The action of the overlap-Dirac operator on a vector is typically implemented
in directly through a multi-shift conjugate gradient solver. The compute-time
this takes to evaluate depends upon the condition number $\kappa$ of the matrix
that is used as the overlap kernel. We examine the low-lying spectra of various
candidate kernels in an effort to optimise $\kappa$, thereby speeding up the
overlap evaluation.Comment: 5 pages, 8 figure

A.G. Williams

Bonnet

D. Adams

D.B. Leinweber

Edwards

Neuberger

W. Kamleh

English

arXiv

Copyright © 2002 Published by Elsevier Science B.V. All rights reserved. Copyright © 2001. Submitted to Cornell University’s online archive www.arXiv.org in 2001 by Waseem Kamleh. Post-print sourced from www.arxiv.org.The action of the overlap-Dirac operator on a vector is typically implemented
in directly through a multi-shift conjugate gradient solver. The compute-time
this takes to evaluate depends upon the condition number $\kappa$ of the matrix
that is used as the overlap kernel. We examine the low-lying spectra of various
candidate kernels in an effort to optimise $\kappa$, thereby speeding up the
overlap evaluation.W. Kamleh, D. Adams, D. B. Leinweber and A. G. Williamshttp://www.elsevier.com/wps/find/journaldescription.cws_home/505717/description#descriptio

Kamleh, W.

Adams, D.

Leinweber, D.

Williams, A.

Adelaide Research & Scholarship

arXiv:hep-lat/0112042v1  20 Dec 2001ADP-01-47/T479Improving the low-lying spectrum of the overlap kernelW. Kamleha∗, D. Adamsa†, D.B. Leinwebera‡, A.G. Williamsa§aSpecial Research Centre for the Subatomic Structure of Matter (CSSM) and Department of Physicsand Mathematical Physics, University of Adelaide 5005, Australia.The action of the overlap-Dirac operator on a vector is typically implemented indirectly through a multi-shiftconjugate gradient solver. The compute-time this takes to evaluate depends upon the condition number κ of thematrix that is used as the overlap kernel. We examine the low-lying spectra of various candidate kernels in aneffort to optimise κ, thereby speeding up the overlap evaluation.1. IntroductionOverlap fermions are a realisation of chiralsymmetry on the lattice. Given some reasonableHermitian-Dirac operator H , we can deform Hinto a chiral action through the overlap formal-ism,Do =12(1 + γ5ǫ(H)), ǫ(H) =H√H2.(1)Unfortunately, the matrix sign function ǫ(H) isdifficult to evaluate and is typically approximatedby a sum over poles [1] which can be evaluatedusing a multi-shift conjugate gradient (CG) solver[2]. This is an iterative approximation where thenumber of iterations for a given accuracy dependsupon the condition number of the kernel, κ(H) =|λmax/λmin|.Usually the Hermitian Wilson-Dirac operatoris used as the overlap kernel. Its low-lying spec-trum is characterised by a handful of isolatedeigenmodes which can be very small, increasingκ unacceptably. These eigenmodes can be pro-jected out of the basic operator, reducing its con-dition number to a numerically acceptable level,and then dealt with explicitly[3]. Unfortunately,as the spectrum rapidly becomes dense, project-ing out low-lying modes can only help one so far.An alternative is to use a kernel with an im-proved spectrum, that is, where the region of∗wkamleh@physics.adelaide.edu.au†dadams@physics.adelaide.edu.au‡dleinweb@physics.adelaide.edu.au§awilliam@physics.adelaide.edu.audense modes is shifted away from zero. In thispaper we consider six different actions as poten-tial kernels. The first three actions are the stan-dard Wilson with and without the addition of theclover term at both tree-level and with a mean-field improved coefficient. The last three actionsare derived from the first three actions by APE-smearing the irrelevant terms. The details of theactions are given in [4], in particular, the FLIC(Fat Link Irrelevant Clover)[5] action is definedthere.2. Spectral Flow ComparisonIn order to test the merits of each of ourproposed actions, we calculate the spectral flowof each of them to see if our reasoning regard-ing their low-lying spectra is correct. From thequadratic form of the lower bounds as a functionof m, and based upon results given in [6], we ex-pect there to be some peak value of m for whichthe gap around zero is the largest. We calculatedthe flow of the lowest 15 eigenvalues as a functionof m for two ensembles of 10 L = 83 × 16 mean-field improved Symanzik configurations, one fineensemble at β = 4.80 and one coarse ensemble atβ = 4.38. The following flow graphs allow us tosee the m value for the biggest gap, and also al-low us to compare the different actions. As we areinterested in the magnitude of the low-lying val-ues rather than their sign, we plot |λ| vs m. Thesmall physical extent of the fine lattices leads toqualitatively different behaviour in the spectralflow from the coarse lattice.2Figure 1. Spectral flow of the Wilson action atβ = 4.80 (top) and 4.38 (bottom).We begin by examining the flow of the Wilsonaction, in Figure 1. We see on the fine latticesthat there is a reasonable gap away from zero anda handful of isolated low-lying eigenmodes beforethe spectra becomes dense. On the coarse lat-tices the Wilson spectra is very poor, with a highdensity of very small eigenmodes.The addition of the clover term (at csw = 1)provides some improvement (Figure 2), shiftingthe flow upwards on the fine lattices and movingthe peak gap to m = 1 as expected. The cloveraction still performs poorly on the coarse latticeshowever, as the presence of many small eigen-modes persists, although their density is clearlyFigure 2. Spectral flow of the clover action atβ = 4.80 (top) and 4.38 (bottom).reduced.Mean field improvement assists the basic cloveraction somewhat (Figure 3), spreading the spec-trum upwards, although the lowest modes arenot raised significantly. The mass value at whichthe peak gap occurs has moved significantly fromm = 1 to m = 0.6. As mentioned earlier, es-sentially all MFI does in this case is to changethe value of csw to 1.0/u30, pushing it towards itsnon-perturbative value.Modifying the Wilson action by smearing theirrelevant operators provides a considerable im-provement (Figure 4), enhancing the gap aroundzero on the fine lattices significantly. While there3Figure 3. Spectral flow of the MFI clover actionat β = 4.80 (top) and 4.38 (bottom).are still some small modes present on the coarselattices, their density has been greatly reduced,and the spectral flow now has a clear division be-tween the isolated low-lying modes and the modeswhere the spectral density becomes high. The lat-ter are well separated from zero. Smearing wasperformed with α = 0.7 and nape = 12 smearingsweeps.The spectral flow of the fat clover action clearlydemonstrates the superiority of clover-improvedactions (Figure 5). The gap around zero is en-hanced again over the fat Wilson action, and thenumber of isolated low-lying modes is reduced.The effect is very pronounced on the coarse lat-Figure 4. Spectral flow of the fat Wilson actionat β = 4.80 (top) and 4.38 (bottom).tice.As the fat links are already close to unity, theaddition of mean field improvement only affectsthe fat clover flow slightly (Figure 6), raising thegap around zero a little and spreading the eigen-values upwards slightly also. The low-lying den-sity is again good in this case and far superior tothat of the Wilson in Figure 1. On the fine lat-tices we note that we have a peak gap at m = 1.4of 0.5! On the coarse lattices the region wherethe eigenmodes start becoming dense occcurs atabout 0.35 at the same mass value, indicatingthat the FLIC action should be extremely well-conditioned on both ensembles, especially so on4Figure 5. Spectral flow of the fat clover actionat β = 4.80 (top) and 4.38 (bottom).the coarse lattice if we project out the isolatedlow-lying modes.Additionally, we tested the dependence of thefat clover and FLIC actions upon the amount ofsmearing done (Figures 7,8). As stated in [7], weonly effectively need to vary the product αnape,so we fix α at 0.7 and vary nape between 0 and 12.We observe that in both the case of the fat cloverand FLIC actions the initial 4-6 sweeps have asignificant effect, but past 6 sweeps the effect ismarginal, with the low lying density remainingconstant and the eigenvalues being compressedvery slightly downwards.Figure 6. Spectral flow of the FLIC action atβ = 4.80 (top) and 4.38 (bottom).3. ConclusionThe addition of the clover-term at tree-level orwith a mean field improved coefficient does notdrastically improve the low-lying spectrum of theWilson action at an optimal choice of m. How-ever, APE smearing the irrelevant-terms in theWilson action both with and without the cloverterm provides a marked improvement in the low-lying spectrum, reducing the number of isolatedlow-lying modes and shifting the spectrum awayfrom zero. The combination of APE-smearingand the clover term at a mean-field improved co-efficient provides the best overall low-lying spec-5Figure 7. Dependence of the fat clover spec-trum at β = 4.80 (top) and 4.38 (bottom) on thenumber of APE smearing sweeps.tra. Thus we conclude that the FLIC action is thebest candidate for the overlap kernel out of thesix actions tested. These results are built upon in[4] where a comprehensive numerical comparsionis undertaken.REFERENCES1. H. Neuberger, Phys. Rev. Lett. 81 (1998)4060, hep-lat/9806025.2. R.G. Edwards, U.M. Heller and R.Narayanan, Nucl. Phys. B540 (1999)457, hep-lat/9807017.Figure 8. Dependence of the FLIC spectrum atβ = 4.80 (top) and 4.38 (bottom) on the numberof APE smearing sweeps.3. R.G. Edwards, U.M. Heller and R.Narayanan, (1999), hep-lat/9905028.4. W. Kamleh et al., (2001), hep-lat/0112041.5. J.M. Zanotti et al., (2001), hep-lat/0110216,and these proceedings.6. R.G. Edwards, U.M. Heller and R.Narayanan, Nucl. Phys. B535 (1998)403, hep-lat/9802016.7. F.D.R. Bonnet et al., Phys. Rev. D62 (2000)094509, hep-lat/0001018.

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