Benchmarking the Ising Universality Class in $3 \le d < 4$ dimensions

Errors in user-supplied markup (flagged; corrections coming soon)

Dear editors,

We thank the referees for the careful reading of our manuscript, and
for their comments. We are resubmitting our paper with a fair number
of modifications, following the referees' suggestions. However, we did
not consider a major rewriting, and in the following we would like to
motivate this choice. More specific answers to queries follow
afterwards.

=======

First referee.

A major revision was requested for better explaining
the resummation methods. We think that extensive additions are not
necessary, because:

i) there is a huge literature on this historically established
subject. In our paper we provide a rather simple introduction in
Sect. 3.2 and discuss a toy model in appendix A; in the revised
version, we address the reader to the paper by one of the authors, Ref. [38],
where a specific resummation, very close to the one used in our paper, is nicely presented.
After these readings, the main Refs. [40-41] for our work
should be directly accessible.

ii) the aim of our paper is the comparison of two rather different
research topics: numerical bootstrap and perturbative expansion.
Attempts to reviewing either method would end up in a very lengthy
paper: actually, while the bootstrap approach is much less explained than
resummation methods, the referees only asked on details of the latter.

- Corrections implemented: We added remarks in several parts
of the paper, for pointing to review material, and for better motivating
the used methods. See later for answers to specific questions
on this issue (and the separate list of corrections).

==========

Second referee.

The request a for major revision was motivated by the
relation of our work with Ref. [21] (Ref. [20] in the previous version of the manuscript),
presenting results obtained by
the 3-correlator bootstrap with the navigator method.
The Referee argued that these results are more precise than ours, that
our approach is outdated, and that we should rewrite completely
our analysis using the data from Ref. [21].
However, we believe that a major rewriting of our paper is not necessary:

i) The data of [21] are certainly very precise, but unfortunately they
come without error estimates, so they cannot be directly used for fits,
which are the baseline for our comparison with perturbative
methods. Furthermore, results for the structure constants were not
provided in [21], while our data are even more precise than those
for conformal dimensions.

ii) We would like to point out that while the data of Ref. [21] may be more precise than ours,
it is always good to have an independent verification, and different questions asked.
We view Ref. [21] and our paper as complementary in this respect.
Actually, in our
paper we show a rather satisfactory agreement in the low-lying
part of the conformal spectrum: the data agree, up to very tiny
differences. The disagreement for higher-dimensional
states was discussed in Sect. 4.3, where we made clear that
the navigator results are better.

Corrections implemented:

- we gave more credit to Ref. [21] and the navigator method in several
parts of the revised paper, including Introduction, Sect. 2.1, and
Conclusions. We said that our method is inferior, yet useful for the problem at hand,
since it is computationally much cheaper.

- we updated the comparison of data for the two approaches in Fig. 4:
we propose an estimate for the navigator errors, observed the small offset in our data,
remarked that the navigator approach correctly identifies the unitarity island.
We then enlarged our error estimates to account for this offset
(see gray areas in the plots of Fig. 4).

- we added navigator data to Figs. 7, 9 and 13, where the comparison
with resummed perturbative results is made. This allows us to estimate
the potential error of our fits.

- we added the reference to earlier work [1602.02810]; we are sorry
that we forgot to mention it, but actually its data have errors
too large to be used in our fits.

=============

Answers to referees' questions

- First referee

1) Could you give some more details on how the resummation was
computed? How were the errors estimated? What value does the
parameter a take?

Answer was given above. The parameter $a$ is defined by the asymptotics (3.2); the instanton calculus provides a semiclassical estimate.


2) You mention another resummation technique (Hypergeometric
resummation of ref. 55). Could you compare against this technique
and give recommendations based on the comparison? Also, it appears
like the self-consistent resummation for $3<d<4$ was only performed
for $\Delta_\epsilon$ and not the other quantities.

The use of Hypergeometric functions is yet at a primitive stage; in
our paper, we did not discussed it, because a whole paper needs to be written before its precision could properly be addressed; we added a short sentence on page 13.

As explained in Ref. [41], the self-consistent resummation gave small
errors only for the quantity $1/\nu^3$, which has a good limit for $d\to 2$
into an exactly-known expression. A similar strategy was not found for
the other exponents. We added a short sentence on page 15.

3) Footnote 8 comments on the inclusion of seventh-order results,
saying that they give larger errors than sixth-order results. Could
you provide more details?

The seventh-order perturbative terms have not been independently confirmed, and there are some concerns about their validity in the community. Attempts
by two of the authors to use them in the resummations were not satisfactory.
Here too we prefer to be rather brief, to not convey any criticism or indulge into speculation.

4) Comparing with the "Navigator" of ref. 20 (red triangles in
the draft) there appear to be systematic errors of the same size, or
larger, than the errors given.

This point was answered above and is now discussed in the revised version.

5) What does 190 components correspond to in terms of the value
$\Lambda$ that is commonly given in the bootstrap literature?

It corresponds to $\Lambda = 18$.
We added a footnote in Sec. 2.1 to clarify this point.

6) In table 1, comparing the size of the errors in d=3.875 and
d=3.75, precision for spin 4 is much higher in d=3.875, and for spin
0 and spin 2 much higher in d=3.75. Is there any explanation for
this?

In the 1-correlator bootstrap, the states higher up in the conformal
spectrum change rapidly within the range of parameters identified as
the Ising point (see Sect. 2.1). The low-lying operators, more
precisely the leading twists, are more stable. This is a known fact. In
the case of $\Delta_{\epsilon'}$ and $\Delta_{\epsilon''}$ this
instability is parametrically sharper when passing from $d=3.75$ to
$d=3.875$, thus explaining the increase in errors. Instead, the
spin-four field $C$ is a leading twist and is not affected. We added
a footnote on page 20.

7) In appendix B, you mention that convergence for $|z|<1$ is
difficult to prove. Does it not follow from the assumptions made on
the analytic structure in the $t$ plane?

The problem is that the analytic structure in the $t$ plane in not known
for general field theories. In the toy model discussed in App.B, analytic
properties are explicit.

----------

Second referee

1) Write a fit of the data of [20] for all observables for which
this is available...... 2) Change the origin of all plots to the one
provided by the fit of [20] (when available).

Already replied: in general, a fit of data without errors is only
qualitative; nevertheless, in the revised versions we added navigator
data to our fits for comparison.

3) Throughout the paper it should be made clear that the
1-correlator results are not the best ones available in the
literature, but that they are simpler to achieve.....

Already replied. The suggested modifications have been implemented.

4) In figure 4 (and similar) we can see that the data of [20]
lies consistently below the shaded area of the fit of [BCKOW]. This
suggests that the bootstrap data points of [BCKOW] produced by
c-minimization have a systematic error, which is not taken into
account in the paper. If it is possible it would be useful to
estimate such error. An option is to simply increase the error in
order to contain the points predicted by [20]........

Already replied. This issue of the offset is discussed in the revised
version; it is now included in the error estimates.

5) Since one of the most valuable parts of the paper is the
comparison with resummation techniques, it would be useful to expand
on the latter. Can you review the details of the various resummation
methods used in the paper and how the errors are estimated?

This point was also raised by the first referee and it is answered above.

6) Did you try to use other non-polynomial fits of the bootstrap
data? For example, inspired by the Pade' approximations of the
perturbative series, one could try rational functions. It would be
instructive to know if other fits have advantages.

Pade' approximations are used for representing functions that have a
finite radius of convergence. As explained in Sect. 3.1, after (3.6),
the bootstrap non-perturbative data we are approximating by polynomials
are expected to be analytic functions in dimension $d<4$, and to have a branch cut singularity for $d>4$.
A Pade' approximant could not properly model this.

Best regards,

The Authors (Bonanno, Cappelli, Okuda, Kompaniets, Wiese)

- Sec. 1, 3rd paragraph, added

"The conformal spectrum for varying dimension 4 > d >= 2.6
has also been obtained in Ref. [21] by using the more advanced bootstrap technique of the
navigator method [37]. We use these very precise results in combination with ours to
obtain a consistent description of the low-lying spectrum."

- Sec. 1, 5th par., rewritten

"The analysis is done on the dimensions of the conformal fields sigma, epsilon, epsilon^prime, respectively corresponding
to spin, energy and subleading energy, which determine the critical exponents
eta, nu, omega. The precision of bootstrap data can be summarized by the d-independent value of the
relative error Err(gamma)/gamma = O(10^(-3) ) for the anomalous dimensions gamma for the conformal fields
sigma, epsilon. As the anomalous dimensions are very small for d \approx 4, the precision for the conformal
dimensions delta_sigma, delta_epsilon is actually higher in this region. Regarding the subleading energy, the
relative error Err(gamma_epsilon^\prime)=gamma_epsilon^\prime stays at three digits, as explained later. Furthermore, some of
the structure constants are determined with even better O(10^4) accuracy."

- Sec. 2.1, 1st par. added footnote

"This corresponds to the parameter Lambda = 18 counting the number of derivatives in the approximation of
the functional basis."

- Sec. 2.1, 2nd par., added references [16, 45, 46] -> [16, 19, 21, 47, 48] and rewritten

"Our 1-correlator numerical bootstrap approach has been surpassed by more recent implementations
[16,19,21,47,48], but we find it convenient for determining the low-lying spectrum
with modest computing resources."

- Sec. 2.2, last 3 pars., rewritten

"Next, we compare these results with those recently obtained by solving the 3-correlator
bootstrap with the navigator method [21]. In Fig. 4 our data, given in earlier figures (blue
circles), are shown on a finer scale, together with the estimated error of the fit (cyan shaded
area). The red triangles are the navigator values: they come with no errors and thus cannot
be directly used for the fits. A first observation is the fairly good agreement between the
two different bootstrap approaches at our level of precision..
We propose to estimate the error of navigator data as follows. We suppose that they
are roughly of the same size as those found in other 3-correlator studies at d = 3 (rigorous
bounds) [48, 50], which are plotted in Fig. 4 as black diamonds (gamma_sigma and gamma_epsilon), and a grey
rightward triangle (epsilon^prime). Assuming these very small uncertainties for each value of d, there
seems to be a negative offset with respect to our data, in particular for "0. This could be a
systematic error due to our approximate identification of the Ising point within the unitarity
region (Section 2.1), while the navigator method rigorously determines it within a unitarity
island [37]. However, other explanations are possible.

In conclusion, taking into account these considerations, we enlarge the error estimate of
our fits to the shaded gray bands in Figs. 4, which correspond to the following bound:"

- Sec. 2.2, last par., added footnote

"Earlier results of Ref. [19] are not considered here due to their large errors."

- Sec. 2.2, Fig. 4 modified by adding new error bounds (in gray)

- Sec. 3.2, 4th par., added

"In our work, the resummed data are obtained by extending the setup of Refs. [40, 41] to noninteger
dimension. A complete account of these methods is too long to be presented here:
nonetheless, our introduction, App. B and the paper [38] provide enough background for
accessing the original work."

- Sec. 3.2, 5th par., added

"As here we could not give justice to their influence, we exclude this resummation method."

- Sec. 3.2, 6th par., added

"Finally, Fig. 7 and
later plots comparing the dimensions delta_epsilon, delta_epsilon^prime also report the result of navigator bootstrap for
d = 3:5 (red triangle). This allows to assess the negligible difference between the two sets of
bootstrap data in the comparison with the epsilon-expansion.

- Fig. 7, 9, 13 modified by adding navigator point at d=3.5, and added to the caption

"Results from earlier works [3] have been omitted due to their large error bars."

- Sec. 3.2, 9th par., added

"A similar constraint does not seem to be possible for the other
critical exponents, as discussed in Ref. [41]."

- Sec. 3.2, 13th par., added sentence

"The red triangle at d = 3.5 again indicates the offset with respect to the navigator
bootstrap data."

and footnote

"The growth of the error when passing from d = 3.75 to d = 3.875 is due to the instability of the higher
part of the spectrum when approaching d = 4. This issue will be further discussed in Sect. 4.3."

- Sec. 4.2, 3rd par., modified

"Data from the navigator method are unfortunately only available for f_{sigma sigma epsilon} [21]."

- Sec. 4.3, 1st par., added

"which are definitely more accurate for the higher spectrum than our results."

- Sec. 5, 2nd par., added

"For these low-lying states of the conformal
spectrum, our results are in very good agreement with those of more advanced 3-correlator
bootstrap techniques [21, 37, 47, 48], with a small offset included in the error estimate."

- Sec. 5, 5th par., rewritten

"We were able to compute bootstrap data for the conformal dimensions of higher-order
fields in 4 > d >= 3, including the lowest-lying spinful fields T^prime (l = 2) and C (l = 4), with a
precision comparable to that of spinless operators. The central charge and OPE coefficients
of low-lying fields were obtained with even higher precision than that of the corresponding
anomalous dimensions. The structure constants agree well with those of the 3-correlator
bootstrap, where available (mostly in d = 3), and with perturbation theory for d -> 4."