Dislocations under gradient flow and their effect on the renormalized coupling
DDislocations under gradient flow and their effect on the renormalized coupling
Anna Hasenfratz ∗ and Oliver Witzel † Department of Physics, University of Colorado, Boulder, Colorado 80309 (Dated: April 3, 2020)Non-zero topological charge is prohibited in the chiral limit of gauge-fermion systems because anyinstanton would create a zero mode of the Dirac operator. On the lattice, however, the geometric Q geom = (cid:104) F ˜ F (cid:105) / π definition of the topological charge does not necessarily vanish even when thegauge fields are smoothed for example with gradient flow. Small vacuum fluctuations (dislocations)not seen by the fermions may be promoted to instanton-like objects by the gradient flow. Wedemonstrate that these artifacts of the flow cause the gradient flow renormalized gauge couplingto increase and run faster. In step-scaling studies such artifacts contribute a term which increaseswith volume. The usual a/L → β function. I. INTRODUCTION
The net instanton charge Q = n + − n − is a topo-logically protected quantity in continuum gauge-fermionsystems. On the lattice, however, Q is not protectedand different definitions of the topological charge, likethe number of zero modes of the Dirac operator Q ferm = 12 tr( γ D ) , (1)or the geometric definition Q geom = 132 π (cid:90) dx tr( F µν ( x ) ˜ F µν ( x )) (2)may not agree. The latter definition is particularly trou-blesome as the ultraviolet (UV) fluctuations of the gaugefield can dominate Q geom . Smoothing the gauge fieldwith smearing or gradient flow (GF) reduces the problem,but the fate of small instanton-like objects, dislocations,depends on the details. These may grow to topologicalmodes | Q | ≈ Q geom . There is no unique definition of the topologicalcharge on the lattice. Different definitions are expectedto agree only in the continuum limit [1–8].Dynamical configurations with net topological charge Q ferm have | Q ferm | zero modes in the spectrum of themassless Dirac operator assuming the lattice fermions arechirally symmetric [9, 10]. As a consequence, non-zero Q ferm configurations are excluded in the chiral limit. Thisis one of the rare instances where theoretical argumentsrigorously constrain the value of Q even at finite cutoff. ∗ [email protected] † [email protected] Residual chiral symmetry breaking or effects due to the finitevolume could potentially allow configurations with nonzero topo-logical charge. This effect is however negligible.
The fermions restrict Q ferm = 0 and any Q geom (cid:54) = 0 sig-nals a lattice artifact of the smoothing algorithm or theoperator used in the definition of Q geom . Even though itis a lattice artifact, Q geom (cid:54) = 0 can have significant effectson the renormalized gradient flow gauge coupling and thefinite volume step scaling function β c,s ( g c ).The GF gauge coupling at energy scale µ = 1 / √ t is g GF ( t ; L, β ) = N t (cid:104) E (cid:105) where (cid:104) E (cid:105) is the energy density, β = 6 /g is the bare coupling, L refers to the linear sizeof the system, and the normalization factor N is cho-sen to match g MS at one-loop [11–13]. Lattice studiesshow that at large flow time g GF ( t ) exhibits only mild,approximately linear or weaker, dependence on t . There-fore the energy density (cid:104) E (cid:105) decreases ∝ /t or faster.While GF removes vacuum fluctuations and instantonpairs, some instantons can survive the flow and become(quasi-)stable. At large flow time Q geom approaches in-teger values and (cid:104) Q (cid:105) is frequently used to define thelattice topological susceptibility [7, 8, 14, 15]. To simplifythe notation we will from now on refer to Q geom simplyby using Q .The action of a single instanton is S I = 8 π in thecontinuum. On the lattice this value depends on theinstanton size and the lattice action, but at large flowtime smooth instantons increase the energy of the con-figuration by ≈ S I [3]. The net number of instantons Q = n + − n − is expected to scale with the square rootof the number of lattice sites V /a , even if they arisefrom vacuum fluctuations as artifacts of the GF. The in-stanton contribution to the energy density is therefore ∝ / √ a V . If instanton-antiinstanton pairs are present,this contribution is even larger.In step scaling studies the GF flow time is connectedto the lattice size as t = ( cL ) /
8, where the constant c defines the renormalization scheme and g c refers to thegradient flow renormalized coupling at the correspondingflow time g GF ( t ; L, β ). The discrete lattice β function of a r X i v : . [ h e p - l a t ] A p r scale change s is defined as [16] β c,s ( g c ; L, β ) = g c ( sL ; β ) − g c ( L ; β )log s . (3)In volumes V = L the contribution of the instantons tothe discrete β function is ∝ t / √ a V ∝ L /a , β c,s ( g c ; L, β ) = β c,s ( g c ) Q =0 + C ( β, c ) L /a , (4)where β c,s ( g c ) Q =0 is the step scaling function in the Q = 0 sector and C depends on the bare coupling β and the renormalization scheme c but is independent ofthe lattice size L . When the simulations are performedwith chirally symmetric fermions in the chiral limit, theterm C ( β, c ) L /a is a lattice artifact, the consequenceof the GF promoting vacuum fluctuations to topologicalobjects.Even on Q = 0 configurations β c,s ( g c ) Q =0 has cutoffeffects. These are typically removed by an a /L → g c [16–23].If the data does not follow a /L dependence, higher or-der ( a/L ) terms can be included [24, 25]. However, inthe strong coupling limit with a non-negligible instantondensity, Eq. (4) implies that the correct continuum ex-trapolation should include an ( L/a ) term instead or atleast in addition to ( a/L ) . Practically such an L → ∞ extrapolation is not viable. This is a reflection of the non-perturbative nature of instantons and shows that theireffect cannot be removed by perturbatively motivated ex-trapolations. The effect of instanton-like objects in thecontinuum prediction could be substantial, especially inslowly running systems near or within the conformal win-dow where the coupling β = 6 /g barely changes as thecontinuum limit is taken on available lattice volumes. Aclean way to avoid this issue is to choose a flow whereinstantons are not generated even on coarse lattices.In this paper we re-analyze simulations performed atstrong coupling where the gauge fields are rough and dis-locations frequent. Such simulations are necessary to ex-plore the step scaling function of (near) conformal sys-tems. This phenomena might also affect scale setting[12, 26, 27] in strongly coupled beyond the StandardModel systems (see e.g. [28–41]) or even quantum chro-modynamics (QCD) simulations at coarse lattice spac-ings necessary to achieve a large physical box needede.g. to study multi-particle interactions [42].We consider two different systems to illustrate the is-sue. In both cases we study two different gradient flowkernels, Wilson and Symanzik flow. We start with ourrecent 10-flavor SU(3) domain wall simulations where wefirst observed the effect of non-zero topological charge[22]. An accompanying paper discusses the step-scalingfunction of this most likely conformal system and pro-vides further details [43]. Next we analyze configurationsgenerated for an older study of the SU(3) 8-flavor systemwith staggered fermions [18]. We chose these two systemsbecause both simulations have been pushed toward verystrong coupling where the contamination from topolog-ical modes can be significant. Our results demonstrate these lattice artifacts are more severe for Symanzik thanfor Wilson flow. In Section IV we demonstrate how asmall modification of the flow kernel results in a gradientflow that is better at smoothing out local dislocations re-sulting in fewer configurations with nonzero topologicalcharge. The lattice discretization errors of such a modi-fied gradient flow will need to be explored in the future.Finally we briefly summarize our findings. II. SU(3) WITH N f = 10 FLAVORSA. Details of the simulations
In this part of our study we utilize existing gauge fieldconfigurations generated with ten degenerate and mass-less flavors of three times stout-smeared [44] M¨obius do-main wall (DW) fermions [45–47] with Symanzik gaugeaction [48, 49]. The configurations are generated us-ing
Grid [50] and we choose symmetric volumes with V = L where the gauge fields have periodic, the fermionsantiperiodic boundary conditions in all four space-timedirections. The bare input quark mass is zero and forthe domain wall fermions we choose the domain wallheight M = 1 and the extent of the fifth dimension L s = 16. Configurations are generated using the hy-brid Monte Carlo update algorithm [51] choosing trajec-tories of length two molecular time units (MDTU) andwe use configurations saved every 10 MDTU. Our statis-tical data analysis is performed using the Γ-method [52]which estimates and accounts for integrated autocorrela-tion times. For the L/a = 32 ensembles at strong cou-pling considered here autocorrelations range from threeto five measurements.Due to the finite extent of the fifth dimension, DWfermions exhibit a small, residual chiral symmetry break-ing which conventionally is parametrized by an additivemass term am res . We determine am res numerically us-ing the ratio of midpoint-pseudoscalar and pseudoscalar-pseudoscalar correlator. At strong coupling am res de-pends on the bare coupling β and increases from am res =2 × − at β = 4 .
15 to 6 × − at β = 4 .
02. To demon-strate that am res is sufficiently small and not the originof nonzero topological charges, we compare results for β = 4 .
05 from ensembles with L s = 16 and L s = 32below. B. Effects of nonzero topological charge
We illustrate the effects of Q (cid:54) = 0 instanton-like ob-jects on the gradient flow coupling in Fig. 1 where weshow the flow time dependence of the topological charge https://github.com/paboyle/Grid -202 Q trajectory 700 Wilson flowSymanzik flow flow time t/a g G F WWWCSWSC -202 Q trajectory 575 Wilson flowSymanzik flow flow time t/a g G F WWWCSWSC -202 Q trajectory 2965 Wilson flowSymanzik flow flow time t/a g G F WWWCSWSC -202 Q trajectory 2100 Wilson flowSymanzik flow flow time t/a g G F WWWCSWSC -202 Q trajectory 2255 Wilson flowSymanzik flow flow time t/a g G F WWWCSWSC -202 Q trajectory 845 Wilson flowSymanzik flow flow time t/a g G F WWWCSWSC
FIG. 1. The flow time history of the topological charge (upper panels) and the gradient flow coupling (lower panels) on sixselected gauge field configurations of our ten flavor (
L/a ) ensembles with L/a = 32, L s = 16 at bare coupling β ≡ /g = 4 . -404 Q W L=32, L s =16, =4.15c=0.25c=0.30 -404 Q W L=32, L s =16, =4.10c=0.25c=0.30 -404 Q W L=32, L s =16, =4.05c=0.25c=0.30 flow time t/a -404 Q W L=32, L s =16, =4.02c=0.25c=0.30 -404 Q S L=32, L s =16, =4.15c=0.25c=0.30 -404 Q S L=32, L s =16, =4.10c=0.25c=0.30 -404 Q S L=32, L s =16, =4.05c=0.25c=0.30 flow time t/a -404 Q S L=32, L s =16, =4.02c=0.25c=0.30 FIG. 2. Dependence of the topological charge Q on the flow time t/a for ( L/a ) = 32 ensemble with L s = 16 and at barecouplings β = 4 .
15, 4.10, 4.05, and 4.02. Each panel show the flow time histories for the first thermalized 100 configurations ofeach ensemble. The left (right) panels shows the flow time histories using Wilson (Symanzik) gradient flow. Lattice artifactsin the form of nonzero topological charges Q increase in the strong coupling limit (decreasing β ) and are more pronounced forSymanzik than for Wilson flow. Q and the GF coupling g GF on six individual configura-tions. We use the clover operator to approximate F (cid:101) F inEq. (2). The upper panels in each sub-figure show theflow time evolution of the topological charge both withWilson (W) and Symanzik (S) flows. The lower panelsshow the the renormalized g GF coupling evaluated withboth the Wilson plaquette (W) and clover (C) operatorsfor both flows. The six configurations were chosen toillustrate the difference between Q = 0 and Q (cid:54) = 0. Theyare part of our N f = 10 DW ensemble at β = 4 .
02, thestrongest bare coupling we consider, on 32 volumes [43].The topological charge shows large fluctuations at smallflow time but settles to a near-integer value by t/a (cid:38) . Q for t/a > g GF at large flow time, as isshown on the lower panels. The first letter shorthand notation indicates the gradient flow(W or S), the second letter the operator (W or C).
At trajectory Q = 0 but Symanzik flow identifiestopological charge Q = 2 and -2, respectively. With Wil-son flow, g GF shows a flat, slowly decreasing behaviorwith flow time, similar to what is observed at trajectory Q = 0. Symanzik flow, however, shows g GF increasing roughly linearly with the flow time, similar totrajectory Q = −
1, although the slope is larger,consistent with two topological objects on the configura-tions. Different operators are still consistent within eachflow.At trajectory Q = 0 but with Symanzik flow we see a rapid change atlarger flow time. At trajectory Q = − → Q = 0 around t ≈
26. Correspondingly g GF changes from a linearly increasing flow time dependenceto a flat/decreasing form. At trajectory Q = 0 → Q = −
1, suggesting that the configuration atflow time t/a <
12 had an instanton-antiinstanton pair.The instanton is annihilated by the flow at t/a ≈ g GF follows the expected behavior. Its linearrise with the flow time slows at t/a ≈
13 but remainslinear, similar to what is observed at trajectory Q is an artifact of the gradient flowin simulations with massless chirally symmetric fermions.The panels of Fig. 1 verify that on Q (cid:54) = 0 configurations g GF receives a contribution that increases approximately c -1.2-1-0.8-0.6-0.4-0.200.20.40.60.8 c , s ( g c ) =4.02 WW, 16 32WW, 16 32 |Q| < 0.5 c c , s ( g c ) =4.02 SW, 16 32SW, 16 32 |Q| < 0.5
FIG. 3. Effect of non-zero topological charge Q on the value of the step-scaling β c,s function determined from the Wilsonplaquette operator for the 16 →
32 volume pair ( s = 2) at bare coupling β ≡ /g = 4 .
02 as function of the renormalizationscheme parameter c , which is related to the flow time t . Since at β = 4 .
02 Wilson flow exhibits in total only three configurationswhere a nonzero topological charge is measured, filtering configurations with | Q | < . β c,s (left panel). For Symanzik flow (right panel), however, a large number of configurations with nonzero topological charge arefound, resulting in a larger β c,s compared to the analysis using only | Q | < .
5. The discrepancy grows with c . linearly with flow time and | Q | . Next we investigate whatfraction of the configuration ensembles is affected by thislattice artifact. In Fig. 2 we show the flow time evolu-tion of the topological charge defined by Eq. (2) on asubset of our N f = 10, 32 configurations at bare cou-pling β = 4 .
02, 4.05, 4.10 and 4.15. Each panel includes100 configurations, separated by 10 MDTU, analyzingWilson flow data on the left, Symanzik flow data on theright. The vertical lines indicate flow times t/a = 8 . c = 0 .
25 and 0.3 in step-scaling studies.At small flow time, vacuum fluctuations dominate Q .At the weaker couplings, β = 4 .
15, 4.10, most vacuumfluctuations die out by t/a (cid:38)
2, and while Q may notexactly be integer, it is close to an integer (0, ± ± Q = 0. Symanzikflow sustains Q (cid:54) = 0 longer, and one of the 100 β = 4 . Q = − t/a = 32, ourmaximal flow time. The picture changes rapidly to-wards strong coupling. At our strongest gauge coupling β = 4 .
02 even Wilson flow has several Q (cid:54) = 0 configu-rations at t/a ≈
10, some surviving even at t/a = 32.Symanzik flow enhances this lattice artifact even further.The topological charge distribution of Symanzik flowedconfigurations resemble QCD at finite mass. Many QCDsimulations would be relieved to see such a rapid changeof the topological charge, yet here Q (cid:54) = 0 signals onlylattice artifacts.At large flow time it is possible to filter configurationsaccording to different topological sectors. Analyzing onlythose with Q = 0 and contrasting the predictions with the full data set provides information on the effect of Q (cid:54) = 0. In Fig. 3 we compare the finite volume step-scaling β c,s ( g c ) functions defined in Eq. (4) with andwithout topological filtering. The plots show β c,s pre-dicted by the lattice volumes L/a = 16 →
32 on the β = 4 .
02 configuration set as the function of c = √ t/L with Wilson flow (left) and Symanzik flow (right), usingthe Wilson plaquette operator to predict the energy den-sity. While in the Wilson flow analysis filtering on thetopology has only a minimal effect, the Q = 0 subsetwith Symanzik flow predicts a significantly slower run-ning step-scaling function . This is consistent with theobservation we made in connection with Fig. 1 where wepointed out that Q (cid:54) = 0 configurations have faster run-ning gauge coupling g GF . This effect weakens at weakgauge coupling, but we expect that step-scaling functionscould overestimate the running of the gauge coupling inthe strong coupling, especially with Symanzik flow. Inthe accompanying paper [43] we show details of our anal-ysis.We close our discussion with Fig. 4 where we com-pare g c for c=0.300 as predicted by configurations with | Q | = 0, 1 and 2 on our β = 4 .
02 data set. As expectedbased on Eq. (4) and Figs. 1 and 3, g c increases with | Q | . On the right side panel of Fig. 4 we show the rel-ative weight of the different topological sectors. In thecase of Symanzik flow we analyze 371 measurements in We define the integer topological charge as the integer part of( | Q geom | + 0 .
5) where Q geom is the value predicted by the clover F ˜ F operator. At large flow time Q geom is close to an integer,apart from the regions where the topological charge undergoes arapid change. topo. charge |Q| g c L=32, Ls=16=4.02, c=0.300Wilson operator S y m an z i k f l o w W il s on f l o w FIG. 4. On the left: renormalized coupling g c as predictedby Symanzik (red/orange/yellow triangles) and Wilson (greensquares) flows and Wilson operator on configurations with | Q | = 0, 1, and 2 at c = 0 .
300 (GF flow time t = 11 . | Q | sector when using Symanzik and Wilson, respec-tively. In total 371 (372) configurations enter the presentedresults for Symanzik (Wilson) flow. total, 211 with | Q | = 0, 143 with | Q | = 1 and 17 with | Q | = 2 but do not show one measurement with | Q | > | Q | = 0 and 19 with | Q | = 1. Differences betweenWW and SW determinations of g c indicate cutoff effectswhich are only supposed to disappear after taking thecontinuum limit.Measuring the total Q = n + − n − does not giveinformation on possible instanton–antiinstanton pairs.However the change of the slopes of g GF observed inFig. 1 suggests that most Q (cid:54) = 0 configurations haveonly isolated instantons and not many pairs. We wantto strongly emphasize that our analysis filtering on thetopological charge is not an alternative method to predictthe running coupling and the step-scaling function. Wesolely use it to show the expected change due to latticeartifacts created by Q (cid:54) = 0 configurations. C. Finite value of L s Stout smeared M¨obius domain wall fermions with L s =16 have a small residual mass, am res < − even at ourstrongest gauge coupling. We check for possible effectsdue to non-vanishing residual mass by generating a sec-ond ensemble at bare coupling β = 4 .
05 with L s = 32.The numerical cost of generating an L s = 32 trajectory ismore than five times greater compared to the simulationwith L s = 16. Thus we have fewer L s = 32 trajectories(about 1 /
3) than for L s = 16. In Fig. 5 we show the flow -404 Q W L=32, L s =16, =4.05c=0.25 c=0.30 flow time t/a -404 Q W L=32, L s =32, =4.05c=0.25 c=0.30 -404 Q S L=32, L s =16, =4.05c=0.25 c=0.30 flow time t/a -404 Q S L=32, L s =32, =4.05c=0.25 c=0.30 FIG. 5. Comparison of flow time histories of the topologicalcharge Q for the first 100 thermalized configurations of the β = 4 .
05 ensembles with L s = 16 and L s = 32. The uppertwo panels show Q W determined with Wilson flow (W), thelower two panels show Q S determined with Symanzik flow (S).In each case, the L s = 16 data are shown above the L s = 32data. time histories for the topological charge Q for the first100 configurations of each ensemble. While Wilson flowidentifies very few configurations with nonzero Q on ei-ther ensembles, the same ensembles exhibit more nonzerotopology under Symanzik flow. Surprisingly, the rela-tive number of configurations with nonzero Q more thantriples under Symanzik flow when L s increases from 16 to32. This observation again indicates that non-vanishingtopology is an artifact of the flow and not due to thesmall residual mass.Next we determine the renormalized coupling g c forthe renormalization scheme c = 0 .
300 on both ensembleswhere we again separate configurations according to thevalue of | Q | . The outcome is shown in Fig. 6. On both en-sembles Wilson flow (green squares) predominantly findszero topological charge and identifies too few configura-tions with | Q | = 1 to reliably estimate an uncertainty on g c . Hence we show the values for | Q | = 0 with statisticalerror bar but indicate only the central values at | Q | = 1.The prediction on the L s = 16 ensemble (open symbol)is in perfect agreement with the L s = 32 ensemble (filledsymbol). For Symanzik flow (red/orange/yellow trian-gles) we find several configurations with | Q | = 0 and 1plus one configuration with | Q | = 2 on the L s = 16 en- topo. charge |Q| g c L=32 Ls=16Ls=32=4.05, c=0.300Wilson operator S y m an z i k f l o w L s = S y m an z i k f l o w L s = FIG. 6. Renormalized coupling g c for the renormalizationscheme c = 0 .
30 determined on
L/a = 32 ensembles at β =4 .
05 for ensembles with L s = 16 (open symbols) and L s = 32(filled symbols). In case of Wilson flow (green squares) onlya value for configurations with | Q | = 0 can be determined,whereas for Symanzik flow (red/orange/yellow triangles) g c for | Q | = 0 , L s = 16 and inaddition | Q | = 2 for L s = 32. All g c values at the same | Q | agree perfectly which strongly implies effects due to L s = 16are not resolved within our statistical errors. For Symanzikflow, however, each | Q | sector predicts a statistically differentvalue of g c . The relative distribution of the | Q | sectors forSymanzik flow are shown in the small panel on the right. For L s = 16 in total 372 measurements are analyzed, for L s = 32112. semble and several configurations in all three sectors for L s = 32. The g c values clearly resolve a dependence on Q . At the same time, we observe good agreement for g c predicted at the same value of Q on ensembles with dif-ferent L s . The latter strongly implies that the effect ofchoosing L s = 16 vs. L s = 32 is negligible within our sta-tistical uncertainties. The relative distribution of the | Q | sectors for Symanzik flow are shown in the small panelon the right of Fig. 6. For L s = 16 in total 372 measure-ments are analyzed and 90% have Q = 0. For L s = 32 weanalyze 112 measurements but only 70% have | Q | = 0.Since | Q | > g c , the average of the renor-malized coupling increases with increasing L s . However,this is an artifact of the flow and implies larger latticeartifacts for larger L s . III. SU(3) WITH N f = 8 FLAVORSA. Details of the simulations
In this part of our study we utilize existing gauge fieldconfigurations generated with eight degenerate and mass-less flavors of staggered fermions with nHYP smeared links [53, 54] and gauge action that combines plaque-tte and adjoint plaquette terms [18]. The configura-tions have symmetric volumes, V = L , where the gaugefields have periodic boundary conditions and the fermionsantiperiodic boundary conditions in all four space-timedirections. Apart from the boundary conditions thisis the same action used in the large scale studies ofRefs. [34, 55].Staggered fermions have a remnant U(1) chiral symme-try that protects the fermion mass from additive massrenormalization. On the other hand taste breaking ofstaggered fermions split the eigenmodes of the Dirac op-erator. Smooth, isolated instantons have four near-zeroeigenmodes for the four staggered species, but they aresplit into two positive, two negative imaginary eigenvaluepairs. The determinant of the Dirac operator is not ex-actly zero, topologically non-trivial configurations are notprohibited. In the continuum limit taste symmetry is re-covered and Q (cid:54) = 0 configurations should be suppressed.Therefore it is reasonable to consider all Q (cid:54) = 0 as latticeartifact — either from the action or from the flow. B. Effects of nonzero topological charge
Our discussion and analysis here follows that of Sec. IIwith domain wall fermions. The strongest gauge cou-pling of the simulations with one level of nHYP smear-ing is β = 5 .
0, and the largest volume has
L/a = 30. InFig. 7 we show the evolution of the topological chargewith Wilson and Symanzik flow on 50 thermalized con-secutive configurations at β = 5 .
0, 5.4 and 5.8. Simi-lar to the DW result, we observe the emergence of more Q (cid:54) = 0 configurations at strong coupling. We also ob-serve rapid changes in Q at large flow time, and againmore | Q | > c = 0 .
300 renormalization scheme for the different topo-logical sectors. As in Fig. 4, we see a clear increase in g c as | Q | increases. Since the fraction of Q (cid:54) = 0 configu-rations is much larger with Symanzik than Wilson flow,this implies that step scaling studies using Symanzik flowmay overestimate β c,s ( g ) at strong gauge coupling.We note however the investigation in Ref. [53] studiedthis system using only Wilson flow. It would be veryinteresting to re-analyze the existing configurations notonly with Symanzik flow, but also with a flow that sup-presses the topology even further that Wilson flow. IV. GRADIENT FLOW WITH IMPROVEDTOPOLOGY SUPPRESSION
The flow kernel of Symanzik flow is a combination ofa 1 × × c × = 5 / c × = − /
12. Wilson flow isperformed only with the plaquette term i.e. c × = 1, c × = 0. Apparently the negative c × term increases -404 Q W L=30, 1nHYP =5.80c=0.25 c=0.30 -404 Q W L=30, 1nHYP =5.40c=0.25 c=0.30 flow time t/a -404 Q W L=30, 1nHYP =5.00c=0.25 c=0.30 -404 Q S L=30, 1nHYP =5.80c=0.25 c=0.30 -404 Q S L=30, 1nHYP =5.40c=0.25 c=0.30 flow time t/a -404 Q S L=30, 1nHYP =5.00c=0.25 c=0.30
FIG. 7. Flow time histories for the N f = 8 data set with one level of nHYP smeared staggered fermions on ( L/a ) = 30 lattices at bare gauge coupling β = 5 . , . , .
00. Similar to N f = 10 DW fermions (Fig. 1) we observe an increase of nonzerotopology as β decreases and the suppression of topology is inferior for Symanzik compared to Wilson flow. topo. charge |Q| g c L=30, 1nHYP=5.00, c=0.300Wilson operator S y m an z i k f l o w W il s on f l o w FIG. 8. Renormalized coupling g c determined for differenttopological sectors using the strongest coupling of the N f = 8staggered fermion ensemble at β = 5 .
00 and c = 0 . g c increases with the charge Q , similar to DW in Fig. 4. Greensquares correspond to Wilson flow, red/orange/yellow trian-gles to Symanzik flow. The relative fraction of each Q sectoris shown by the panel on the right. Similar to the DW re-sults, Symanzik flow creates more Q (cid:54) = 0 configurations thanWilson flow. the probability of Q (cid:54) = 0 in Symanzik flow. This sug-gests that a positive c × term might lead to a bettersuppression of this lattice artifact. To test the idea weimplemented an alternative gradient flow (A) where weset the coefficients to c × = 2 / c × = 1 /
24 (5) and demonstrate its effect on the topological charge Q using our N f = 10 domain wall ensemble at bare cou-pling β = 4 .
02. In Fig. 9 we show how the suppressionof the topological charge is improved w.r.t. Wilson andSymanzik flow. Whether or not this alternative gradientflow is a viable candidate to perform step-scaling studiesat strong coupling will however require further investiga-tions using multiple volumes and a range of bare coupling β . Only that will allow to estimate discretization effectsto be removed by the continuum limit extrapolation. V. SUMMARY
In this paper we demonstrate that gradient flow mea-surements on rough gauge field configurations can pro-mote lattice dislocations to instanton-like topological ob-jects. The number of these instanton-like objects dependon the gradient flow kernel. In the case of step-scalingcalculations of the lattice β function, the simulations arecarried out in the chiral limit where a nonzero instan-ton number is suppressed. Hence instanton-like objectscreated by the gradient flow are lattice artifacts. Our in-vestigations reveal a clear correlation between a nonzerotopological charge seen by the gradient flow and an in-crease in the value of gradient flow renormalized cou-pling. We further demonstrate that this also results inan overestimate of the step-scaling β -function. By inves-tigating the N f = 10 system simulated with domain wallfermions and the N f = 8 system studied with staggeredfermions, we show that this artifact is not related to thelattice actions used in the simulations but an artifact ofthe gradient flow which arises at (very) strong coupling.In both systems we also observe that the effect is morepronounced when using Symanzik compared to Wilson -404 Q A L=32, L s =16, =4.02c=0.25 c=0.30 -404 Q W c=0.25 c=0.30 flow time t/a -404 Q S c=0.25 c=0.30 FIG. 9. Demonstration of the topology suppressing featuresof our alternative flow with positive coefficient for the 2 × N f = 10 domain wall lattice with L/a = 32 at β = 4 . flow.Since this effect becomes only noticeable at very strongcoupling, it may explain why it has not been reportedearlier. In the case of our N f = 12 simulations, wechecked that both step-scaling calculations using domainwall [22, 23, 56] or staggered fermions [21] do not includeensembles exhibiting more than one or two configurationswhere a gradient flow finds nonzero topological charge.These simulations have simply been performed at weakercoupling.Similarly to step-scaling calculations, continuous β function determinations [57–59] at (very) strong couplingmight also be affected by nonzero topological charge oc-curring as part of the gradient flow. Our studies of the N f = 2 and 12 systems, however, do not extend into theproblematic range and are therefore not affected. ACKNOWLEDGMENTS
We are very grateful to Peter Boyle, Guido Cossu,Anontin Portelli, and Azusa Yamaguchi who developthe
Grid software library providing the basis of thiswork and who assisted us in installing and running
Grid on different architectures and computing centers. Wethank David Schaich for his support to run additionalSymanzik flow measurements on the N f = 8 staggeredconfigurations of Ref. [18]. We benefited from many dis-cussions with Thomas DeGrand, Ethan Neil, and Ben-jamin Svetitsky. A.H. and O.W. acknowledge supportby DOE grant DE-SC0010005. A.H. would like to ac-knowledge the Mainz Institute for Theoretical Physics(MITP) of the Cluster of Excellence PRISMA+ (ProjectID 39083149) for enabling us to complete a portion of thiswork. O.W. acknowledges partial support by the Mu-nich Institute for Astro- and Particle Physics (MIAPP)which is funded by the Deutsche Forschungsgemeinschaft(DFG, German Research Foundation) under Germany’sExcellence Strategy – EXC-2094 – 390783311.Computations for this work were carried out in part onfacilities of the USQCD Collaboration, which are fundedby the Office of Science of the U.S. Department of Energyand the RMACC Summit supercomputer [60], which issupported by the National Science Foundation (awardsACI-1532235 and ACI-1532236), the University of Col-orado Boulder, and Colorado State University. Thiswork used the Extreme Science and Engineering Dis-covery Environment (XSEDE), which is supported byNational Science Foundation grant number ACI-1548562[61] through allocation TG-PHY180005 on the XSEDEresource stampede2 . This research also used resourcesof the National Energy Research Scientific ComputingCenter (NERSC), a U.S. Department of Energy Office ofScience User Facility operated under Contract No. DE-AC02-05CH11231. We thank Fermilab, Jefferson Lab,NERSC, the University of Colorado Boulder, TACC, theNSF, and the U.S. DOE for providing the facilities es-sential for the completion of this work. [1] T. Sch¨afer and E. V. Shuryak, Rev. Mod. Phys. , 323(1998), arXiv:hep-ph/9610451 [hep-ph].[2] T. A. DeGrand and A. Hasenfratz, Phys. Rev. D64 ,034512 (2001), arXiv:hep-lat/0012021 [hep-lat].[3] T. A. DeGrand, A. Hasenfratz, and T. G. Kovacs, Phys.Rev.
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