The topological charge is studied on lattices of large physical volume and
fine lattice spacing. We illustrate how a parity transformation on the SU(3)
link-variables of lattice gauge configurations reverses the sign of the
topological charge and leaves the action invariant. Random applications of the
parity transformation are proposed to traverse from one topological charge sign
to the other. The transformation provides an improved unbiased estimator of the
ensemble average and is essential in improving the ergodicity of the Markov
chain process.Comment: 4 pages, 2 figure

Allés

Anthony G. Williams

Aoki

Atiyah

Bilson-Thompson

Crewther

Dawson

de Forcrand

Del Debbio

Derek B. Leinweber

Di Vecchia

Frank X. Lee

Hart

Iwasaki

Jian-Bo Zhang

Leinweber

Leutwyler

Luscher

Orginos

Takaishi

Veneziano

Witten

Physics Letters B

English

arXiv

Copyright © 2004 Elsevier B.V. All rights reserved. Printed in U.S.A. Submitted to Cornell University’s online archive www.arXiv.org in 2004 by Jianbo Zhang. Post-print sourced from www.arxiv.org.The topological charge is studied on lattices of large physical volume and fine lattice spacing. We illustrate how a parity transformation on the SU(3) link-variables of lattice gauge configurations reverses the sign of the topological charge and leaves the action invariant. Random applications of the parity transformation are proposed to traverse from one topological charge sign to the other. The transformation provides an improved unbiased estimator of the ensemble average and is essential in improving the ergodicity of the Markov chain process.Derek B. Leinweber, Anthony G. Williams, Jian-Bo Zhang and Frank X. Leehttp://www.elsevier.com/wps/find/journaldescription.cws_home/505706/description#descriptio

Leinweber, D.

Williams, A.

Zhang, J.

Lee, F.

Adelaide Research &amp; Scholarship

arXiv:hep-lat/0312035v2  13 Feb 2004ADP-03-143/T578Topological Charge Evolution in the Markov-Chain of QCDDerek B. Leinweber,∗ Anthony G. Williams,† and Jian-bo Zhang‡Department of Physics and Special Research Centre for the SubatomicStructure of Matter, University of Adelaide 5005, AustraliaFrank X. Lee§Center for Nuclear Studies, Department of Physics,The George Washington University, Washington, D.C. 20052 andJefferson Lab, 12000 Jefferson Avenue, Newport News, VA 23606The topological charge is studied on lattices of large physical volume and fine lattice spacing. Weillustrate how a parity transformation on the SU(3) link-variables of lattice gauge configurationsreverses the sign of the topological charge and leaves the action invariant. Random applications ofthe parity transformation are proposed to traverse from one topological charge sign to the other. Thetransformation provides an improved unbiased estimator of the ensemble average and is essential inimproving the ergodicity of the Markov chain process.PACS numbers: PACS numbers: 11.15.Ha 12.38.Aw 21.10.HwI. INTRODUCTIONThe Markov chain process is the corner stone of mod-ern importance-sampling techniques for generating fieldconfigurations in the numerical simulation of quantumfield theories. Central to the process is ergodicity; theability to move through configuration space from anyparticular field configuration to any other field configura-tion with finite probability. Vacuum expectation valuesof observable operators are estimated from the average offield configurations selected with a probability given bythe exponentiation of the Euclidean action, exp(−SE),governing the quantum field theory.It is essential that these field configurations are selectedbased on the action alone, unbiased by the field configu-ration used to initiate the Markov chain. In practice thisis done by updating the field configurations for thousandsof sweeps through the lattice, monitoring the autocorrela-tion of various observables, and selecting a new represen-tative field configuration only after significant evolutionthrough configuration space.The topological charge of a gauge configuration in lat-tice QCD has already been identified as a particularly dif-ficult quantity to evolve in the Markov chain process, dis-playing unusually long autocorrelation times [1, 2, 3, 4].Some gauge actions such as the renormalization-groupblock-transform based DBW2 action [5, 6] are notoriousfor locking in the topological charge at fine lattice spac-ings [3, 4]. Difficulties associated with the Iwasaki gaugeaction [7, 8] are presented here.In this study we address an aspect of the topological∗Electronic address: dleinweb@physics.adelaide.edu.au;URL: http://www.physics.adelaide.edu.au/theory/staff/leinweber/†Electronic address: anthony.williams@adelaide.edu.au‡Electronic address: jzhang@physics.adelaide.edu.au§Electronic address: fxlee@gwu.educharge evolution problem that will present itself for anygauge field action. In particular, we address a problemassociated with the approach to the infinite-volume con-tinuum limit in numerical supercomputer simulations ofSU(3) gauge theory and QCD in general.As the physical volume, V , of the lattice increases, theaverage value of the squared topological charge increases,in accord with the topological susceptibility, given byχ =〈Q2〉V. (1)In quenched QCD, χ is related to physical hadron massesvia the large Nc Witten-Veneziano relation [9, 10]χ =f2pi2Nf(m2η +mη′ − 2m2K). (2)In full QCD, the quark mass dependence of the topolog-ical susceptibility may be related to pseudoscalar mesonproperties via the Gell-Mann–Oakes–Renner relation as[11, 12, 13, 14]χ =f2pim2pi2Nf+O(m4pi) . (3)On a sufficiently large volume lattice, the distribution ofthe topological charge is expected to be Gaussian. Re-gions of non-trivial topological charge density are uncor-related for sufficiently large separations and a normal dis-tribution will result. The distribution of Q for the Wil-son gauge action has been found to be Gaussian to a verygood approximation for both quenched QCD [15, 16] andfull QCD where the Wilson gauge action is complementedby the Wilson fermion action [1] or the Wilson-clover ac-tion [17].Alternate distributions of the topological charge canoccur in a finite volume if the correlation length of thetopological charge density for a particular lattice gaugeaction approaches the length of the lattice dimensions.2FIG. 1: Histogram illustrating the number of configurations inour ensemble of 250 configurations having a particular topo-logical charge, Q. A suppression of Q ∼ 0 configurations isobserved.Figure 1 displays a histogram of the topological chargefrom an ensemble of 250 quenched Iwasaki [7, 8] gaugeconfigurations at β = 9.1674 on a 283× 44 lattice, wherethe lattice spacing a is 0.113 fm. The topological charge iscalculated using the highly-improved, three-loop O(a4)-improved lattice field-strength tensor [18] on 10-sweepcooled configurations obtained with a three-loop O(a4)-improved action. The gauge configurations representedin Fig. 1 are separated by 1000 pseudo-heatbath sweeps inan attempt to obtain uncorrelated configurations. Fig. 2illustrates the time evolution of the topological chargeplotted as a function of simulation time, represented bythe configuration number. Acceptable movement of thetopological charge is indicated by the rapidity and am-plitude of the oscillations. To the best of our knowl-edge, this is the first time a double-peaked structurein the probability distribution of the topological chargehas been revealed for a renormalization-group improvedgauge action. It would be interesting to examine thisdistribution for the DBW2 action.For any lattice action, the distribution of the topo-logical charge will broaden as the volume of the latticeincreases, such that〈Q2〉 = χV . (4)As the infinite-volume continuum limit is approached andcorrelation lengths diverge, it will become increasinglydifficult to evolve the topological charge over the broadrange demanded by Eq. (4). In particular, a symmet-ric distribution about Q = 0 is required to preserve thesymmetries of QCD.Recognizing the need for improvement in the evolutionof the topological charge in the Markov chain process,we present a simple transformation in the following sec-tion, that may be applied to a gauge-field configuration ofquenched QCD or dynamical-fermion QCD that changesFIG. 2: The time evolution of the topological charge plottedas a function of simulation time as represented by the config-uration number. 1000 pseudo-heatbath sweeps are performedbetween each gauge field sample.the sign of Q while leaving the action invariant. Thetransformation provides an improved unbiased estimatorof the ensemble average and is essential in improving theergodicity of the Markov chain process.II. GAUGE-LINK TRANSFORMATIONThe index theorem [19] relates the topological chargeQ to the chirality index of the Dirac operator D on acontinuum 4-torus asQ = index(D) ≡ n− − n+ . (5)Here n+ and n− are the number of exact zero eigen-modes,Dψ = 0, with positive, γ5ψ = +ψ, (right-handed)and negative, γ5ψ = −ψ, (left-handed) chiralities respec-tively.This link between the topological charge and the chi-rality (or helicity) of zero eigenmodes of the Dirac op-erator identifies parity as the transformation for chang-ing the sign of the topological charge while leaving theaction invariant. Helicity transforms as a pseudoscalarunder rotations, whereas the action of QCD, designed toconserve parity, transforms as a scalar under rotations.For the improper rotation of the parity transformation,the right-handed modes will become the left-handed, andvice versa, changing the sign of Q.In deriving the transformation of the links under par-ity, we begin with the transformation of the gauge po-tential. Under the parity transformationAi(~x, t)→ −Ai(−~x, t) , and ∂i → −∂i , (6)for the spatial indices i = 1, 2, 3 andA4(~x, t)→ A4(−~x, t) , and ∂4 → ∂4 . (7)3Here the 3×3 colour matrix degrees of freedom of Aµ(x)are implicit. The non-abelian field strength tensor Fµν isFµν = ∂µAν − ∂νAµ + i g [Aµ, Aν ] . (8)Under parity transformationsFij → Fij , F4j → −F4j , and Fi4 → −Fi4 . (9)Hence the actionS =12∫d4xTr (Fµν(x)Fµν(x)) , (10)is invariant under parity while the topological chargeQ =g232π2∫d4x ǫµνρσ Tr (Fµν(x)Fρσ(x)) , (11)changes sign due to the presence of ǫµνρσ ensuring thepresence of one and only one time component in the fieldstrength tensor product.The gauge-field links are related to the gauge potentialviaUµ(x) = exp(i g∫ a0Aµ(x+ λ µˆ) dλ), (12)where x ≡ (~x, t). Under the parity transformation, thespatial components, i = 1, 2, 3, of the links transform asP Ui(x)P† = exp(−i g∫ a0Ai(−x˜− λ ıˆ) dλ), (13)= U †i (−x˜− aıˆ) , (14)where− x˜ ≡ (−~x, t) . (15)Similarly, the time components transform asP Ut(x)P† = exp(i g∫ a0At(−x˜+ λ tˆ) dλ), (16)= Ut(−x˜) . (17)The exact nature of the parity transformation on thelinks follows from the fact that parity is an exact sym-metry on the hypercubic lattice.It is interesting to examine the manner in which thelattice action density and lattice field-strength tensortransform under parity. Consider the product of linksabout an elementary plaquette located at space-timepoint x, in the µ - ν planeUµν(x) ≡ Uµ(x)Uν(x+ µˆ)U†µ(x + νˆ)U†ν (x) , (18)and its transformation under parity. For spatially ori-ented plaquettes, the untransformed plaquette Uij(x)originates from x, and loops counter-clockwise in the pos-itive ıˆ, ˆ directionUi(x)Uj(x+ ıˆ)U†i (x+ ˆ)U†j (x) . (19)Under the parity transformation Uij(x) transforms toU†i (−x˜− ıˆ)U†j (−x˜− ıˆ− ˆ)Ui(−x˜− ˆ− ıˆ)Uj(−x˜− ˆ) , (20)which is the spatial plaquette originating from −x˜, loop-ing in a counter-clockwise orientation again, this time inthe negative ıˆ, ˆ direction. The space-time oriented pla-quettes originating from x and looping counter-clockwisein the positive ıˆ, tˆ directionUi(x)Ut(x+ ıˆ)U†i (x+ tˆ)U†t (x) , (21)transform toU†i (−x˜− ıˆ)Ut(−x˜− ıˆ)Ui(−x˜− ıˆ+ tˆ)U†t (−x˜) , (22)which is the space-time oriented plaquette originatingfrom −x˜ but this time looping clockwise in the negativeıˆ, positive tˆ direction.The connection between these link products and thelattice field strength tensor is provided byg Fµν(x) =18 i[(Oµν(x)−O†µν(x))−13Tr(Oµν(x) −O†µν(x))], (23)where Oµν(x) is the sum of 1× 1 link paths orientedabout x in the µ - ν planeOµν(x) = Uµ(x)Uν(x + µˆ)U†µ(x+ νˆ)U†ν (x)+ Uν(x)U†µ(x+ νˆ − µˆ)U†ν (x− µˆ)Uµ(x− µˆ)+ U †µ(x− µˆ)U†ν (x− µˆ− νˆ)Uµ(x− µˆ− νˆ)Uν(x − νˆ)+ U †ν (x − νˆ)Uµ(x− νˆ)Uν(x+ µˆ− νˆ)U†µ(x) . (24)Taking the Hermitian conjugate of Oµν(x) changes theorientation of the link products from counter-clockwise toclockwise. Noting further that Fµν(x) is odd under suchtransformations, we see that the orientations of the link-product parity transformations of Eqs. (20) and (22) areprecisely those required to recover the continuum trans-formations of Fµν(x) in Eq. (9). While the topologicalcharge density undergoes a parity transformation, themagnitude of the topological charge remains invariant,as all lattice sites are summed over. Similarly, the latticeactionS = β∑xµ<ν16Tr(2− Uµν(x) − U†µν(x)), (25)is even under the orientation transformation Uµν(x) →U †µν(x), and since all sites are summed over, the latticegauge action is invariant under the lattice parity trans-formation.III. NUMERICAL RESULTSThe parity transformations of the gauge links inEqs. (14) and (17) have been coded in Fortran 90 [20]4and tested on a 123×24 lattice with the tadpole-improvedLu¨scher-Weisz gauge action [21]. We use three methodsto examine the properties of the parity transformed con-figurations.The 3-loop improved field strength tensor [18] accom-panied by the associated 3-loop improved cooling actionare used to provide a determination of the topologicalcharge Q and action S for a cooled gauge configurationand its parity transformed partner. The transformed ac-tion and topological charge magnitude agree to machineprecision.The low-lying spectral flow of the hermitian Wilson-Dirac operator HW = γ5DW is examined. Under theparity transformation, the low-lying eigenvalues of HWchange sign such that the slope of the spectral flow (pro-portional to the topological charge giving rise to the ze-romode) changes sign. Since HW is of even dimension(lattice volume × number of colors × number of Diracindices), det(HW ) = det(γ5)× det(DW ) is unaffected bythe parity transformation, as required to preserve parityin dynamical fermion QCD simulations.Finally, the topological charge is determined by count-ing the exact zeromodes of the massless overlap Diracoperator. Under a parity transformation, the number ofexact zeromodes with positive (right-handed) and nega-tive (left-handed) chiralities exchange as expected.IV. DISCUSSION AND CONCLUSIONSThe manner in which the parity transformation is usedin practice will depend on the application at hand forthe gauge fields. Often, exact parity can be enforcedduring the construction of correlation functions by av-eraging opposite-sign nontrivial momenta of the latticeGreen’s functions [22]. In other applications, exact par-ity can be enforced by doubling the number of gauge fieldconfigurations, averaging each field configuration with itsparity transformed partner. Finally, the transform canbe applied in the Markov chain process itself, applyingthe transform randomly with a 50% probability prior towriting the gauge configuration to disk, or equivalently,following the reading of the configuration to be evolved,from disk.Motivated by the lattice index theorem, we have illus-trated how a parity transformation on the SU(3) link-variables of lattice gauge configurations reverses the signof topological charge while leaving the action invariant.The transformation provides an improved unbiased esti-mator of the ensemble average and is essential in improv-ing the ergodicity of the Markov chain process.We thank Philippe De Forcrand, Alistair Hart and Et-tore Vicari for beneficial correspondence. Supercomput-ing support from the Australian National Facility for Lat-tice Gauge Theory, The South Australian Partnershipfor Advanced Computing (SAPAC) and the AustralianPartnership for Advanced Computing (APAC) is grate-fully acknowledged. This research is supported by theAustralian Research Council.[1] B. Alle´s et al., Phys. Rev. D58, 071503 (1998).[2] C. Dawson [RBC Collaboration],[arXiv:hep-lat/0310055].[3] K. Orginos [RBC Collaboration], Nucl. Phys. B (Proc.Suppl.) 106, 721 (2002).[4] Y. Aoki et al., [arXiv:hep-lat/0211023].[5] T. Takaishi, Phys. Rev. D 54, 1050 (1996).[6] P. de Forcrand et al. [QCD-TARO Collaboration], Nucl.Phys. B 577, 263 (2000) [arXiv:hep-lat/9911033].[7] Y. Iwasaki, Nucl. Phys. B 258, 141 (1985).[8] Y. Iwasaki, University of Tsukuba report UTHEP-118(1983), unpublished.[9] E. Witten, Nucl. Phys. B 156, 269 (1979).[10] G. Veneziano, Nucl. Phys. B 159, 213 (1979).[11] R. J. Crewther, Phys. Lett. B 70, 349 (1977).[12] P. Di Vecchia and G. Veneziano, Nucl. Phys. B 171, 253(1980).[13] H. Leutwyler and A. Smilga, Phys. Rev. D 46, 5607(1992).[14] A. Hart and M. Teper [UKQCD Collaboration], Phys.Lett. B 523, 280 (2001) [arXiv:hep-lat/0108006].[15] L. Del Debbio, H. Panagopoulos and E. Vicari, JHEP0208, 044 (2002) [arXiv:hep-th/0204125].[16] L. Del Debbio, H. Panagopoulos and E. Vi-cari, Nucl. Phys. Proc. Suppl. 119, 661 (2003)[arXiv:hep-th/0208208].[17] A. Hart [UKQCD and QCDSF Collaborations],arXiv:hep-lat/0401015.[18] S. O. Bilson-Thompson, D. B. Leinweber andA. G. Williams, Annals Phys. 304, 1 (2003)[arXiv:hep-lat/0203008].[19] M. F. Atiyah, I.M. Singer, Ann. Math. 87, 484 (1968);ibid. 93, 119 (1971).[20] http://www.physics.adelaide.edu.au/cssm/lattice/code/parityTransform.f[21] M. Luscher and P. Weisz, Commun. Math. Phys. 97, 59(1985) [ibid. 98, 433 (1985)].[22] D. B. Leinweber, R. M. Woloshyn and T. Draper, Phys.Rev. D 43, 1659 (1991).

Topological charge evolution in the Markov chain of QCD

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Topological Charge Evolution in the Markov-Chain of QCD

Topological Charge Evolution in the Markov-Chain of QCD

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Topological Charge Evolution in the Markov-Chain of QCD

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