A comparison of improved cooling and hypercubic smearing for topology on dynamical Asqtad lattices

PROCEEDINGS SUPPLEMENTS Nuclear Physics B (Proc. Suppl.) 119 (2003) 769-771

ELSEVIER

A comparison on dynamical

of improved cooling and hypercubic Asqtad lattices

smearing

for topology

MILC Collaboration: C. Bernarda, T. Burchb, T. DeGrandC, C.E. DeTard, S. Gottliebe, E.B. Gregory”, A. HasenfratzC, U.M. Heller f, J.E. Hetrickg* J. Osbornd, R. Sugarh, and D. Toussainth aDepartment

of Physics, Washington

bDepartment

o f Physics, University

University, of ilrizona,

St. Louis, MO 63130, USA Tucson, AZ 85721, USA

‘Physics Department,

University

of Colorado, Boulder, CO 80309, USA

dPhysics Department,

University

of Utah, Salt Lake City, UT 84112, USA

eDepartment

of Physics, Indiana University,

fCSIT, Florida State University, guniversity hDepartment

Bloomington,

IN 47405, USA

Tallahassee, FL 32306-4120, USA

of the Pacific, Stockton, CA 95211, USA of Physics, University

of California,

Santa Barbara, CA 93106, USA

We present a comparison of two highly optimized methods for extracting the topological charge density of a lattice based on smoothing away non-topological fluctuations. The methods are Improved Cooling and Hypercubic Smearing. We find that while the two methods differ significantly in their philosophy and formulation, they produce surprisingly similar results after a few sweeps. This supports the notion that, at least after t,he initial sweeps, the surviving topological information they produce is physical.

1. INTRODUCTION Understanding the dynamics of topological excitations in QCD is an important aspect of our understanding of the theory. Instantons are responsible for chiral symmetry breaking and thus for hadronic masses, and there is much debate as to their role in confinement. However, in lattice simulations it is difficult to identify the physical topological excitations. Furthermore, the discretization of QCD on the lattice introduces both measurement difficulties and algorithmic effects in Monte Carlo updates related to instantons. In order to extract the local topological structure and explore the behavior of individual instantons, one needs to compute q(x) = FP,,hV, the local topological charge density. Over the years a number of methods have been put for*Presented by J.E. Hetrick

ward, and here we focus on “gold panning” methods [l], which iteratively smooth the gauge fields so that only topological excitations remain. In this report, we use two such methods and compare the properties of the topological charge densities they produce. Topological susceptibility results are in the contribution of DeTar [4]. The first method is Improved Cooling (hereafter ICOOL) [2] in which a generalized gauge field action is iteratively minimized. This action involves plaquette loops of up to 3 x 3 links, with coefficients chosen such that minimization should preserve the size and shape of instantons with radius above N 2~. Along with the action, this method uses a corresponding definition of FPV@’ for measuring topological charge. ICOOL iterates until reaching a plateau in Q. The second method, called Hypercubic Smearing (hereafter HYP), was introduced in 131. HYP

0920-5632/03/$- see front matter 0 2003 Elsevier Science B.V. All rights reserved. doi: 10.1016/SO920-5632(03)01679-7

110

C. Bernard et al./Nuclear

Physics B (Proc. Suppl.) 119 (2003) 769-771

smearing is a sequence of APE link fattenings that mix gauge links only within hypercubes attached to the original link, and thus is very local. It too has a corresponding Fpy~~” constructed from smeared links. To maintain locality, HYP smearing is meant to be applied only a few times. The lattices examined have moderately large volumes, 203 x 64 with a = 0.13 fm (- 150 fm4), 2+1 flavors of dynamical quarks, and improved chiral symmetry using the Asqtad action developed by MILC [5]. For this study two ensembles were examined: p = 6.76 and 6.79, with (up=down, strange) quark masses (0.01, 0.05) and (0.02, 0.05) respectively. 2. GLOBAL

TOPOLOGICAL

p = 6.76,m,d = o.ol,m, = 0.05: 210 configs =82f8 Improved cooling Hypercubic smearing < Q2 >= 79 f 10 ,lb?= 6.‘79,m,d = OSE,m, = 0.05: 187 configs =68f7 Improved cooling Hypercubic smearing < Q2 >= 68 f 7 The global topological charge average is essentially the same, within errors, for the two methods on these two lattice sets. In Fig. 1, we show the global topological charge measured with the two methods for a set of lattices in the first sequence. It is clear that the methods track each other, with an average difference here, of IAQl = 2.1. It is important to realize that these configurations, each have some 200 isolated “topological objects” remaining after smoothing. TOPOLOGICAL

I 1100

1500

1300 1400 1200 Configuration number

1600

Figure 1. Lattice by lattice comparison of ICOOL (x) and HYP (+) methods for global Q

CHARGE

Although we are interested in the local topological density, the global charge is what is used in computing the topological susceptibility. Thus a first step is to see by how much the methods differ in this global quantity. For this comparison we use 6 sweeps of HYP smearing, and about 20-30 sweeps of ICOOL.

3. LOCAL TURE

I 1000

STRUC-

Next we examine the effects of the methods on the local topological structure by comparing the

q(s) produced by each smoothing sweep. Since the two methods are rather different in their algorithmic motivation, we would like to have some idea of how one sweep of HYP smoothing compares with one sweep of ICOOL. To this end, we plot in Fig. 2 a histogram of Q(Z) values for one sweep of each method, on the same lattice. From this graph we see that the HYP method 1 ICOOL step -

I

12000 10000 $ .si z

8000

i E z’

6000 4000 2000 0 -50

-40

-30

-20

-10 q(x)

l

0 IO volume

20

30

40

50

Figure 2. q(z) histograms for one sweep of each method on the same lattice. HYP is the narrower distribution.

reduces fluctuations more effectively per sweep. This is to be expected since one HYP sweep involves a hierarchy of three successive smearing steps on links within hypercubes connected to the

C. Bernard et ul. /Nuclear

Physics B (Proc. Suppl.) II 9 (2003) 769-771

original link. It is important to understand the nature (physical or lattice artifact) of the objects being erased by both methods, and we hope to pursue this elsewhere. We visualize the smoothing process of the two methods by displaying in Fig. 3, time slice t = 0 on a typical lattice. The topological charge density is displayed when [q(x)1 is larger than qmax, where qmax is the value where the histogram of q(z) (as shown in Fig. 2) drops to a fraction (- 0.05) of the central value. The filamentary “mist” shows where q(z) = 0. Fig. 3 shows how similar the structure of q(z) is after only five sweeps of each method. While there is a difference in the number of very small objects, the overall locations and sizes of the instantons are remarkably similar. 4. CONCLUSIONS l The average < Q2 > is indistinguishable on the two ensembles studied. l Both HYP and ICOOL stabilize to a Q plateau after about the same number of sweeps. (not shown here). l Histogramming the q(z) values shows that HYP smearing narrows the distribution of q(x) throughout the lattice more rapidly. l Visualizing the smoothing sequence shows that ICOOL and HYP track each other very closely. While there are some differences at the cutoff level, the overall topological structure, q(z), remaining after just a few sweeps A full set of images is remarkably similar. of the smoothing sequence can be viewed at http://physics.sci.uop.edu/-hetrick/lat02. l The fact that both methods produce such similar results after the first few sweeps, supports the notion that the topological information they produce is physical. Computations were performed at NERSC, PSC, SDSC, and UOP with funding from the DOE and NSF.

REFERENCES 1. B. Berg, Phys. Lett. B104B (1981) 475; Y. Iwasaki and T. Yoshie, Phys. Lett., 131B

Figure 3. Five sweeps of HYP (upper) and ICOOL (lower). The two shades represent &q(z).

(1983) 159; S. Itoh, Y. Iwasaki and T. Yoshie, Phys. Lett., 147B (1983) 141; M. Teper, Phys. Lett. B162 (1985) 357; J. Hoek, M. Teper and J. Waterhouse, Nucl. Phys. B288 (1987) 889. 2. Ph. de Forcrand, M. Garcia Perez, and I.-O. Stamatescu, Nucl. Phys. B499 (1997) 409. 3. A. Hasenfratz and F. Knechtli, Phys. Rev. D64 (2001) 034504 4. C. Bernard et al. (MILC collaboration), these proceedings (presented by C. DeTar). 5. C. Bernard et al. (MILC collaboration), Phys Rev. D61 (2000) 111502