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Investigating the roots of the nonlinear Luttinger liquid phenomenology
by L. Markhof, M. Pletyukhov, V. Meden
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Submission summary
Authors (as registered SciPost users):  Lisa Markhof · Volker Meden 
Submission information  

Preprint Link:  https://arxiv.org/abs/1904.06220v4 (pdf) 
Date submitted:  20190827 02:00 
Submitted by:  Meden, Volker 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
The nonlinear Luttinger liquid phenomenology of onedimensional correlated Fermi systems is an attempt to describe the effect of the band curvature beyond the TomonagaLuttinger 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 twoparticle 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 singleparticle 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) twoparticle potential. This dependence was recently shown to spoil power laws in the singleparticle spectral function which previously were believed to be part of the TomonagaLuttinger liquid universality. Although our second order results for the structure factor are consistent with powerlaw scaling, this raises doubts that the conjectured nonlinear Luttinger liquid phenomenology can be considered as universal.
Current status:
Reports on this Submission
Report #1 by Anonymous (Referee 7) on 2019910 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1904.06220v4, delivered 20190910, doi: 10.21468/SciPost.Report.1161
Strengths
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
powerlaw behaviour near the loweredge 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 "singleparticle spectral function"'s powerlaw 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' powerlaw 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 nearestneighbor 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 spin1/2 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
powerlaw 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 momentumfrequency plane, of the dynamic response functions starting from their series of multiple integral representations.
The next twosentences "This in turn yielded powerlaw behavior in dynamical response functions,
and the exponents agreed with the nonlinear Luttinger liquid prediction. The calculation relies on twodimensional 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 powerlaw 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
noninteger powerlaw 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 waveoff the results of an exact rigorous analysis by some heuristic like argument. Finally,
the sentence "The same type of twodimensional asymptotic analysis is a crucial ingredient of the form factor approach to
the LiebLiniger 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 lowenergy 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 twopoint 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.