Dislocations under gradient flow and their effect on the renormalized coupling

Non-zero topological charge is prohibited in the chiral limit of gauge-fermion systems because any instanton would create a zero mode of the Dirac operator. On the lattice, however, the geometric $Q_\text{geom}=\langle F{\tilde F}\rangle /32\pi^2$ definition of the topological charge does not necessarily vanish even when the gauge 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. We demonstrate that these artifacts of the flow cause the gradient flow renormalized gauge coupling to increase and run faster. In step-scaling studies such artifacts contribute a term which increases with volume. The usual $a/L\to 0$ continuum limit extrapolations can hence lead to incorrect results. In this paper we investigate these topological lattice artifacts in the SU(3) 10-flavor system with domain wall fermions and the 8-flavor system with staggered fermions. Both systems exhibit nonzero topological charge at the strong coupling, especially when using Symanzik gradient flow. We demonstrate how this artifact impacts the determination of the renormalized gauge coupling and the step-scaling $\beta$ function.


I. INTRODUCTION
The net instanton charge Q = n + − n − is a topologically protected quantity in continuum gauge-fermion systems. On the lattice, however, Q is not protected and different definitions of the topological charge, like the number of zero modes of the Dirac operator or the geometric definition may not agree. The latter definition is particularly troublesome as the ultraviolet (UV) fluctuations of the gauge field can dominate Q geom . Smoothing the gauge field with 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 topological modes |Q| ≈ 1 with some but not with other definitions of Q geom . There is no unique definition of the topological charge on the lattice. Different definitions are expected to agree only in the continuum limit [1][2][3][4][5][6][7][8].
Dynamical configurations with net topological charge Q ferm have |Q ferm | zero modes in the spectrum of the massless Dirac operator assuming the lattice fermions are chirally symmetric [9,10]. As a consequence, non-zero Q ferm configurations are excluded in the chiral limit. This is one of the rare instances where theoretical arguments rigorously constrain the value of Q even at finite cutoff. 1 * anna.hasenfratz@colorado.edu † oliver.witzel@colorado.edu 1 Residual chiral symmetry breaking or effects due to the finite volume could potentially allow configurations with nonzero topological charge. This effect is however negligible.
The fermions restrict Q ferm = 0 and any Q geom = 0 signals a lattice artifact of the smoothing algorithm or the operator used in the definition of Q geom . Even though it is a lattice artifact, Q geom = 0 can have significant effects on the renormalized gradient flow gauge coupling and the finite volume step scaling function β c,s (g 2 c ).
The GF gauge coupling at energy scale µ = 1/ √ 8t is g 2 GF (t; L, β) = N t 2 E where E is the energy density, β = 6/g 2 0 is the bare coupling, L refers to the linear size of the system, and the normalization factor N is chosen to match g 2 M S at one-loop [11][12][13]. Lattice studies show that at large flow time g 2 GF (t) exhibits only mild, approximately linear or weaker, dependence on t. Therefore the energy density E decreases ∝ 1/t or faster. While GF removes vacuum fluctuations and instanton pairs, some instantons can survive the flow and become (quasi-)stable. At large flow time Q geom approaches integer values and Q 2 geom is frequently used to define the lattice topological susceptibility [7,8,14,15]. To simplify the notation we will from now on refer to Q geom simply by using Q.
The action of a single instanton is S I = 8π 2 in the continuum. On the lattice this value depends on the instanton size and the lattice action, but at large flow time smooth instantons increase the energy of the configuration by ≈ S I [3]. The net number of instantons Q = n + − n − is expected to scale with the square root of the number of lattice sites V /a 4 , even if they arise from vacuum fluctuations as artifacts of the GF. The instanton contribution to the energy density is therefore ∝ 1/ √ a 4 V . If instanton-antiinstanton pairs are present, this contribution is even larger.
In step scaling studies the GF flow time is connected to the lattice size as t = (cL) 2 /8, where the constant c defines the renormalization scheme and g 2 c refers to the gradient flow renormalized coupling at the corresponding flow time g 2 GF (t; L, β). The discrete lattice β function of arXiv:2004.00758v1 [hep-lat] 2 Apr 2020 scale change s is defined as [16] β c,s (g 2 c ; L, β) = In volumes V = L 4 the contribution of the instantons to the discrete β function is where β c,s (g 2 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 of the lattice size L. When the simulations are performed with chirally symmetric fermions in the chiral limit, the term C(β, c)L 2 /a 2 is a lattice artifact, the consequence of the GF promoting vacuum fluctuations to topological objects.
Even on Q = 0 configurations β c,s (g 2 c ) Q=0 has cutoff effects. These are typically removed by an a 2 /L 2 → 0 extrapolation at fixed renormalized coupling g 2 c [16][17][18][19][20][21][22][23]. If the data does not follow a 2 /L 2 dependence, higher order (a/L) 4 terms can be included [24,25]. However, in the strong coupling limit with a non-negligible instanton density, Eq. (4) implies that the correct continuum extrapolation should include an (L/a) 2 term instead or at least in addition to (a/L) 4 . Practically such an L → ∞ extrapolation is not viable. This is a reflection of the nonperturbative nature of instantons and shows that their effect cannot be removed by perturbatively motivated extrapolations. The effect of instanton-like objects in the continuum prediction could be substantial, especially in slowly running systems near or within the conformal window where the coupling β = 6/g 2 0 barely changes as the continuum limit is taken on available lattice volumes. A clean way to avoid this issue is to choose a flow where instantons are not generated even on coarse lattices.
We consider two different systems to illustrate the issue. In both cases we study two different gradient flow kernels, Wilson and Symanzik flow. We start with our recent 10-flavor SU(3) domain wall simulations where we first observed the effect of non-zero topological charge [22]. An accompanying paper discusses the step-scaling function of this most likely conformal system and provides further details [43]. Next we analyze configurations generated for an older study of the SU(3) 8-flavor system with staggered fermions [18]. We chose these two systems because both simulations have been pushed toward very strong coupling where the contamination from topological modes can be significant. Our results demonstrate these lattice artifacts are more severe for Symanzik than for Wilson flow. In Section IV we demonstrate how a small modification of the flow kernel results in a gradient flow that is better at smoothing out local dislocations resulting in fewer configurations with nonzero topological charge. The lattice discretization errors of such a modified gradient flow will need to be explored in the future. Finally we briefly summarize our findings. In this part of our study we utilize existing gauge field configurations generated with ten degenerate and massless flavors of three times stout-smeared [44] Möbius domain wall (DW) fermions [45][46][47] with Symanzik gauge action [48,49]. The configurations are generated using Grid 2 [50] and we choose symmetric volumes with V = L 4 where the gauge fields have periodic, the fermions antiperiodic boundary conditions in all four space-time directions. The bare input quark mass is zero and for the domain wall fermions we choose the domain wall height M 5 = 1 and the extent of the fifth dimension L s = 16. Configurations are generated using the hybrid Monte Carlo update algorithm [51] choosing trajectories of length two molecular time units (MDTU) and we use configurations saved every 10 MDTU. Our statistical data analysis is performed using the Γ-method [52] which estimates and accounts for integrated autocorrelation times. For the L/a = 32 ensembles at strong coupling considered here autocorrelations range from three to five measurements.
Due to the finite extent of the fifth dimension, DW fermions exhibit a small, residual chiral symmetry breaking which conventionally is parametrized by an additive mass term am res . We determine am res numerically using the ratio of midpoint-pseudoscalar and pseudoscalarpseudoscalar correlator. At strong coupling am res depends on the bare coupling β and increases from am res = 2 × 10 −5 at β = 4.15 to 6 × 10 −4 at β = 4.02. To demonstrate that am res is sufficiently small and not the origin of nonzero topological charges, we compare results for β = 4.05 from ensembles with L s = 16 and L s = 32 below.

B. Effects of nonzero topological charge
We illustrate the effects of Q = 0 instanton-like objects on the gradient flow coupling in Fig. 1 where we show the flow time dependence of the topological charge  Q and the GF coupling g 2 GF on six individual configurations. We use the clover operator to approximate F F in Eq. (2). The upper panels in each sub-figure show the flow time evolution of the topological charge both with Wilson (W) and Symanzik (S) flows. The lower panels show the the renormalized g 2 GF coupling evaluated with both the Wilson plaquette (W) and clover (C) operators for both flows. 3 The six configurations were chosen to illustrate the difference between Q = 0 and Q = 0. They are part of our N f = 10 DW ensemble at β = 4.02, the strongest bare coupling we consider, on 32 4 volumes [43]. The topological charge shows large fluctuations at small flow time but settles to a near-integer value by t/a 2 5.0. We observe occasional change in Q for t/a 2 > 5 but these tend to be quick as topological objects are annihilated by the flow.
At large flow time we expect different flows and operators to converge. That is indeed the case at trajectory #700 and #575 (top left and top right in Fig. 1) where, as shown in the upper panels, Wilson and Symanzik flows find the same topological charge at large flow time. Both Wilson and Symanzik flows and Wilson and clover operators predict consistent g 2 GF at large flow time, as is shown on the lower panels. 3 The first letter shorthand notation indicates the gradient flow (W or S), the second letter the operator (W or C).
At trajectory #2965 and #2100 (middle of Fig. 1) Wilson flow predicts Q = 0 but Symanzik flow identifies topological charge Q = 2 and -2, respectively. With Wilson flow, g 2 GF shows a flat, slowly decreasing behavior with flow time, similar to what is observed at trajectory # 700 with Q = 0. Symanzik flow, however, shows g 2 GF increasing roughly linearly with the flow time, similar to trajectory # 575, Q = −1, although the slope is larger, consistent with two topological objects on the configurations. Different operators are still consistent within each flow.
At trajectory #2255 (bottom left) and #845 (bottom right) the topological charge with Wilson flow is Q = 0 but with Symanzik flow we see a rapid change at larger flow time. At trajectory #2255 this corresponds to Q = −1 → Q = 0 around t ≈ 26. Correspondingly g 2 GF changes from a linearly increasing flow time dependence to a flat/decreasing form. At trajectory #845 the change is Q = 0 → Q = −1, suggesting that the configuration at flow time t/a 2 < 12 had an instanton-antiinstanton pair. The instanton is annihilated by the flow at t/a 2 ≈ 13, leaving the anti-instanton unpaired. The renormalized coupling g 2 GF follows the expected behavior. Its linear rise with the flow time slows at t/a 2 ≈ 13 but remains linear, similar to what is observed at trajectory #575.
Any non-vanishing Q is an artifact of the gradient flow in simulations with massless chirally symmetric fermions. The panels of Fig. 1 verify that on Q = 0 configurations g 2 GF receives a contribution that increases approximately  4 . This is consistent with the observation we made in connection with Fig. 1 where we pointed out that Q = 0 configurations have faster running gauge coupling g 2 GF . This effect weakens at weak gauge coupling, but we expect that step-scaling functions could overestimate the running of the gauge coupling in the strong coupling, especially with Symanzik flow. In the accompanying paper [43] we show details of our analysis.
We close our discussion with Fig. 4 where we compare g 2 c for c=0.300 as predicted by configurations with |Q| = 0, 1 and 2 on our β = 4.02 data set. As expected based on Eq. (4) and Figs. 1 and 3, g 2 c increases with |Q|. On the right side panel of Fig. 4 we show the relative weight of the different topological sectors. In the case of Symanzik flow we analyze 371 measurements in Measuring the total Q = n + − n − does not give information on possible instanton-antiinstanton pairs. However the change of the slopes of g 2 GF observed in Fig. 1 suggests that most Q = 0 configurations have only isolated instantons and not many pairs. We want to strongly emphasize that our analysis filtering on the topological charge is not an alternative method to predict the running coupling and the step-scaling function. We solely use it to show the expected change due to lattice artifacts created by Q = 0 configurations.

C. Finite value of Ls
Stout smeared Möbius domain wall fermions with L s = 16 have a small residual mass, am res < 10 −3 even at our strongest gauge coupling. We check for possible effects due to non-vanishing residual mass by generating a second ensemble at bare coupling β = 4.05 with L s = 32. The numerical cost of generating an L s = 32 trajectory is more than five times greater compared to the simulation with 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 time histories for the topological charge Q for the first 100 configurations of each ensemble. While Wilson flow identifies very few configurations with nonzero Q on either ensembles, the same ensembles exhibit more nonzero topology under Symanzik flow. Surprisingly, the relative number of configurations with nonzero Q more than triples under Symanzik flow when L s increases from 16 to 32. This observation again indicates that non-vanishing topology is an artifact of the flow and not due to the small residual mass. Next we determine the renormalized coupling g 2 c for the renormalization scheme c = 0.300 on both ensembles where we again separate configurations according to the value of |Q|. The outcome is shown in Fig. 6. On both ensembles Wilson flow (green squares) predominantly finds zero topological charge and identifies too few configurations with |Q| = 1 to reliably estimate an uncertainty on semble and several configurations in all three sectors for L s = 32. The g 2 c values clearly resolve a dependence on Q. At the same time, we observe good agreement for g 2 c predicted at the same value of Q on ensembles with different L s . The latter strongly implies that the effect of choosing L s = 16 vs. L s = 32 is negligible within our statistical uncertainties. The relative distribution of the |Q| sectors for Symanzik flow are shown in the small panel on the right of Fig. 6. For L s = 16 in total 372 measurements are analyzed and 90% have Q = 0. For L s = 32 we analyze 112 measurements but only 70% have |Q| = 0. Since |Q| > 0 predict larger g 2 c , the average of the renormalized coupling increases with increasing L s . However, this is an artifact of the flow and implies larger lattice artifacts for larger L s . In this part of our study we utilize existing gauge field configurations generated with eight degenerate and massless flavors of staggered fermions with nHYP smeared links [53,54] and gauge action that combines plaquette and adjoint plaquette terms [18]. The configurations have symmetric volumes, V = L 4 , where the gauge fields have periodic boundary conditions and the fermions antiperiodic boundary conditions in all four space-time directions. Apart from the boundary conditions this is the same action used in the large scale studies of Refs. [34,55].
Staggered fermions have a remnant U(1) chiral symmetry that protects the fermion mass from additive mass renormalization. On the other hand taste breaking of staggered fermions split the eigenmodes of the Dirac operator. Smooth, isolated instantons have four near-zero eigenmodes for the four staggered species, but they are split into two positive, two negative imaginary eigenvalue pairs. The determinant of the Dirac operator is not exactly zero, topologically non-trivial configurations are not prohibited. In the continuum limit taste symmetry is recovered and Q = 0 configurations should be suppressed. Therefore it is reasonable to consider all Q = 0 as lattice artifact -either from the action or from the flow.

B. Effects of nonzero topological charge
Our discussion and analysis here follows that of Sec. II with domain wall fermions. The strongest gauge coupling of the simulations with one level of nHYP smearing is β = 5.0, and the largest volume has L/a = 30. In Fig. 7 we show the evolution of the topological charge with Wilson and Symanzik flow on 50 thermalized consecutive configurations at β = 5.0, 5.4 and 5.8. Similar to the DW result, we observe the emergence of more Q = 0 configurations at strong coupling. We also observe rapid changes in Q at large flow time, and again more |Q| > 0 with Symanzik than with Wilson flow. In Fig. 8 we compare the renormalized GF coupling in the c = 0.300 renormalization scheme for the different topological sectors. As in Fig. 4, we see a clear increase in g 2 c as |Q| increases. Since the fraction of Q = 0 configurations is much larger with Symanzik than Wilson flow, this implies that step scaling studies using Symanzik flow may overestimate β c,s (g 2 ) at strong gauge coupling.
We note however the investigation in Ref. [53] studied this system using only Wilson flow. It would be very interesting to re-analyze the existing configurations not only with Symanzik flow, but also with a flow that suppresses the topology even further that Wilson flow.

IV. GRADIENT FLOW WITH IMPROVED TOPOLOGY SUPPRESSION
The flow kernel of Symanzik flow is a combination of a 1 × 1 plaquette and a 2 × 1 rectangle term, with coefficients c 1×1 = 5/3 and c 2×1 = −1/12. Wilson flow is performed only with the plaquette term i.e. c 1×1 = 1, c 2×1 = 0. Apparently the negative c 2×1 term increases  the probability of Q = 0 in Symanzik flow. This suggests that a positive c 2×1 term might lead to a better suppression of this lattice artifact. To test the idea we implemented an alternative gradient flow (A) where we set the coefficients to c 1×1 = 2/3 and c 2×1 = 1/24 (5) and demonstrate its effect on the topological charge Q using our N f = 10 domain wall ensemble at bare coupling β = 4.02. In Fig. 9 we show how the suppression of the topological charge is improved w.r.t. Wilson and Symanzik flow. Whether or not this alternative gradient flow is a viable candidate to perform step-scaling studies at strong coupling will however require further investigations using multiple volumes and a range of bare coupling β. Only that will allow to estimate discretization effects to be removed by the continuum limit extrapolation.

V. SUMMARY
In this paper we demonstrate that gradient flow measurements on rough gauge field configurations can promote lattice dislocations to instanton-like topological objects. The number of these instanton-like objects depend on the gradient flow kernel. In the case of step-scaling calculations of the lattice β function, the simulations are carried out in the chiral limit where a nonzero instanton number is suppressed. Hence instanton-like objects created by the gradient flow are lattice artifacts. Our investigations reveal a clear correlation between a nonzero topological charge seen by the gradient flow and an increase in the value of gradient flow renormalized coupling. We further demonstrate that this also results in an overestimate of the step-scaling β-function. By investigating the N f = 10 system simulated with domain wall fermions and the N f = 8 system studied with staggered fermions, we show that this artifact is not related to the lattice actions used in the simulations but an artifact of the gradient flow which arises at (very) strong coupling. In both systems we also observe that the effect is more pronounced when using Symanzik compared to Wilson flow.
Since this effect becomes only noticeable at very strong coupling, it may explain why it has not been reported earlier. In the case of our N f = 12 simulations, we checked that both step-scaling calculations using domain wall [22,23,56] or staggered fermions [21] do not include ensembles exhibiting more than one or two configurations where a gradient flow finds nonzero topological charge. These simulations have simply been performed at weaker coupling.
Similarly to step-scaling calculations, continuous β function determinations [57][58][59] at (very) strong coupling might also be affected by nonzero topological charge occurring as part of the gradient flow. Our studies of the N f = 2 and 12 systems, however, do not extend into the problematic range and are therefore not affected.