Investigating the roots of the nonlinear Luttinger liquid phenomenology

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

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.

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.

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.

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.

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 effect 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.

See last line of the above 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.

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