# Investigating the roots of the nonlinear Luttinger liquid phenomenology

### Submission summary

 As Contributors: Lisa Markhof · Volker Meden Arxiv Link: https://arxiv.org/abs/1904.06220v4 (pdf) Date submitted: 2019-08-27 02:00 Submitted by: Meden, Volker Submitted to: SciPost Physics Discipline: Physics Subject area: Condensed Matter Physics - Theory Approach: Theoretical

### 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 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.

### Ontology / Topics

See full Ontology or Topics database.

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

### Submission & Refereeing History

Resubmission scipost_201909_00006v1 on 25 September 2019

Resubmission 1904.06220v4 on 27 August 2019
Resubmission 1904.06220v3 on 2 August 2019
Resubmission 1904.06220v2 on 7 June 2019
Submission 1904.06220v1 on 15 April 2019

## Reports on this Submission

### Anonymous Report 1 on 2019-9-10 Invited Report

• Cite as: Anonymous, Report on arXiv:1904.06220v4, delivered 2019-09-10, doi: 10.21468/SciPost.Report.1161

none

### Weaknesses

Numerious. See report.

### Report

The paper "Investigating the roots of the nonlinear Luttinger liquid phenomenology" by L. Markhof, M. Pletyukhov and V. Meden
discusses some aspects related to the universality of the nonlinear Luttinger liquid phenomenology.

The first part of the paper is devoted to a second order perturbation theory calculation for the dynamical structure factor in a model of spinless fermions
so as to test the validity of the nonlinear Luttinger liquid based predictions. Not astonishingly, the authors find a matching between their perturbative calculations and the
power-law behaviour near the lower-edge as predicted by the use of the nonlinear Luttinger liquid (NLLL).

The second part of the paper raises criticisms towards the nonlinear Luttinger liquid universality. The main back up for this criticism are results issuing from
old works of a subset of the authors, Ref. [7] and [8]. As such, the criticisms are thus very disconnected from the first part of the paper.
The authors try to argue that "single-particle spectral function"'s power-law behaviour might not be grasped by a nonlinear Luttinger liquid (or that it might not even exist!).
The strategy employed by the authors to defend their point is very strange. If they believe so, they should have simply computed the second order perturbation theory result for these spectral functions
and then check that it agrees or show explicitly that there is a discrepancy. Yet this is not done.

Instead, the author give some vague arguments and try to critically assess the existing literature on the subject. The author's main point is that the behaviour of the correlators in real space along the directions
$x \pm v t=0$, with $v$ the Fermi velocity, spoils the dynamic response functions' power-law behaviour on the edges of a given model's spectrum. However, it is a rather easy exercise to convince oneself that this behaviour may only affect
the behaviour of dynamic response functions close to edges of the spectrum's curve $(k,\mathcal{E}(k))$ whose dispersion relation $\mathcal{E}(k)$ satisfies to the constraint $\mathcal{E}^{\prime}(k)=v$.
These are very special points and I am not aware of anyone serious in the business ever claiming
that NLLL grasps the behaviour of dynamic response functions for these special cases. I will not elaborate further on the other criticisms of the authors of the mobile impurity model since
these issues were already discussed in the previous reports.

However, I would like to focus on the authors criticisms of the analysis carried out on integrable systems in the works [29,31].
This part is very mind blowing to me in that they criticise the content and results that are \textit{not} established in these works!
The authors write "the lattice model of spinless fermions with nearest-neighbor interaction, which is equivalent to the XXZ
Heisenberg model, was studied and the form factors of the model were examined analytically." However, as it is suggested from that paper's title "On singularities of dynamic response functions in the massless regime
of the XXZ spin-1/2 chain", the work actually deals with the XXZ chain. Moreover, no form factors are analysed there. In fact the starting point of that paper
is a series of multiple integral representation for the XXZ chain's response functions.

Then the authors write "After resummation, the correlation functions in the limit of large x and t were found to exhibit
power-law behavior." Again, this is not done at all in [29]. Rather, the work [29] develops a rigorous method allowing one to analyse directly
the behaviour in, the momentum-frequency plane, of the dynamic response functions starting from their series of multiple integral representations.
The next two-sentences "This in turn yielded power-law behavior in dynamical response functions,
and the exponents agreed with the nonlinear Luttinger liquid prediction. The calculation relies on two-dimensional asymptotic analysis in real space and time." are also nonsense.
The asymptotic analysis carried out in the work [29] does actually deal with multidimensional integrals.
The authors write "In particular, the special directions x = ±vt, with the appropriate renormalized velocity of the elementary excitations
v, are not considered separately." Again, the authors do seem to exhibit a certain lack of understanding of the paper's content. The two branches do have to be analysed jointly
(what is done in [29]) and it is their mutual interaction that does produce the power-law behaviour. The work [29] provides a precise control on the corrections and, in particular,
on the potential effects that could be induced from the real space correlator's behaviour along the two special lines $x = ±vt$. These are shown not to contribute to the leading
non-integer power-law behaviour close to the edges of the spectrum (be it single or multi species edges). Independently of a total lack of connection between the author's criticisms and
the content of [29], it sounds to me pretty strange to try to wave-off the results of an exact rigorous analysis by some heuristic like argument. Finally,
the sentence "The same type of two-dimensional asymptotic analysis is a crucial ingredient of the form factor approach to
the Lieb-Liniger model [31]." also does translate the author's missunderstanding of the work's content.

It is also important to stress that, thanks to the recent progress on the analysis of spectral functions in the XXZ chain, there is almost no room for the NLLL paradigm to fail.
Indeed, a consequence of the work K. K. Kozlowski et J. M. Maillet, {\it Microscopic approach to a class of 1D quantum critical models}, J. Phys. A: Math. $\&$ Theor. , {\bf 48}, 484004, (2015),
is that the matrix elements of local operators in a model falling into the universality class of a Luttinger liquid take a very specific form between low-energy states.
In their turn, as follows from decades of calculations carried out in quantum field theories or condensed matter physics, and by means of various approaches,
the matrix elements of such local operators taken between finite energy states are described by a density of form factors, in the large volume limit.
Taken these two facts and assuming a natural parametrisation of the model's spectrum in terms of particle species, one may repeat the reasonings explained in
K. K. Kozlowski, {\it On the thermodynamic limit of form factor expansions of dynamical correlation functions in the massless regime of the XXZ spin 1/2 chain.},
J. Math. Phys. {\bf 59} (9), 091408 (2018), so as to provide an explicit functional form for the form factor expansion of two-point functions in such a model.
I stress that with the facts given as above, there is no need for the model to be integrable. Of course, some of the building blocks of such a series will remain unknown, so that
only the overall functional form of a massless form factor series will be available in such a model. This is explained in broader details in the mentioned works.
However, it is the functional form of the massless form factor expansion that does fix the edge behaviour of the response functions. Indeed, this is the only input that is needed so as to implement the asymptotic analysis
carried out in [29].

To summarise, I do not think that the scientific discussion in the second part of the paper makes the work fit for publication, be it in SciPost or any other journal.
Thus I strongly recommend to reject the paper.

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