Physics Letters B 695 (2011) 337–342

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Physics Letters B

www.elsevier.com/locate/physletb

Wilson mass dependence of the overlap topological charge density

Peter J. Moran a,c, Derek B. Leinweber a,∗, J.B. Zhang a,b

a Special Research Center for the Subatomic Structure of Matter (CSSM) and Department of Physics, University of Adelaide 5005, Australiab ZIMP and Department of Physics, Zhejiang University, Hangzhou 310027, People’s Republic of Chinac CSIRO Mathematics, Informatics and Statistics, Private Bag 33, Clayton South, VIC 3169, Australia

a r t i c l e i n f o a b s t r a c t

Article history:Received 6 July 2010Received in revised form 17 October 2010Accepted 2 November 2010Available online 6 November 2010Editor: G.F. Giudice

Keywords:OverlapFat-link fermionsStout-linkSmearingTopological charge densityTopologyVacuum structure

The dependence of the overlap Dirac operator on the Wilson-mass regulator parameter is studied throughcalculations of the overlap topological charge densities at a variety of Wilson-mass values, using aLüscher–Weisz gauge action. In this formulation, the Wilson-mass is used in the negative mass regionand acts as a regulator governing the scale at which the Dirac operator is sensitive to topological aspectsof the gauge field. We observe a clear dependence on the value of the Wilson-mass and demonstratehow these values can be calibrated against a finite number of stout-link smearing sweeps. The overlaptopological charge density is also computed using a pre-smeared gauge field for the input kernel. Weshow how applying the overlap operator leads to further filtering of the gauge field. The results suggestthat the freedom typically associated with smearing algorithms, through the variable number of sweeps,also exists in the overlap operator, through the variable Wilson-mass parameter.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

The topological structure of the QCD vacuum has been the sub-ject of many lattice investigations over the years. Local patternsin topological charge fluctuations represent a significant aspect ofthis structure. Moreover, important physical phenomena such as alarge η′ mass, θ dependence, and possibly spontaneous chiral sym-metry breaking are directly related to vacuum fluctuations of thetopological charge. By the axial anomaly, matrix elements or cor-relation functions involving the topological charge density operatorq(x) can be related to relevant quantities of hadronic phenomenol-ogy.

Lattice QCD enables non-perturbative studies of the strong in-teraction from first principles, and should prove useful for study-ing the important topological structure of the vacuum. Unfortu-nately, obtaining a lattice discretization for studying topology isnot completely straightforward, e.g. naively discretizing the topo-logical charge density generally leads to non-integer values for thetopological charge. Physical hadronic interactions also observe an

* Corresponding authorE-mail addresses: peter.moran@alumni.adelaide.edu.au (P.J. Moran),

derek.leinweber@adelaide.edu.au (D.B. Leinweber), jbzhang08@zju.edu.cn(J.B. Zhang).

0370-2693/$ – see front matter © 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.physletb.2010.11.005

approximate chiral symmetry that is described by the theory ofQCD, where in the massless limit, an exact chiral symmetry is re-alized. Unfortunately, naive transcriptions of the continuum theoryexplicitly break chiral symmetry at finite lattice spacing a.

The Wilson–Dirac operator [1],

DW =∑μ

(γμ∇μ − 1

2ra�μ + m

), (1)

contains the irrelevant Wilson term, r�μ/2, that explicitly breakschiral symmetry at O(a) in order to remove fermion doublers. Thislattice discretization is often improved through the introduction ofa clover term [2], however issues with chiral symmetry breakingstill exist.

One technique that has recently been used to successfully re-produce the light hadron spectrum [3], is to filter the gauge linksprior to applying the Dirac operator. These types of fermion ac-tions are typically referred to as UV-filtered or fat-link actions. Theterm “fat-link” comes from the smeared, i.e. fat, links that are usedto construct the Dirac operator. One can smear either all links[4–8], only the irrelevant terms [9–12], or even just the relevantterms [13]. Incorporating at least some amount of UV-filtering hasbeen shown to reduce the effects of chiral symmetry breaking [4,7,14,11,12,15]. Unfortunately, there is no firm prescription for de-

338 P.J. Moran et al. / Physics Letters B 695 (2011) 337–342

termining the correct amount of smearing to apply to the gaugebackground. One must find a balance between speeding up con-vergence of the Dirac operator, reducing chiral symmetry breakingeffects, and removing short-distance physics from the gauge field.Of course, when using a fixed number of smearing sweeps nsw ,with a constant smearing parameter α, the smearing procedureonly introduces irrelevant terms to the action. The fat-link actiontherefore remains in the same universality class of QCD. Neverthe-less, this freedom, in the number of smearing sweeps that can beapplied to the gauge field, can sometimes be regarded as a draw-back to fat-link fermion actions.

The difficulties with implementing exact chiral symmetry onthe lattice are summarized by the well-known Nielsen–Ninomiyano-go theorem [16]. The no-go theorem forbids the existence of alocal lattice Dirac operator, with exact chiral symmetry, and is freeof doublers. However, in 1982, Ginsparg and Wilson [17] showedthat the physical effects of chiral symmetry will be preserved ifone can find a lattice Dirac operator, D , satisfying the Ginsparg–Wilson relation,

Dγ5 + γ5 D = aD Rγ5 D, (2)

where R is a local operator. Lüscher later showed [18] that any D ,which is a solution of (2), obeys an exact chiral symmetry. A popu-lar solution to the Ginsparg–Wilson relation is the Neuberger Diracoperator [19,20],

D = m

a

(1 + DW (−m)√

D†W (−m)DW (−m)

), (3)

which satisfies Eq. (2) with R = 1/m. Here we consider the stan-dard choice of input kernel, D w(−m), the Wilson Dirac operatorwith a negative Wilson-mass term. To produce an acceptable Diracoperator m must lie in the range 0 < m < 2. For m < 0 there areno massless fermions, while for m > 2 doublers appear [21]. Vary-ing the choice of m within the allowed range results in a flow ofD-eigenvalues, and facilitates a scale-dependent fermionic probeof the gauge field [20]. Any value of m in the range (0,2) shouldyield the same continuum behavior [22,23]. However, simulationsare performed at a finite lattice spacing a, and empirical studiesprefer m � 0.9 [24].

The overlap Dirac operator is extremely useful for studies ofQCD vacuum structure because it satisfies the Atiyah–Singer in-dex theorem, and will always give an exact integer topologicalcharge. However, the value is not always unique and depends onthe value of the Wilson-mass parameter [19,24–26]. Studies of thetopological susceptibility χ = 〈Q 2〉/V , have also observed this de-pendence [24,27]. In particular, the study of Ref. [27] found thatχ varied with m for small values of β , but that this dependencedecreased as the continuum limit was approached.

In the following, we extend these previous studies to include ananalysis of the topological charge density q(x), Q ≡ ∫

d4x q(x), asm is varied. In performing an analysis of the topological chargedensity, rather than χ , we have access to a greater amount ofinformation than that which is learnt from the susceptibility.A change in χ can be due to a change in the mean-square ofthe topological charge 〈q2(x)〉, or to a more fundamental shift inthe long-range structure of the vacuum. As such, it is not possibleto understand the underlying change in the topological structurefrom a calculation of χ .

A calculation of the topological charge density is also a usefulprobe of the gauge field, due to its strong correlation with low-lying modes of the Dirac operator [28,29], which strongly influencehow quarks propagate through the vacuum. Also, while our focusis on the topological charge density, all hadronic observables onthe lattice are impacted as we are examining the properties of a

lattice fermion action. In recent years, the available compute re-sources and algorithm enhancements have reached a point wherecalculations of q(x) using the overlap operator have become feasi-ble [29–31].

We visualize the topological density as this is currently themost effective way to view the extra information. Our analysiswill focus on a comparison between the gluonic topological chargedensity that is calculated following the application of a smearingalgorithm (see Section 3). Here our decision is motivated by thegrowing relevance of fat-link fermion actions. By studying differ-ent smearings, we are also able to provide a direct quantitative linkto the negative-mass Wilson renormalization parameter of overlapfermions. We gain useful insights into the similarities and differ-ences between these smeared actions and the overlap action, andtheir relative effectiveness for studies of topological vacuum struc-ture. A central conclusion of this study, is that the “smoothness”of the gauge field, as seen by the overlap operator, depends on thevalue of the Wilson-mass parameter

2. Simulation details

Due to the high computational effort involved in a full cal-culation of the overlap topological charge density, we consider asingle slice of representative 163 × 32 lattice configurations. Theconfigurations were generated using a tadpole improved, plaquetteplus rectangle (Lüscher and Weisz [32]) gauge action through thepseudo-heat-bath algorithm, with β = 4.80 giving a lattice spacingof a = 0.093 fm.

Five values of the Wilson-mass in the range (1,2) are used tocalculate the overlap topological charge density,

qov(x) = − tr

(γ5

(1 − a

2mD

)). (4)

Results are reported in terms of the input parameter κ , which attree level is related to m by

κ = 1

2(−m)a + 8r, (5)

with the standard choice r = 1. Note that the allowed range forκ is 1/8 < κ < 1/4, and in the interacting theory renormalizationleads one to consider 1/6 � κ < 1/4. A single calculation of qov(x)for one time-slice will contain 162 × 32 = 8192 sites of informa-tion that must be analyzed, and this most easily achieved throughdirect visualizations. In all figures, we represent regions of posi-tive topological charge density by the color red fading to yellow,for large to small qov(x) respectively. Similarly, regions of negativetopological charge are colored blue fading to green. A cutoff is ap-plied to the topological charge density, below which no charge isrendered. This allows one to observe the underlying structure ofthe field.

3. Dependence on the Wilson-mass parameter

The topological charge densities, for the five choices of κ , arepresented in Fig. 1. A clear dependence on κ is apparent from thefigures, with larger values of κ revealing greater amounts of topo-logical charge. This is consistent with expectations since as κ isincreased the Dirac operator becomes more sensitive to smallertopological objects. When using smaller values of κ these objectswill not be felt by the Dirac operator.

The removal of non-trivial topological objects as κ is decreased,bears a striking resemblance to the well-tested cooling [33–37,26]and smearing [38–42] algorithms. In these procedures, the links onthe lattice are systematically updated such that the gauge field is

P.J. Moran et al. / Physics Letters B 695 (2011) 337–342 339

Fig. 1. The overlap topological charge density qov (x) calculated with five choices for the Wilson hopping parameter, κ . Positive regions of topological charge are colored redto yellow, and negative regions are shown as blue to green. From left to right, we have κ = 0.23, 0.21, and 0.19 on the first row, with 0.18, and 0.17 on the second. There isa clear dependence on the value of κ used, with larger values revealing a greater amount of topological charge density. (For interpretation of the references to color in thisfigure, the reader is referred to the web version of this Letter.)

driven towards a more classical state. This results in a removal oftopological charge density, as the action is decreased.

The over-improved stout-link smearing algorithm [43] is a mod-ification of the original stout-link algorithm [42]. Instead of thestandard single plaquette, a combination of plaquettes and rectan-gles are used, with the ratio between the two tuned to preservetopology. In every sweep through the lattice, all links are replacedby the smeared links Uμ(x) [42]

Uμ(x) = exp(i Q μ(x)

)Uμ(x), (6)

with

Q μ(x) = i

2

(Ω

†μ(x) − Ωμ(x)

) − i

6Tr

(Ω

†μ(x) − Ωμ(x)

), (7)

and

Ωμ(x) =( ∑

νν �=μ

ρΣ†μν(x)

)U †

μ(x), (8)

where Σμν(x) denotes the sum of the plaquette and rectangularstaples touching Uμ(x) which reside in the μ–ν plane. The ratio ofplaquette to rectangular staples is controlled by a new parameterε [43]. In the following we use the suggested value of ε = −0.25,which has yielded good results in other studies [29,44]. For thesmearing parameter we select a relatively weak value of ρ = 0.01.This should be compared with the maximum value possible forthis combination of plaquettes and rectangles, ρ ≈ 0.06. Whilst inthe standard stout-link smearing algorithm, 0.1 is the commonlyused value. After smearing, the gluonic topological charge densitycan be calculated,

qsm(x) = g2

32π2εμνρσ F ab

μν(x)F baρσ (x). (9)

In order to fairly compare the two definitions for the topolog-ical charge density one usually applies a multiplicative renormal-ization to the gluonic qsm(x) [29],

qsm(x) → Zqsm(x). (10)

This is because after a relatively small amount of smearing the to-tal gluonic topological charge is typically non-integer valued dueto the presence of quantum field renormalizations. By matchingto the overlap topological charge density we can alleviate thisbias.

For this study we have a single slice of the topological chargedensity and thus cannot match the total topological charge. Insteadthe renormalization factor is chosen such that the structure of thetwo field densities can be best compared. The best match to theoverlap qov(x) is then found by calculating,

min∑

x

(qov(x) − Zqsm(x)

)2, (11)

as the number of smearing sweeps is varied. Two methods for cal-culating Z are considered;

• Zcalc ≡ ∑x |qov(x)|/∑

x |qsm(x)|,• Zfit, where the renormalization factor is calculated such

that (11) is minimized.

The first definition is motived by our aim of comparing the struc-ture of the two field densities. The second choice was consideredto see if the matching could be improved beyond the first defini-tion. We also compare with an alternative matching procedure [45,46] in which one calculates,

ΞAB = χ2AB

χA AχB B, (12)

with

χAB = (1/V )∑

x

(qA(x) − qA

)(qB(x) − qB

), (13)

where q denotes the mean value of q(x), and in our case qA(x) ≡qov(x), qB(x) ≡ qsm(x). Here the best match is found when ΞAB isnearest 1. In this case, the ratio eliminates any dependence on therenormalization factor, Z .

340 P.J. Moran et al. / Physics Letters B 695 (2011) 337–342

Fig. 2. The best smeared matches (right) compared with the overlap topological charge densities (left) in order of decreasing κ , where qsm(x) is renormalized using Zcalc .Positive regions of topological charge are colored red to yellow, and negative regions are shown as blue to green. There is a clear relationship between κ and nsw , withsmaller κ values requiring a greater number of smearing sweeps to reproduce the topological charge density. (For interpretation of the references to color in this figure, thereader is referred to the web version of this Letter.)

We first consider Zcalc. The overlap topological charge densities,along with the corresponding best matches, for three choices of κare shown in Fig. 2. We see that as κ is decreased, and non-trivialtopological charge fluctuations are removed, a greater number ofsmearing sweeps are needed in order to recreate the topologi-cal charge density. Again this agrees with expectations since theoverlap operator becomes less sensitive to small objects as κ isdecreased, and it is these objects that are removed by the smear-ing algorithm. Comparing the different definitions in Fig. 2 showsgood agreement in the topological structures revealed.

The two methods for calculating the renormalization con-stant Z , together with the values for Ξ , are compared in Table 1.As we move down the table there is a monotonically increasingtrend in the number of sweeps required to match the value of κ .We note that despite some minor variation in nsw , it is possible tocorrelate the number of sweeps to the value of the Wilson hop-ping parameter. We note that the average renormalization factor

Table 1The number of smearing sweeps, nsw , needed to match the overlap topologicalcharge density calculated with the listed value of κ . The three methods used tofind the best match are detailed in the text.

κ nsw Zcalc nsw Zfit nsw ΞAB

0.17 28 0.56 29 0.47 29 0.76

0.18 26 0.70 27 0.61 27 0.78

0.19 25 0.82 25 0.68 25 0.77

0.21 23 0.91 23 0.76 23 0.75

0.23 22 0.89 23 0.76 23 0.73

Z ∼ 0.7, reflecting the fact that with ρ = 0.01 the gauge fieldsremain rough after ∼ 25 sweeps of smearing. The value for Ξ re-mains approximately constant around ∼ 0.75, suggesting that afterrenormalizing the level of agreement between the smeared topo-logical charge density and the overlap density is consistent.

P.J. Moran et al. / Physics Letters B 695 (2011) 337–342 341

Fig. 3. A comparison of the overlap topological charge density qov (x) computed using κ = 0.19 (left), with qUVov (x) calculated using the same κ , on the same configuration, after

first applying 25 sweeps of smearing (right). Positive regions of topological charge are colored yellow, and negative regions are shown as blue to green. (For interpretation ofthe references to color in this figure, the reader is referred to the web version of this Letter.)

Fig. 4. The overlap charge density calculated on a configuration filtered by 25 stout-link smearing sweeps, compared with qsm(x) after 45 sweeps of smearing. Positive regionsof topological charge are colored yellow, and negative regions are shown as blue to green. There is a strong correlation between the objects observed. It appears as thoughthe overlap operator has again “smoothed” the configuration. (For interpretation of the references to color in this figure, the reader is referred to the web version of thisLetter.)

4. UV-filtered overlap

Let us now consider the effect of evaluating the overlap op-erator on a pre-smeared gauge field. This is of some relevance toUV-filtered overlap actions [15,47–49], in which all links of a gaugefield are smeared prior to applying the overlap operator. As alreadyseen in Fig. 2, applying the overlap operator is in some respectssimilar to smearing the gauge field. Of interest here is whetherthe overlap operator, acting on a smeared gauge field, will reveal atopological charge density close to the input smeared gauge field,or whether further smearing will be needed to match the calcu-lated qov(x).

To make comparisons clear, we denote the overlap topologicalcharge density, calculated using a smeared configuration as input,by qUV

ov (x). We consider the third Wilson-mass, where κ = 0.19and the best smeared match was provided by nsw = 25. Fig. 3shows the original qov(x) along with the new UV-filtered qUV

ov (x).Far less topological charge density is observed in the pre-filteredcase. Given the previous results, it is clear that a far greater num-ber of smearing sweeps will be required to reproduce q(x) usingthe gluonic definitions.

Repeating the same calculation as before we find that 45sweeps of over-improved stout-link smearing provides the bestmatch to the overlap topological charge density. A comparison be-tween qUV

ov (x) and the smeared qsm(x) is shown in Fig. 4, whereZcalc = 0.85. This is approximately double the original 25 sweepsrequired to match the overlap topological charge density, onceagain revealing the smoothing aspect of the overlap operator.These results indicate that the filtering that occurs in the overlapoperator is independent of the input gauge field.

5. Conclusion

Using direct visualizations of the topological charge density, wehave analyzed the dependence of the overlap Dirac operator onthe Wilson-mass regulator parameter m. As was hinted at by pre-vious studies of the topological susceptibility [24,27], systematicdifferences appear in the topological structure of the gauge fieldas m is varied. By comparing qov (x) with the gluonic definition ofthe topological charge density, resolved with a topologically sta-ble smearing algorithm, a direct correlation between m and thenumber of sweeps is revealed. Smaller values of κ reveal topolog-ical charge densities that are similar to using a greater number ofsmearing sweeps.

From these observations, one can conclude that the “smooth-ness” of the gauge field, as seen by the overlap operator, dependson the value of the Wilson-mass parameter. This is similar to fat-link fermion actions in which the smoothness is directly dependentupon the number of applied smearing sweeps. These results in-dicate that the freedom typically associated with fat-link fermionactions, through the number of smearing sweeps, is also presentin the overlap formalism, through the freedom in the Wilson-massparameter.

We also considered the application of the overlap operator to asmeared gauge field, which is of relevance to UV-filtered overlapactions. We demonstrated that, regardless of the input gauge fieldto the overlap operator, UV-filtering still occurs via the overlap op-erator. The strength of the filtering is of a comparable strength tothat of the overlap acting on a hot, unfiltered configuration. Whencreating a UV-filtered overlap action, one must therefore take careto preserve the short-distance physics of the gauge field.

342 P.J. Moran et al. / Physics Letters B 695 (2011) 337–342

The topological charge density revealed by the overlap opera-tor is similar to that revealed after 20 to 30 sweeps of stout-linksmearing with smearing parameter ρ = 0.01, or 2 to 3 sweeps atthe standard value of ρ = 0.1. In this light, it is important to con-tinue investigations into the extent to which the properties andphenomenology of the overlap operator can be obtained throughthe use of an efficient Wilson-clover action on smeared configura-tions.

Future work could also include gauge configurations generateddirectly using the overlap Dirac operator, or possibly with an al-ternate overlap definition based on staggered fermions [50], whichmay prove more computationally efficient than the usual Wilson-based overlap operator.

Acknowledgements

This research was undertaken on the NCI National Facility inCanberra, Australia, which is supported by the Australian Com-monwealth Government. We also acknowledge eResearch SA forgenerous grants of supercomputing time which have enabled thisproject. This research is supported by the Australian ResearchCouncil. J.B. Zhang is partly supported by Chinese NSFC-GrantNos. 10675101 and 10835002.

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