Investigating the roots of the nonlinear Luttinger liquid phenomenology

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.