A Possible Resolution to Troubles of SU(2) Center Vortex Detection in Smooth Lattice Configurations

Round 1

Reviewer 1 Report

    I think this article addresses a topic of interest to those who may be interested in identifying center vortex configurations in lattices generated by lattice Monte Carlo, where it is known that there is an undercount of vortices when the configurations are cooled.  An interesting table is presented at the end of the paper.  This table gives a range of couplings beta, at various lattice extensions and cooling steps, in which the vortex estimate of the string tension will show increasingly good agreement with the asymptotic result, as beta is increased within that range.

      However, the results are obscured by the presentation.  The main problem is that the article leans heavily on the concept of center regions, and the identification of such regions, but this concept  and the method of identification is nowhere defined in the article, and without understanding the general idea, the article makes no sense.  In connection with center regions, the authors only refer back to their previous work [15-17].  I think this is very unfair to the reader, who cannot be expected to have read those articles, or to go back and read them to understand this article.  Some exposition regarding center regions is required, and if this means repeating some material found in those previous articles, so be it.   Also, on page 5 the authors state an estimate in fm of where finite size effects will be expected, but it is unclear to me exactly how they arrive at this estimate.   

    Finally, I am puzzled why the authors restrict themselves to such tiny lattices, with extensions of only 8 and 10 lattice spacings.  This was perhaps the state of the art of lattice Monte Carlo simulations in the 1980's, but with a bit of patience it is not a problem to go up to, say, 24 or even 30 lattice spacings with even a modest modern workstation.  And that brings up the obvious question:  by going to larger lattices, the authors ought to be able to verify some of the predictions they have made in Table 5.  Why have they not done so?

    Before I can clear this article for publication, I think the authors ought to go back and address the issues just mentioned.

 

Author Response

Dear reviewer,

Regarding your criticism concerning missing details about non-trivial center regions, where we refer to our previous publications only: it is difficult to harmonize your request with the requirements of the editorial board to prevent redundancies between publications. The open access policy implies, that the relevant open access references are accessible with a single click. This effort can certainly be demanded from an interested reader. Despite that we managed to add an additional page in which we give the fundamental details about the algorithms relevant for the detection of non-trivial center regions. 
Details regarding the estimated size at which we expect finite size effects are now given in several lines on page 6.

The tiny lattices we use are justified by two aspects:

1.: We are looking at finite-size effects. With bigger lattices they occur at larger \beta, but with larger \beta the vortex detection gets increasingly difficult. We strive to clarify the causes for these difficulties. This can only be done starting from a situation where the vortex detection works and proceeding to situations, where problems arise. 

2.: We use data acquired from previous simulations instead of producing new configurations. As the gauge fixing procedure guided by non-trivial center regions is quite resource intensive with respect to computation, it would take more time to generate new configurations than we can afford at the moment.    

It would be indeed interesting to see, whether a full or partly restoration of the string tension occurs for sufficiently large lattices and sufficiently large values of \beta. With our computational resources we could, in theory, strive for lattice sizes up to about 26^4 before the implementations of our algorithms become troublesome. We know that our method of vortex detection gets difficult at higher values of \beta, hence we would need to implement first some improvements before we can run simulations for verification at noticeably larger lattice sizes.

With best regards,
Rudolf Golubich and Manfried Faber

Reviewer 2 Report

`The present work is devoted to study of the center projection in SU(2) gluodynamics.
A particular problem of the center projection after cooling is addressed. It is known
that the center projection is problematic after cooling since projected string tension is too low.
The authors undertake careful study of the center vortex properties after cooling
and obtain important results which should help to solve this problem.
Below are my suggestion/questions concerning improvement of the presentation.

 

p.2 
inverse coupling beta covering an interval from beta = 2.1 to beta = 3.6

My comment: 

1) It is known that the range  beta < 2.2 is the strong coupling range and is not usually 
used in simulations. It is necessary to explain this fact and also to explain the motivation to
use it in this work.
2) It is clear that finite size effects on the lattices 8^4 and 10^4 considered in this work
are large at, say, beta > 2.6. It is then necessary to explain why much larger beta values 
are needed to be studied in this work.

p. 2 
and complemented by an extrapolation according to the asymptotic renormalization group equation

My comment: 
1)Please indicate beta values  range where eq. (1) was used to compute the string tension
2) Please show the source for the Lambda value used in eq.(1).

p.2 
The analysis is performed on lattices of size 84 and 104

My comment: 
The lattices used these days in the studies of SU(2) LGT are much larger than 10^4. It is necessary
to explain why so small lattices are used in this work.

p. 3
The string tension s is determined via Creutz ratios 

My comment: 
1)It is normal to compute the static potential V(R) and then to extract the string tension from it.
Please, explain why the Creutz ratio is used in this work instead.
2) It is necessary to provide references where it was proved that T=1, 2 is enough to obtain
the T \to \infty limit for the Creutz ratios after center projection. 

My comment: 
p. 4
For different lattice sizes a sudden decrease of the string tension occurs at
different values of beta. The respective beta-values are compatible for different numbers of
cooling steps.

My comment: There are only two lattice sizes. The respective beta values should be given for both
lattices. This will make it easier to understand both Fig.4 and its discussion.

p.4
We do not observe a dependency on the lattice size in the low beta-regime

My comments: 1) Please specify 'low beta-regime'. 
2) The simple explanation of the lattice size independence at low beta might be that even 8^4 
lattice is large enough. Please comment on this possibility.

p.5 
Based on the deviations of the string tensions for different lattice sizes we expect finite 
size effects to occur at length scales between 1.1 fm and 1.3 fm, independent of cooling.

My comment: 
1)Please provide more details on how the length scale was determined (beta and lattice size
values). 
2) 1.1 fm for 8^4 lattice and 1.3 fm for 10^4 lattice imply beta values between 2.3 and 2.4. This 
rises a question about relevance of data on these lattices at much higher beta, i.e. with much smaller
lattice size.

p.5
With an increased number of cooling steps the compatibility of different lattice sizes gets 
reduced but it is still given for sufficiently low numbers of cooling steps.

My comment: 
1) The explicit number of cooling steps should be given. e.g. at N_cool up to 5 the two lattices results
are compatible, at N_cool=10 they are not compatible.
2) Fig.6 indicates that results for two lattices are not compatible even for N_cool=0. Please comment
on this.

 
p.6
We restrict the further analysis to the lattice of size 10^4 with beta <= 2.3 and at most 5 cooling 
steps.

My comment: 
From the very beginning it is not clear why the authors introduce some data for extremely large beta 
while the lattice size is rather small. I recommend either to explain why data for very high beta
(say, beta > 2.7) are necessary or remove them completely.
 

p.6
lattice constant

My comment: 
Should be 'lattice spacing', I think

p.7
depicted in Figure 9 for average cross-sections.

My comment: should be 'Figure 8'

p. 7-9

In Tables 2, 3 and 4 the parameters t-statistic and p-value are presented. 

My comment: In the lattice literature one usually uses a standard error and chi^2/N_dof to 
verify the quality of the fit. I suggest to follow this tradition or to explain in the text 
why  t-statistic and p-value are more preferable in the given work.
p. 9
Depicted is the vortex density

My comment: English should be corrected in this and similar sentences.

p. 7-9
In Figures 8-10 the data sets are present without explicit information about respective 
beta values. 

My comment: The information on beta values should be provided in one way or another. 

p. 9

In eq. (12) d_center(0) and A_max(0) are used.

My comment: Is this correct? Why?

p. 10
From this we derive two relevant limits for the lattice spacing a and the lattice extent L

My comment: This statement looks strange. It seems to me that the restriction for L is not 
related to s_flux.

p.10 
Assuming a vanishing minimal flux tube size

My comment: On the lattice the minimal flux tube size is equal to a.

p. 10

In eqs. (14) and (15)  the numerical constants are given without errors.

My comment: The errors should be added. The explanation of the respective fitting procedure
should be added, in particular the fitting range should be specified.

 
p.10
 The upper limit of a for N_cool=0 is not clear from Figure 11(left)
 
My comment: 
I suggest to change the Y-axes tics to make it possible to estimate the upper limit of a 
for N_cool=0.

p. 11
The original direct maximal center gauge minimizes a gauge functional which takes only
those eight links into account that are connected to the respective lattice point. By
taking also those links into account that are connected to neighbouring lattice points the
troubles arising from the spread of the center flux could be counteracted.

My comment: 
This statement should be formulated more clearly. The gauge functional used in the center 
gauge already is a sum over all lattice sites.  

Author Response

Dear reviewer,

Thank you for the very detailed and constructive feedback!

p.2 
inverse coupling beta and lattice size 

We have inserted a sentence at the beginning of Materials and Methods, where we added arguments for the small lattice sizes and the \beta-range: we are interested in finite size effects and an increasing lattice size would push them towards higher values of \beta where the vortex detection gets difficult. We carry the higher values of \beta along the ride to allow a check of the quality of our model and to see, how far we can go until deviations arise. As we are also interested in discretization effects, we have to start at low values of \beta.

Eq. (1) is now complemented with the details concerning the value of \Lambda: it results from a fit to the values from Table 1 with \beta >= 2.6.

p.3
Creutz ratios and string tension

In previous works concerning center vortices it can be seen that the Creutz ratios taken from small loop sizes already give a good estimate of the string tension - there are only tiny variations of the Creutz ratios with increasing loop size. We have listed the corresponding references.
We took the data resulting from a previous study where we did not store the required Wilson-loop data to extract the static potential, but only those, needed for the Creutz ratios. As our algorithms are quite computation intensive, it would take more time to redo the calculations than we can afford at the moment and we would not learn anything new from it.

p.4
For different lattice sizes a sudden decrease of the string tension occurs

The sudden decrease of the string tension occurs at \beta ~2.3 for the lattice of extent 8 and between 2.35 and 2.4 for the lattice of extent 10. "low beta regime" just says "small enough to restrain from these effects". We have complemented the text with some more details about that.

p.4
We do not observe a dependency on the lattice size

Regarding the possibility of the 8-lattice being big enough: it might be on the verge. Our assumptions of circular-cross sections is an approximation that might over- or underestimate the vortex thickness,  resulting in an over or underestimation of the limits especially when no cooling is applied.  Our final results for the lattice of extent 8 restrict \beta to an interval from 2.12 to 2.32, corresponding to an extent of about 2.15 fm and 1.24 fm.

p.5

A short discussion of the length scales is now added, listing the respective \beta-values. The higher values of \beta are mainly of interest to check, how long the model gives plausible results and see, where and which deviations occur. We have added this argument on page 6 now.

Concerning the compatibility of the different lattice sizes at the maximal cross-sections, you are right: strictly spoken, they are not compatible in the same sense as the average cross-sections are. This is now corrected: of importance is, that the behaviour with respect to cooling is qualitatively reproduced, which allows to determine another estimate of the growth rate g_cool. Thanks a lot for the important comment!

p.6

As from the beginning we do not know, how good the model describes the data, we carry along the higher values of \beta, which allows a check of plausibility. In fact, we observe that our model reproduces the data until quite large values of \beta.  

"lattice constant" was indeed a mistake and your suggestion is correct, also on the next page the figure number was wrong, as you realized.

p.7-9
In the tables the standard error is given in brackets, t-statistics and p-value are what many statistic programs and Mathematica, which came to use, do usually return. The more relevant measure of quality is given by the compatibility outside the fit-range: here, the higher values of \beta jump in. 

In the figures you mentioned we have now listed the \beta-values for all data.

Concerning Eq. (12): Yes, the 0 is correct. With an N_cool we would look at the average distance between piercing when already a loss of the vortex density is caused by the cooling. We admit, that the equation would look more elegant with symmetric parameters. We have added additional explanations:
"We use $d_\text{center}(0)$, the average distance between piercings when no loss of the vortex density occurred, and subtract the average diameter of the flux tubes $d_\text{flux}(N_\text{cool})$ with cooling applied."

p.9
Depicted is the vortex density
-> we have corrected this to "The vortex density is depicted". Similar changes have been made in several places. 

p.10
Regarding the two limits presented in Eq. (13): the Limit on L is indeed independent of s_flux, it is in fact a variant of the limit given in Eq. (6). Our wording mistakenly suggested, that both limits are based on s_flux. We have corrected the wording.

p.10 
Assuming a vanishing minimal flux tube size

The assumption of a vanishing minimal flux tube size is just an approximation justified by the dominance of the larger cross-sections. We have clarified that in the text now. 

In Eq. 14 and 15 we now show also the errors of the fit and mention the \beta-interval for the fit.

In Fig. 11 (now Fig. 12) the limit without cooling is now explicitly marked by an additional tick.

p.11
Concerning the DMCG we modified the explanation of our suggested improvement:
"In the original direct maximal center gauge the contribution to the gauge functional at a given site x is determined by its attached links only. By taking farther links into account the troubles arising from the spread of the center flux may be counteracted."

We thank for the constructive feedback and remain

with best regards,
Rudolf Golubich and Manfried Faber

 

Reviewer 3 Report

The presented article is to be placed in the research field which is already long-standing and of great importance in Physics as a whole: understanding of the mechanism of quark confinement. At the same time, the manuscript itself discusses studies which can be considered rather technical and apparently targeted at the readers pursuing the same or a very closely related topic. It is therefore understandable, that the Authors restrained from defining all of the concepts which they are using (but suggestions of some inclusions are listed below.) That said, the presentation of the material and the line of argumentation is very clear and well structured. Worth mentioning is also the style of the writing which is correct, clear and concise. The Authors may consider including the following suggestions, which may bring mainly further improvement to the style of the paper:

1. Line 10: To change "Center", "Closed" to "center", "closed".
2. Lines 19-20: "the path integral" - to say more specifically in a few words which path integral, calculation of what, using path integral.
3. Line 27: Defining of "non-trivial center regions" still would be helpful.
4. Lines 31-32: For completeness it would be also good to define the meaning of the notation "P-".
5. Line 60: Full period symbol is doubled.
6. Lines 79-81: Is it possible to provide some quantification of the statement, especially as it is mentioned again in the Discussion section? It is difficult to see this from the (quite crowded) figure.
7. Line 103: To change "given" to "present".
8. Line 104: To change "Taking a look" to "Looking" (to avoid repetition.)
9. Line 197: To change "guaranty" to "guarantee".
10. For completeness, perhaps at the beginning of Section 2 the reader should be referred e.g. to Ref. [3] for the definitions of relevant concepts and vocabulary.
11. If the right part of Fig. 1 is to be shown at all, then it is suggested to discuss it in more detail. It is not clear just from the graphics, without some additional text what is the "mechanics" of the figure: what is the relation between the "lattice plaquette" (?) in the middle and the four cubes around? What is the relation between the cubes themselves? Does the dot always symbolise the same point in the lattice? What is symbolised by the straight lines joining two of the cubes with the ellipsoidal surface? It is also not clear what is the connection between the feature which the right part of the figure shows and the studies presented in the paper.
12. For Figures 3-7: To try splitting each them into two adjacent panels sharing the common scale and axis, perhaps one of the panels for L=8 and another one for L=10. Perhaps this way it would be easier to look at the figures.
13. Figure 3: Is it correctly understood that for L=8 and 10 cooling steps there are only two points? Would the statement in lines 62-63 more convincing if at least one more point was shown?
14. Still Figure 3: Is the point for L=10, 10 cooling steps and beta=25.5 missing? Is this for the reason discussed in the caption (logarithmic scale?) If so, why this particular point jumps so suddenly.
15. Figures 8-10: Is it possible to display the values of beta and a for other sets of points?
16. Figure 11: Is it possible, besides the explanation in the figure caption, to include some legend showing where the weaker and stronger limits come from?

Author Response

Dear reviewer,

Thank you for the detailed and constructive feedback! The improvements concerning wording, you suggested in comments 1,5,7,8 and 9 are implemented, to your further comments will be referred to now:

Ad 2) We have changed to: "They also point out that the thickness of vortices is of importance for the extraction of  properties related to confinement."

Ad 3 and 4) We now provide more detailed information regarding non-trivial center regions and how they are detected from pages 2 to 4. A new Figure 2 depicts the relevant procedure. The relation between “thick vortices”, “P-vortices” and “center regions” is shown in a new Figure 3.

Ad 6) We inserted a quantification of the improvement concerning the string tension: "the $10^4$-lattice starts at 10 cooling steps with an underestimation of the asymptotic string tension of $50\% \pm 1\%$ at $\beta=2.1$, improving to $40.8\% \pm 0.4\%$ at $\beta=2.25$."

Ad 11) We have complemented the caption of Figure 1 by more details regarding the “mechanics”. The closed surface resulting from the flux line evolving in time (related to the four cubes) leads us to one of the final limits derived in our work.

Ad 12) We thought about separating the different lattice sizes, but refrained from this: if we show separate figures for the two lattice sizes, deviations could not be seen that easy. For the “Average flux tube size” separating the sizes would even result in one over-crowded figure becoming two over-crowded figures because the two sizes give mostly overlapping lines there. The figures are mainly of interest for the selection of that part of the data that is used for the further model-construction.

Ad 13 and 14) Yes, you are right: the missing point in L=10 with 10 cooling steps at \beta=2.55 results from the logarithmic scale. At this \beta no configurations with less P-plaquettes than non-trivial center regions have been found. At lower \beta all lines start at 0%. Of interest is, at which values of \beta the lines start to deviate from 0%. Already a minor deviation from 0% is of relevance since a good vortex detection should find a lot more P-plaquettes than non-trivial center regions have been used for guidance. We have added more details on that within the figure caption.

Ad 15) We have added the “\beta” and “a” values to all sets of points in the respective figures.

Ad 16) The Figure displaying the limits now comes with an additional legend, showing the respective values of g_cool.

We thank for the constructive feedback and remain

with best regards,
Rudolf Golubich and Manfried Faber

Round 2

Reviewer 1 Report

I am satisfied with the revisions/explanations that the authors have made in response to the remarks in my previous report, and I now recommend publication.