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Investigating the roots of the nonlinear Luttinger liquid phenomenology

by L. Markhof, M. Pletyukhov, V. Meden

This is not the current version.

Submission summary

As Contributors: Lisa Markhof · Volker Meden
Arxiv Link: https://arxiv.org/abs/1904.06220v3
Date submitted: 2019-08-02
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

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Nonlinear Luttinger liquids
Current status:
Has been resubmitted


Author comments upon resubmission

Dear Editor,

Thank you very much for reconsidering our submission and for your
feedback.

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
dropped.

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

Strengths

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

Weaknesses

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

Report

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

Author Volker Meden on 2019-08-27
(in reply to Report 1 on 2019-08-20)

Although the new referee is also critical about our second new result,
in contrast to the first referee, the present one provides scientific
arguments. We are very glad about this as it gives us the opportunity to
argue in an equally scientific way that the criticism is based on a
misunderstanding of our second result and a notation issue.

The main problem is that the referee overlooks that our second new result
is based on two ingredients. The first is that we have shown earlier in
Refs. [7,8] that for a strictly linear single-particle dispersion
(of the fermions) any nonconstant momentum dependence of the
two-particle interaction close to momentum transfer q=0 destroys
power-law (threshold) scaling of the single-particle spectral function
as a function of energy at any finite k-k_F. Dropping the momentum
dependence leads to spurious power laws. In the present manuscript we
show that exactly this type of momentum dependence cannot be kept
when attempting to map the interacting electron gas onto a solvable
mobile impurity model. One can thus not exclude that in analogy to the
Tomonaga-Luttinger model this leads to spurious power laws. In the report
the referee ignores the first part of the argument leading to our final
conclusion that more work on the combined effect of a nonlinear fermionic
single-particle dispersion and a momentum dependent two-particle
interaction is required. In fact, in the reports of three of the four
referees of the different versions of our submission we find indications
that they did not seriously consider this part of our argument (the
exception being the second referee of version 1 of our submission).

We next go into more details. In the second paragraph of the report
the referee states that "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." We have two problems with this sentence. Firstly, in the
Tomonaga-Luttinger model, being the fixed point model of the
Tomonaga-Luttinger liquid universality class, the DSF at small momenta
q does not show any power-law behavior but is rather given by a
delta-function. The referee might have confused the DSF and the
single-particle spectral function. In case the referee does indeed speak
about this spectral function we, secondly, have the problem that there
is nothing like "...the conventional Luttinger liquid theory..." result
for this, at least as long as the (fixed) momentum is different from k_F.
This is what we describe above and have shown in Refs. [7,8]. Therefore,
in the present case not even the limit of small (but finite) momenta is
'universal', or more precisely, the single-particle spectral function
depends on details of the momentum dependence of the two-particle
interaction (which are not kept in the the nonlinear Luttinger liquid
phenomenology). Do the protagonists of the nonlinear Luttinger liquid
theory seriously believe that the single-particle spectral function
shows for sufficiently large fixed |k-k_F| a 'universal' power-law
behavior (as a function of energy) with a momentum dependent exponent,
becomes nonuniversal (no power law; see Refs. [7,8]) for smaller
|k-k_F|, and, eventually, universal again at k-k_F=0 (in the sense of
conventional Tomonaga-Luttinger liquid theory; see Refs. [7,8])? If so
they should show this. This was clearly not done so far, mainly because
the absence of universality of the single-particle spectral function
of the Tomonaga-Luttinger model, or, more generally, of
Tomonaga-Luttinger liquids, at finite k-k_F was mainly ignored. We
furthermore wonder what the origin or nature of the 'universality' at
finite k-k_F might be? In contrast to the Tomonaga-Luttinger liquid
universality, emerging when all energy scales are sent to zero, it is
clearly not based on powerful RG arguments.

In the third paragraph, the referee notes that k1, k2, k3 in Eq. (25)
are all much smaller than q, and criticizes that we would take
k1 \approx q. This is a misunderstanding due to a confusing notation
on our part. We apologize for this, and thank the referee for pointing
this out. k1 refers, at this point, to k1 in Eq. (11) rather than
Eq. (25). We have amended our notation to make clear which momentum we
refer to, and have also added comments for clarification. In this
context, we also noted that there was a typo in the paragraph above
Eq. (25): instead of "... can only be close to the momenta 0, \pm kF,
..." it should be "... 0, \pm q ...". We have corrected this. Just as
in the original considerations of Pustilnik et al., we take the
momentum cutoff of the subbands to be much smaller than q. However, no
calculations can be found in the literature which show that dropping
the momentum dependence, as it is done when 'deriving' the mobile impurity
model, is a controlled approximation, even when focusing on this rather
special limit. In particular, no (powerful) RG arguments, often used when
the 'irrelevance' of perturbations is to be investigated, are available.
Our experience from Refs. [7,8] taught us to be extremely careful when
it comes to dropping the momentum dependence without having rigorous
arguments to do so. To be more specific, even if k_1' in Eq. (25) (revised
notation) is taken much smaller than q (see the third paragraph of the
report) it is by no means obvious that replacing V(q-k_1'+k_2'-k_3') -> V(q)
is a controlled step. All this is strongly linked to the question in which
sense the terms kept in the 'derivation' of the mobile impurity model
are 'the leading ones'. In fact, in the third paragraph the referee
also uses this phrase ("If one really wants to go beyond the leading
approximation..."). We believe that 'the leading ones' should be replaced
by 'the ones which can be kept when aiming at a solvable Hamiltonian'.
As we have mentioned in earlier replies our experience shows that this
weakness of the 'derivation' of the (solvable) mobile impurity model is
not recognized by a sizable part of the community.

Another argument put forward in the third paragraph of the report is
"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]). ... However, this
seems rather unlikely." We emphasize that the referee here has to use
phrases such as "general expectation" and "seems rather unlikely". The same
phrases were used when the effect of the momentum dependence of the
two-particle interaction on the single-particle spectral function of the
Tomonaga-Luttinger model (strictly linear fermionic single-particle
dispersion) was discussed in the past. However, Refs. [7,8] showed that
the general expectation is wrong and the unlikely case is realized in
this model. In analogy to the present case it was obvious that (powerful)
RG arguments cannot be used to argue in favor of universality (as an
energy scale, namely k-k_F, is kept fixed). We agree with the the referee
that a detailed analysis of the relevance of terms dropped when 'deriving'
the mobile impurity model from the interacting electron gas is missing.
We, however, disagree with the referee that we have to "...show that
the perturbations which are higher order in momentum destroy the
threshold singularity." In contrast, the protagonists of the nonlinear
Luttinger phenomenology should have shown already years ago that the
threshold power laws are stable against such terms. In our present
manuscript we take one step in this direction by showing that no
(exactly) solvable model in which relevant parts of the (bulk) momentum
dependence are kept can be derived in a straightforward way and analyze
the reason for this.

We emphasize, that we do not argue against any of the "classic" results
obtained for the x-ray edge problem; see the third paragraph of the
report.

In response to the requested changes we have revised the manuscript
as follows:

(1) We agree with the current referee that a comment on the papers by
Kitanine, Kozlowski, et al.. is important. In fact, an earlier version
of our submission contained such a comment which was, however,
criticized by the Editor. We made the mistake to remove it instead of
rewriting it. We now added a brief discussion on these papers (not only
J. Stat. Mech., P09001 but also arXiv:1811.06076) in the discussion
of Sect. 5 and hope that both the current editor and the current referee
are satisfied with this.

(2) In earlier rounds of reviewing we were asked by the referees and
the editors to shorten our discussion of Refs. [24-29] (Refs. [25-30]
of the revised version) with the argument that the weaknesses are well
known. We agree with the current referee that it might be advantageous
to extend this discussion which would allow us to be more specific.
However, we feel that given the history of our submission we cannot
return to the former version. In any case, we do not believe that
"The criticism of Refs. [24-29] and the conclusion are too biased and
unjustified." We simply mention these important papers and briefly state
their limitations.

(3) We have rewritten our comment on the q-> 0 limit following Eq. (24)
along the lines suggested by the referee.

(4) We have amended our notation in Eq. (25) and added comments to clarify
which momentum we refer to.

We hope that our manuscript can now be accepted for publication.

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