# Investigating the roots of the nonlinear Luttinger liquid phenomenology

### Submission summary

 As Contributors: Lisa Markhof · Volker Meden Arxiv Link: https://arxiv.org/abs/1904.06220v1 Date submitted: 2019-04-15 Submitted by: Markhof, Lisa Submitted to: SciPost Physics Domain(s): Theoretical Subject area: Condensed Matter Physics - Theory

### Abstract

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.

###### Current status:
Has been resubmitted

### Submission & Refereeing History

Resubmission 1904.06220v2 on 7 June 2019
Submission 1904.06220v1 on 15 April 2019

## Reports on this Submission

### Strengths

The second order expansion of the dynamical structure factor is non-trivial and original

### Weaknesses

The original results of the paper are not enough to warrant publication or substantiate the conclusions/claim of the paper.

Most of the paper is written more as a review than as paper focused on its own results.

### Report

This paper computes the dynamical structure factor (DSF) of interacting spinless fermions, in a perturbation expansion of the interaction. Its goal is to analyze the behavior close to the threshold, and to compare with the predictions of the so-called non linear Luttinger liquid (NLL), of a modified, interaction dependent exponent close to threshold.

I have no problems with the derivation given in the present paper, leading to the new result of equation (22) but I don't think that the paper is suitable, at least in its present form to be published in Scipost, for the following reasons:

- The goal of the present paper is to ascertain whether the claim of universality for the behavior close to threshold (essentially introduced in the references [6,17] of the present paper) are valid or not. In that respect the paper fails to give an answer on that point. The conclusion of the authors is that the interaction expansion is compatible with the expansion of a powerlaw (and thus that would be compatible with the claims of NLL) but that taking into account the momentum dependence of the interactions might lead to a different behavior. The might'' comes from the discussion around (27). To quote the authors To summarize, we have neither been able to find a way to keep the full momentum dependence of V nor found a justification for partially neglecting it''.
So considering the results of the present paper (namely the second order expansion of the DSF) I feel that there is not enough material to justify publication in the context of the goal of the present paper (namely the test of the NLL). If the authors want to publish their results purely as a second order calculation (which is indeed non trivial) then this is of course possible but the paper must be rewritten in quite different way.

- The bulk of the paper is more written as a review paper, or as a pedagogical presentation or a discussion on the question of the exponents of the DSF, than a paper presenting original research results. Most of the material before section 3 is a summary of well known facts on the TLL. Section 4 is mostly a reminder of the impurity model. Section 5 does not contain new results or does not discuss the results of the present paper but discusses the various papers in the literature (exactly solvable models, numerical calculations, etc.) and whether these works are conclusive or not for the predictions of the NLL. The conclusion of the paper is We conclude that more research is required to verify the predictions obtained with the mobile impurity model, in particular, for correlation functions other than the dynamical structure factor''. So although this could be a nice paper in the context of lecture notes of a school, I would not recommend publication as an original research article.

In conclusion, I feel that the present paper does not substantiate with enough original material the goals that are announced in the title/introduction. The derivation of the second order expansion of the DSF which is non-trivial and original could be published independently (equivalent of a brief report) in another context.

• validity: low
• significance: low
• originality: low
• clarity: high
• formatting: perfect
• grammar: excellent

Author Lisa Markhof on 2019-06-07
(in reply to Report 3 on 2019-06-01)
Category:

There seems to be a misunderstanding about the goal of our work. We do not claim that we can make any definite statements about the earlier conjectured universal power-law behavior close to the threshold, even when just focusing on the dynamical structure factor (DSF). This holds even more so when it comes to other correlation functions, such as, e.g., the single-particle spectral function. In the title, abstract, and introduction we only promise to investigate the roots of the nonlinear Luttinger liquid phenomenology in several ways. We do exactly this. Some of our results support the conjectured behavior, others question the model--the mobile impurity model--which was argued to be the effective model of the nonlinear Luttinger liquid phenomenology. Our work is thus intended to stimulate a discussion on the roots and the results of the nonlinear Luttinger liquid phenomenology which we believe to be overdue. This holds in particular, as, even after years of research, convincing results confirming the conjectured behavior from other considerations, besides computations within the mobile impurity model, are missing. For details on this, see our discussion of Sect. 5.

We do not regard our analysis of the approximations leading to the mobile impurity model presented in Sect. 4 as review material. Instead, it is a critical assessment of this "derivation" which cannot be found anywhere in the literature. We believe that a major part of the community is unaware of the heuristic nature of many of the crucial steps as this does not become clear from the original literature. We emphasize that several of these steps are not backed-up by general arguments, of, e.g., RG nature or similar. This has to be contrasted with the "construction" of the Tomonaga-Luttinger model as the effective low-energy model of the Tomonaga-Luttinger liquid universality class (even neglecting the momentum dependence for that matter). We emphasize that none of the referees doubted that our assessment of the "derivation" is correct. We have modified Sect. 4 to make this more clear.

We have furthermore rewritten the Abstract and revised the Introduction to make more obvious that we present two novel results: (1) The consistency of second order perturbation theory with the conjectured threshold power law of the dynamical structure factor (DSF) but (2) the incompatibility of a crucial part of the momentum dependence of the two-particle interaction and the mobile impurity model.

The referee's summary of our work also indicates a misunderstanding about our second order calculation for the DSF. In this all the momentum dependence of the two-particle interaction, which is relevant for the leading behavior close to the threshold, is fully considered. Doing so the second order contribution is consistent with the conjectured power law and in this part of our considerations there is no "might" concerning the momentum dependence (only concerning terms to third order and higher). We only question whether the treatment of the momentum dependence of the two-particle interaction is justified when it comes to the construction of the mobile impurity model which was used in the literature to compute other correlation functions. The changes in the Abstract, the Introduction and Sect. 4 are also intended to avoid this misunderstanding.

Also Sect. 5 should not be viewed as a review. Instead, we show that no fully convincing results confirming the conjectured universality have been obtained by other means than computations within the mobile impurity Hamiltonian. Again, such a clear evaluation cannot be found in the literature.

To summarize, we present two original results. The second order results for the DSF, characterized as being nontrivial and important by all three referees, as well as the insight that it is impossible to keep crucial parts of the momentum dependence of the two-particle interaction when constructing the mobile impurity model. We believe that this is sufficient to justify publication. On top, our paper is intended to initiate an overdue discussion on the foundations of the nonlinear Luttinger liquid phenomenology. We believe that SciPost is an appropriate platform to do so.

### Strengths

1. Explicit analysis of a tricky second-order contribution to the dynamic structure factor.
2. Raises awareness about possible limitations of conclusions based on beyond-Luttinger-liquid analyses which neglect the momentum-dependence of the interaction.

### Weaknesses

1. Section 5 assumes a very detailed familiarity of previous work [Refs. 7 and 8] on the effects of the momentum dependence of interactions in Luttinger-liquid-type models. A somewhat more detailed exposition of the key findings of these previous works would greatly aid the readibility of Section 5.

### Report

This paper by Markhof, Pletyukhov and Meden (MPM) addresses two issues in the literature on one-dimensional correlated Fermi systems 'beyond the Tomonaga-Luttinger liquid paradigm', i.e. including the effects of band curvature.

A.
The first issue involves the computation of the dynamic structure factor, $S(q,\omega)$, of interacting one-dimensional spinless fermions with a nonlinear dispersion relation, by Pustilnik, Khodas, Kamenev and Glazman (PKKG) in 2006 [Ref. 17]. These authors pointed out that when computed to first order in the two-particle interaction, a logarithmic threshold singularity is obtained. Interpreting this term as the linear one in an expansion conjectured to sum to a power law, PKKG were able to deduce the exponent of the power law. MPM have now computed the second-order term explicitly and showed that it is consistent with the conjectured power law. This demonstration of consistency is reassuring, though not surprising (had they found an inconsistency, that would have been shocking...). I see the main value of their calculation in its detailed exposition (presented in an appendix), which very instructive: the authors explain carefully how the leading divergence can be extracted cleanly by a suitable shift of integration variables.

B.
The second issue involves a broader question. PKKG introduced an effective 'mobile impurity model', constructed in such a manner that it reproduces the power law in the dynamic structure factor. In 2007, PKKG [Ref. 22] used this mobile impurity model to compute the single-particle spectral function, $\rho(k,\omega)$, obtained via double Fourier transform of the single-particle Green's function $G(x,t)$. They found that $\rho(k,\omega)$ displays a power-law singularity on the hole mass shell, similar to that in the Luttinger liquid, and argued that this power law is universal. By contrast, in Section 5 of the present paper, MPM cautiously raise doubts about the latter statement. Let me attempt to summarize their argumentation, which builds on earlier papers by Meden and coworkers [Ref. 7 and 8], as follows:

(a) The standard Tomonaga-Luttinger liquid description of interacting 1D fermions is obtained by (i) linearizing the dispersion relation and (ii) neglecting any momentum dependence in the two-particle interaction vertex (i.e. assuming a point-like interaction). The resulting model famously is exactly solvable using bosonization. When used to compute the single-particle Green's function, $G(x,t)$, and from that the momentum-integrated single-particle spectral function, $\rho(\omega)$, a power law divergence is obtained for the latter, with an exponent which is 'universal', in that it depends only on the Luttinger parameters $K_\rho$ and $K_\sigma$.

(b) In 1999, Meden [Ref. 7] argued that this universality is lost when assumption (ii) is relaxed and the two-particle interaction is allowed to depend on momentum, $V(q)$ (as needed, e.g., to describe a finite-ranged interaction): Meden showed that then the power-law divergence of the single-particle Green's function, $G(x,t)$, involves exponents which do not depend only on $K_\rho$ and $K_\sigma$ [see Eq. (15) of Ref. 7], and argued that this implies the same for $\rho(\omega)$.

(c) In 2016, Markhof and Meden [Ref. 8] revisited this issue by considering the momentum-resolved spectral function, $\rho(k,\omega)$, for a Tomonaga-Luttinger-liquid model with linear dispersion but momentum-dependent interaction, $V(q)$. They obtained analytic expressions for $G(x,t)$ as a power series expansion in $z = e^{i (2 \pi/L)x}$, with $t$-dependent coefficients defined via recursion relations [see Eq. (43) of Ref. 8], and used these expressions to compute $\rho(k,\omega)$ numerically [see Fig. 2 of Ref. 8]. A careful analysis of the near-threshold behavior when $|k-k_F|$ is finite showed that for some choices of $V(q)$, no clear power-law behavior is found [see purple lines in Fig. 2 of Ref. 8, corresponding to V(q) decaying exponentially with $|q|$, cf. Eq. (55)]. Markhof and Meden summarize this finding by stating: "We provide strong evidence that any curvature of the two-particle interaction at small transferred momentum destroys power-law scaling of the momentum-resolved spectral function as a function of energy."

(d) Finally, in section 5 of the present paper, MPM argue that they expect the findings from (c) to apply also for a Tomonoga-Luttinger model with NONlinear dispersion and momentum-dependent interaction: the latter might destroy power-law scaling of the momentum-resolved spectral function as a function of energy. MPM do not attempt to substantiate this statement with explicit calculations, leaving this as an open issue for further investigations: "The e ffect of band curvature on correlation functions of 1d interacting Fermi systems beyond the low-energy scaling limit (in which the curvature is RG irrelevant) remains an open issue which deserves further investigations."

In my view, the fate of power-law behavior in beyond-Luttinger-paradigm model - including both curvature of the dispersion and $q$-dependence of the interaction - is indeed worthy of further investigations. In that sense, I find the discussion of section 5 of the present paper reasonable. However, I would urge the authors to spell out the findings of Ref. 7 and Ref. 8 in a bit more detail in the early parts of their Section 5, to aid readers not thoroughly familiar with those papers in following their line of argumentation.

In particular, MPM should comment on the following possible objection to Refs. 7 and 8: The starting assumptions of the model of Refs. 7 and 8, namely (i) a linearized dispersion and (ii) a momentum-dependent interaction, seem to be mutually inconsistent, since the latter will, already at Hartree-Fock level, cause the effective dispersion to become nonlinear. Thus, once one has decided to linearize the dispersion, one should also neglect the momentum dependence of the interaction. According to this perspective, Refs. 7 and 8 considered a 'bad model', hence their conclusions may be disregarded.

My take on this point is: First, I have no reason to doubt the technical aspects of the computations from Refs. 7 and 8. Second, the possible objection just raised against the model considered in Refs. 7 and 8 does not apply for (d) below, which involves a combination of non-linear dispersion and momentum-dependent interaction. Hence, MPM's suggestion that the lessons from Refs. 7 and 8 might also apply to (d) strikes me as reasonable and worthy of further investigation.

I thus recommend the paper for publication, provided some comment on the issues raised in the preceding two paragraphs are included.

### Requested changes

See last line of the above report.

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

Author Lisa Markhof on 2019-06-07
(in reply to Report 2 on 2019-05-31)

The referee nicely summarizes our results and intentions when mentioning the strengths of our paper.

The referee is right that we do not give a detailed introduction to Refs. [7] and [8] in Sect. 5. However, the main results of these papers of relevance for the present submission are summarized in the Introduction, Sect. 1. We believe that this is useful as it allows us to put our second main result (inconsistency of crucial parts of the momentum dependence of the interaction and the mobile impurity model) in a proper perspective already in the Introduction. In the light of the third report we are facing a conflict. While the second referee asks us to extend on the summary of earlier work the third one criticize our summary as too detailed. We thus hope that the second referee can go along with our decision to only slightly modify the discussion of Refs. [7] and [8] (as explained in the next two paragraphs).

We would like to emphasize that we do not agree with the referee that the model of Refs. [7] and [8] is a "bad model" with respect to the conclusions drawn from the calculations of these references. The goal of these papers was to present a counterexample of a model which is believed to be part of the Tomonaga-Luttinger liquid universality class, but does not show any universal power law at $k-k_F \neq 0$. We trust that the referee agrees that the Tomonaga-Luttinger model (linear single-particle dispersion) even with momentum dependent interaction should be part of this universality class. From this we concluded that the power laws of the single-particle spectral function found at $k-k_F \neq 0$ for the box potential or within the commonly employed ad hoc regularization are not part of the Tomonaga-Luttinger liquid universality. This is fully consistent with renormalization group arguments pointing at universality that can only be employed if all scales, including $k-k_F$, are sent to zero. We have added a corresponding comment in our paper.

We do not understand why the referee denotes the linear single-particle dispersion (of the fermions) and the momentum dependence of the two-particle interaction as being mutually inconsistent. We agree that in first order perturbation theory for the self-energy the fermionic dispersion could become nonlinear if the a momentum dependence of the two-particle interaction would be kept. The same holds for the bosonic dispersion within the bosonization approach in which the interaction is kept to all orders. We, however, fail to see how this indicates any inconsistency. We added a comment on this to our paper. Note, however, that this issue is irrelevant in the context of the present paper in which the combination of momentum dependence and nonlinearity of the single-particle dispersion is considered. This was already pointed out by the referee.

Anonymous on 2019-06-22
(in reply to Lisa Markhof on 2019-06-07)

I am satisfied with the authors' replies to my comments and believe the paper is ready for publication.

Side remark: the second report of referee 2 also addresses an issue I had raised in my first report: namely that a $q$-dependence of the interaction generates, at mean-field level, a curvature for the dispersion, so that a model assuming linear dispersion but $q$-dependent interaction might be argued to be internally inconsistent. Given the contentious nature of this matter, the authors would be well advised to address this matter head-on in their paper -- formulate the potential objection, ideally using the phrasing of referee 1, and explain their reasons for rejecting it. (I write this as advice, not as a referee requirement.)

### Strengths

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

### Weaknesses

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

### Report

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

none

• 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)
Category: