We present a cooling method controlled by a physical cooling radius that
defines a scale below which fluctuations are smoothed out while leaving physics
unchanged at all larger scales. This method can be generally used as a gauge
invariant low pass filter to extract the physics from noisy MC configurations.
Here we apply this method to study topological properties of lattice gauge
theories where it allows to retain instanton--anti-instanton pairs.Comment: Lattice'99 contribution, 3 pages, 5 figure

Berg

Campostrini

de Forcrand

De Grand

DeGrand

Feurstein

García Pérez

Ion-Olimpiu Stamatescu

Margarita García Pérez

Owe Philipsen

Smith

Teper

English

arXiv

We present a cooling method controlled by a physical cooling radius that defines a scale below which fluctuations are smoothed out while leaving physics unchanged at all larger scales. This method can be generally used as a gauge invariant low pass filter to extract the physics from noisy MC configurations. Here we apply this method to study topological properties of lattice gauge theories where it allows to retain instanton--anti-instanton pairs.We present a cooling method controlled by a physical cooling radius that defines a scale below which fluctuations are smoothed out while leaving physics unchanged at all larger scales. This method can be generally used as a gauge invariant low pass filter to extract the physics from noisy MC configurations. Here we apply this method to study topological properties of lattice gauge theories where it allows to retain instanton--anti-instanton pairs

Garcia Perez, Margarita

Philipsen, Owe

Stamatescu, Ion-Olimpiu

CERN Document Server

1Cooling, Physical Scales and the Vacuum Structure of Y-M Theories.Margarita Garca Pereza, Owe Philipsenb and Ion-Olimpiu StamatescucaDepartamento de Fsica Teorica, Univ. Autonoma de Madrid, Canto Blanco, 28049 Madrid, SpainbTheory Division, CERN, CH-1211 Geneva 23, SwitzerlandcFESt, Schmeilweg 5, D-69118 Heidelberg and ITP, Philosophenweg 16, D-69120 Heidelberg, GermanyWe present a cooling method controlled by a physical cooling radius that denes a scale below which fluctuationsare smoothed out while leaving physics unchanged at all larger scales. This method can be generally used as agauge invariant low pass lter to extract the physics from noisy MC congurations. Here we apply this methodto study topological properties of lattice gauge theories where it allows to retain instanton{anti-instanton pairs.Since MC congurations are noisy at small dis-tances, smoothing is needed to observe the physi-cal structure. One approach is \cooling" [1], a lo-cal minimization of a given lattice action. Coolingworks as a diusion process, smoothing out in-creasingly large regions [2]. Thereby the physicalspectrum is aected, in particular the string ten-sion drops rapidly with cooling. Topological exci-tations may be distorted by bad scaling propertiesof the action and by instanton - anti-instanton(I-A) annihilation (pairs are not minima of theaction). We thus need:1) to use an action with practically scale invariantinstanton solutions and a dislocation threshold toeliminate UV noise (dislocations), and2) to control cooling by means of a physical scalesuch that no monitoring or engineering whichmay introduce uncertainties is necessary.Restricted Improved Cooling (RIC) fullls theserequirements [3]. RIC preserves physics at scalesabove a cooling radius r which can be xed un-equivocally beforehand, while smoothing out thestructure below r. In particular the string ten-sion is preserved beyond r, instantons are stable,dislocations are eliminated, and I-A pairs are re-tained above a threshold dened by r.Properties of RIC. RIC uses the action of theImproved Cooling algorithm (IC) with 5 planar,fundamental Wilson loops [4]. This action is cor-∗Partial support from the European Network “Finite Tem-perature Phase Transitions in Particle Physics”.rect to order O(a6) and has a dislocation thresh-old ρ0  2.3a, below which short range topologi-cal structure is smoothed out (note that ρ0 ! 0in approaching continuum). Above ρ0, instan-tons are stable to any degree of cooling (how-ever, I-A pairs annihilate). The correspondingimproved charge density using the same combina-tion of loops leads to an integer charge alreadyafter a few cooling sweeps and stable thereafter.Here we shall restrict ourselves to SU(2), forthe general case see [3]. Recall that the coolingalgorithm is derived from the equations of motionUµ(x)Wµ(x)y −Wµ(x)Uµ(x)y = 0, (1)where W is the sum of staples connected to thelink Uµ(x) in the action. It amounts to substituteU ! U 0 = V = WjjW jj , jjW jj2 =12Tr(WW y). (2)We dene RIC by the constraint2 that only thoselinks be updated, which violate the equation ofmotion by more than some chosen threshold:U ! V i µ(x)2 = 1a6Tr(1− UV y)  δ2. (3)We have 2µ(x) / −Tr((DνFνµ(x))2) in contin-uum limit [5]. Thus δ controls the energy of thefluctuations around classical solutions and actsas a lter for short wavelengths. Since it uses thesame action RIC has the same scaling properties2We thank F. Niedermayer for this suggestion.2as IC. However, since RIC does not update linksalready close to a solution, it changes fewer linksafter every iteration until the algorithm saturates.(3) denes a constrained minimization and thesmoothing is homogeneous over the lattice.The next step is to relate the parameter δ whichdenes the cooling to a physical scale. For YangMills theory the latter should involve the stringtension σ. We calculate the \eective mass" M(t)from correlation functions of spatial Polyakovloops separated by t steps in time. Asymptoti-cally M(t) ’ Nsσ up to nite size corrections.For the calculations we generate congurationsusing the Wilson action on the lattices:123  36, p.b.c., β = 2.4 (a = 0.12 fm): 800congurations with 100 sweeps separation;244, twist in time, β = 2.6 (a = 0.06 fm): 350congurations with 200 sweeps separation.We do 20000 thermalization sweeps. We use δ =2.89, 4.92, 11.57, 23.15 and 46.30 fm−3.For illustration we present in Fig. 1 M(t) forthe 244 lattice. Dening r(δ) as the distance tat which M(t) on δ-cooled congurations startsto agree with the uncooled value (obtained byfuzzing and tted to a smooth function of t) wearrive at the results in Fig. 2, compatible withr(δ) ’ 0.8 δ−1/3, (4)showing the correct scaling behaviour, as ex-pected since δ has a continuum limit.Figure 1. M(t) for the a = 0.12 fm lattice. Thehorizontal band represents standard results for σ.The behaviour of instantons is similar underRIC and IC whereby the former has a somewhatsmaller, δ-dependent dislocation threshold ρ0(δ).I-A pairs, however, do not annihilate under RICthe way they do for the other cooling methods0.20.30.40.50.60.70.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9Figure 2. The smoothing scale of RIC: r(δ)vs δ−1/3 for the a = 0.12 fm (diamonds) anda = 0.06 fm (squares) lattices (the middle pointsare slightly displaced around the true positions forreadability). Also shown is the fit (4).(including IC): since the distortion of the part-ners in a pair depends on the overlap, RIC pre-serves pairs below some overlap-threshold. Moreprecisely, it turns out that there is a well denedrelation between  and SIAint = 16pi2−SIA, there-fore RIC stabilizes I-A pairs with Sint below athreshold which is a function of δ, hence of r.SU(2) topological properties by RIC. Thetotal, improved topological charge stabilizes tointeger values after only few IC or RIC sweeps.Therefore the topological susceptibility appearsvery stable if δ is small enough to eliminate dis-locations, shows correct scaling and agrees withthe Witten-Veneziano relation.To describe the details of the instanton ensem-ble one needs to recognize the particular I’s andA’s. Since the objects can be very distorted, ourdescription relies on two assumptions: a) instan-tons should appear as local self-dual peaks in theaction and charge density, and b) only pairs withSIAint considerably smaller than 16pi2 should beconsidered as such. We approximate the actionand charge density by a superposition of I’s andA’s parameterized through the BPST formula foran instanton of size ρ. We measure the adequacyof this ansatz through the fraction of the totalaction (charge) reproduced in this way. We rep-resent the data in physical units using the corre-sponding a and the r(δ) obtained from the stringtension analysis without any further t or tuning.The main results of our analysis are:1. The density of instantons increases drasticallywith decreasing smoothing scale r.3Figure 3. Normalized size distributions on thea = 0.12 fm lattice (squares, triangles) and a =0.06 fm lattice (crosses). Vertical dotted (dashed)lines indicate the IC dislocation threshold ρ0 =2.3a for a = 0.12 (0.06) fm.0200040006000800010000120000.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Figure 4. Non-normalized overlap distributionsfor a = 0.06 fm and δ = 2.89 (circles), 4.92(bars), 11.57 (squares) and 23.15 (crosses) fm−3.2. Size distributions depend only weakly on r.3. The typical I-A distance dIA (from an I to thenearest A) seems given by the smoothing scale.4. The overlap (ρI + ρA)/2dIA increases stronglywith decreasing r and the t quality deteriorates.All these results are largely invariant underrescaling of the lattice spacing a. They are sum-marized in the Figures 3, 4 and 5.It appears therefore that there are topologicalfeatures of the vacuum structure, like suscepti-bility and charge distributions and to a fair de-gree also size distributions, which are well dened,show correct scaling and are rather independenton the smoothing scale once the noise has beendamped suciently. However, I(A)-density, over-lap and I-A distance distributions are not of thiskind: although these quantities seem to scale cor-Figure 5. Dependence of the average size (tri-angles), the average I - A distance (diamonds)and the instanton density (squares) on the cool-ing radius r. The dashed curve is hN/V i =hN/V iasymptotic + C/r4. Open (filled) symbolscorrespond to the a = 0.12 (0.06) fm lattice.rectly with the cut o, their behaviour with r sug-gests that looking at smaller and smaller scales acontinuous spectrum of fluctuations emerges anda description in terms of I’s, A’s and I-A pairsbecomes less and less meaningful. This may ex-plain both the agreements and the disagreementsbetween various analyses { see, e.g. [6].Generally RIC proves itself a good instrumentfor probing dierent scales in a controlled wayand extract physics from noisy congurations.REFERENCES1. B. Berg, Phys. Lett. B104 (1981) 475; M. Te-per, Phys. Lett. B162 (1985) 357.2. M. Campostrini, A. Di Giacomo, M. Mag-giore, H. Panagopoulos and E. Vicari,Phys. Lett. B225 (1989) 403.3. M. Garca Perez, O. Philipsen and I.-O. Sta-matescu, Nucl. Phys. B551 (1999) 293.4. P. de Forcrand, M. Garca Perez and I.-O. Stamatescu, Nucl. Phys. Proc. Suppl. 47(1996) 777; Nucl. Phys. B499 (1997) 409.5. M. Garca Perez and P. van Baal, Nucl. Phys.B429 (1994) 451.6. T. DeGrand, A. Hasenfratz and D. Zhu, Nucl.Phys. B475 (1996) 321; M. Feurstein, E. M. Il-genfritz, M. Mu¨ller-Preussker and S. Thurner,Nucl. Phys. B511 (1998) 421; T. De Grand,A. Hasenfratz and T. G. Kovacs, Nucl. Phys.B520 (1998) 301; D. A. Smith and M. Teper,Phys. Rev. D58 (1998) 014505.

Cooling, Physical Scales and the Vacuum Structure of Y-M Theories

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Cooling, physical scales and the vacuum structure of Y-M theories

Cooling, Physical Scales and the Vacuum Structure of Y-M Theories

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