threshold behaviour for correlation at small but finite q. As q decreases, the energy window in the DSF where the effective model can be applied shrinks, and one observes a crossover to the power law behavior of the conventional Luttinger liquid theory. In this manuscript the authors state that "the scale on which the above procedure, if applicable at all, is valid remains open", but the energy scales involved and the resulting crossover have been discussed for the spectral function in Imambekov and Glazman, Science 323, 228 (2009); see also the review in Ref. [6].

As a consequence of the momentum cutoff in the impurity subbands, it only makes sense to use the effective model in the regime where k1, k2, k3 in Eq. (25) are much smaller than q. One cannot take k1 \approx q as the authors do. If one really wants to go beyond the leading approximation, the correct procedure would be to expand the interaction potential in the small momenta within the subbands, generating additional interactions with higher powers of momentum. The authors seem to be bothered by the fact that an impurity model which is not restricted to density-density interactions is no longer solvable. However, such additional terms are allowed by symmetry even in the original context of the x-ray edge singularity in metals where the original microscopic model is not Galilean invariant. The general expectation is that they may renormalize the parameters of the effective theory, but do not remove the power law behavior in the cases where a finite-frequency lower threshold at finite q is guaranteed by kinematic constrains (see Khodas et al., Ref. [27]). If the authors really want to point out a problem with the mobile impurity model of the nonlinear Luttinger liquid theory, they should show that the perturbations which are higher order in momentum destroy the threshold singularity. However, this seems rather unlikely.

In their concluding remarks, the authors also manifest their skepticism of previous papers that provided evidence for the nonlinear Luttinger liquid phenomenology. Of course numerical calculations such as those in Refs. [24] and [25] have limited frequency resolution, but together with the exact solutions of integrable model they do support the whole picture. Even if the authors want to leave out long-range potentials as in the Calogero-Sutherland model, they should acknowledge the example of the Lieb-Liniger model in Kitanine et al., J. Stat. Mech., P09001 (2012), in which the exact calculation of form factors are consistent with the nonlinear Luttinger liquid theory. Judging by what has been presented in this manuscript as compared to the existing literature on this subject, I don't find it reasonable to conclude that "this raises doubts that the conjectured nonlinear Luttinger liquid phenomenology can be considered as universal", as stated in the abstract.

Requested changes

1- Below Eq. (24), the authors should mention the interpretation for the q\to0 limit within the nonlinear Luttinger liquid theory; cite Imambekov and Glazman, Science 323, 228 (2009).
2-Please remove statements about the momentum dependence with k\approx q around Eq. (25).
3-The criticism of Refs. [24-29] and the conclusion are too biased and unjustified. Even if the authors believe that more research is required, they should at least acknowledge the successful results of nonlinear Luttinger liquid theory. I recommend rewriting the conclusion and the last statement in the abstract.

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

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