Domain-wall fermions preserve chiral symmetry up to terms that decrease
exponentially when the lattice size in the fifth dimension is taken to
infinity. The associated rates of convergence are given by the low-lying
eigenvalues of a simple local operator in four dimensions. These can be
computed using the Ritz functional technique and it turns out that the
convergence tends to be extremely slow in the range of lattice spacings
relevant to large-volume numerical simulations of lattice QCD. Two methods to
improve on this situation are discussed.Comment: 14 pages, talk given by P. H. at the workshop on {\em Current
  theoretical problems in lattice field theory}, Ringberg, German

Hernández, P.

Jansen, K.

Lüscher, M.

English

arXiv

Domain-wall fermions preserve chiral symmetry up to terms that decrease exponentially when the lattice size in the fifth dimension is taken to infinity. The associated rates of convergence are given by the low-lying eigenvalues of a simple local operator in four dimensions. These can be computed using the Ritz functional technique and it turns out that the convergence tends to be extremely slow in the range of lattice spacings relevant to large-volume numerical simulations of lattice QCD. Two methods to improve on this situation are discussed.Domain-wall fermions preserve chiral symmetry up to terms that decrease exponentially when the lattice size in the fifth dimension is taken to infinity. The associated rates of convergence are given by the low-lying eigenvalues of a simple local operator in four dimensions. These can be computed using the Ritz functional technique and it turns out that the convergence tends to be extremely slow in the range of lattice spacings relevant to large-volume numerical simulations of lattice QCD. Two methods to improve on this situation are discussed

Hernandez, Pilar

Jansen, Karl

Luscher, Martin

CERN Document Server

CERN-TH/2000-196IFIC-00/41FTUV-00-0712A note on the practical feasibility of domain-wallfermions Pilar Hernandezx, Karl Jansen{ and Martin Lu¨scherkCERN, Theory Division, CH-1211 Geneva 23, SwitzerlandJuly 14, 2000AbstractDomain-wall fermions preserve chiral symmetry up to terms that de-crease exponentially when the lattice size in the fifth dimension is takento infinity. The associated rates of convergence are given by the low-lyingeigenvalues of a simple local operator in four dimensions. These can becomputed using the Ritz functional technique and it turns out that theconvergence tends to be extremely slow in the range of lattice spacings rel-evant to large-volume numerical simulations of lattice QCD. Two methodsto improve on this situation are discussed.∗Talk given by P. Herna´ndez at the workshop on Current theoretical problems in lattice fieldtheory, 2 – 8 April 2000, Ringberg, Germany.§On leave from Departamento de F´ısica Teo´rica, Universidad de Valencia, Spain.¶Heisenberg Foundation Fellow.‖On leave from Deutsches Elektronen-Synchrotron DESY, D-22603 Hamburg, Germany.1IntroductionThe idea to realize 4-dimensional (4D) chiral fermions on the lattice by coupling5D fermions to a 4D domain wall [1] has attracted a lot of attention in the latticecommunity (for a review see [2]). Although originally designed to construct chiralgauge theories, the idea can also be applied to lattice QCD in order to preservethe global chiral symmetry at zero quark mass [3, 4]. In this case, a 5D Wilson{Dirac operator is chosen with N slices in the extra dimension and appropriateDirichlet boundary conditions in the fth dimension. In the limit N !1, chiralzero modes exist as surface modes on the 4D boundaries, even at nite latticespacing.The bulk fermionic degrees of freedom are massive and can be shown to decouplein the continuum limit: the action of the 5D system is equivalent to the onecorresponding to a 4D Dirac operator describing the boundary chiral modes [5];similarly, the propagator of the boundary elds can be obtained from the inverseof the same 4D operator [6]. The chirality of the surface modes in the limitN !1 then follows [6] from the fact that, in this limit, this 4D Dirac operatorsatises the Ginsparg{Wilson (GW) relation [7]{[9], which implies an exact latticechiral symmetry [10]. Thus the 5D domain-wall construction in the limit N !1is completely equivalent to a 4D lattice formulation of Ginsparg{Wilson fermions,and satises all the properties that follow from the exact chiral symmetry [8, 11,12, 10, 13]. Moreover, if the continuum limit is taken in the extra dimension, this4D formulation coincides with that using Neuberger’s fermion operator [14, 9].The introduction of an extra dimension makes domain-wall fermions more de-manding numerically than standard Wilson fermions (the equivalent 4D formu-lation is similarly more demanding, owing to the non-ultralocality of the action).Nevertheless the advantage of preserving an exact chiral symmetry might com-pensate for the higher cost in some cases. An analysis in the free theory showedthat the convergence to the exact operator as a function of N is rapid [3, 15].This gave rise to the hope that also in the interacting case domain-wall fermionscould be used without too much computational overhead. However, in realisticsimulations, there are indications that the convergence rate deteriorates rapidlyat large values of the gauge coupling, and much larger values of N are indeedneeded [16]{[18], leading to a substantial computational cost.2In a recent paper [19], the problems found in practical simulations were tracedback to the appearance of very small eigenvalues of a certain 4D operator, whichcontrol the rate of convergence in N . We have performed an independent studyand conrm the analysis in [19]. In addition, we discuss a new method to improvethe domain-wall fermion operator, which diers from the one proposed in [19] andproves to work better numerically.Five dimensional theoryIn this section we establish our notation and collect some useful formulae, thederivation of which can be found in [3, 4, 5, 6, 20]. The 5D domain wall operatorwe consider here is dened asD = 12fγ5(@s + @s)− as@s@sg+M; (1)where s denotes a lattice site in the fth direction (as  s  asN), as is thecorresponding lattice spacing, and @s and @s are the usual forward and backwardderivatives. The operator M in eq. (1) is obtained from the standard 4D Wilsonoperator byM = Dw −m0 (2)withDw =12γµ(rµ +rµ)− arµrµ}: (3)Here rµ and rµ are the gauge covariant forward and backward derivatives anda is the lattice spacing in the four physical dimensions  = 1; :::; 4. The massparameter m0 obeysm0 > 0; asm0 < 2; am0 < 2: (4)Note that the lattice spacings as and a can be dierent. The boundary conditionsare xed throughP+ (0; x) = P− (asN + as; x) = 0 ; (5)where P  12(1 γ5).3Appropriate interpolating elds for the quarks constructed out of the left andright boundary modes areq(x) = P− (as; x) + P+ (Nas; x) ; (6)q(x) =  (as; x)P+ +  (Nas; x)P−: (7)A mass term can then be introduced by adding to eq. (1) the term12m (as; x)P+ (Nas; x) +  (Nas; x)P− (as; x)}: (8)The two-point function of the quark elds is related to an eective 4D operatorDN by [6]hq(x)q(x)i = 2− aDNaDm,N; (9)withDm,N = (1− 12am)DN +m: (10)In terms of the operators K,K  12 12γ5asM2 + asM; (11)DN is given byaDN = 1 + γ5KN+ −KN−KN+ +KN−: (12)From eq. (12) it is straightforward to show thataD  limN!1aDN = 1 + γ5(K+ −K−) ; (13)which can also be written as [20]aD = 1− A(AyA)−1/2 (14)A = −asM(2 + asM)−1 : (15)It follows easily from this expression that D satises the Ginsparg{Wilson rela-tion, the only dierence to Neuberger’s operator being the dierent denition of4A. Indeed, Neuberger’s operator is readily obtained from eqs. (14) and (15) bytaking the limit as ! 0.Similarly, in the limit N ! 1, the fermion determinant of the 5D formulationcan be written in terms of the determinant of Dm,N , up to local subtractions.The 5D formulation is thus completely equivalent to a 4D lattice formulation ofGinsparg{Wilson fermions satisfying an exact chiral symmetry.A nal necessary condition for this formulation to be an acceptable regularizationof QCD is that the operator of eq. (14) is local. Indeed, it has been shown byKikukawa [21] that both the operators in eqs. (14) and (12) are exponentiallylocalized for smooth enough gauge elds, satisfying a plaquette bound [22, 23].In realistic simulations of domain-wall fermions, N is nite. In this situation,the chiral symmetry is broken by the residual terms D  DN − D. It may bespeculated that one could include a small additive quark mass renormalization, inorder to get rid of these chirality breaking eects. This is, however, only justiedby universality arguments if the subleading corrections in N in the action arelocal. The result of [21] shows that this is indeed the case since D is also local.However it is important to stress that the exponential localization of D onlysets in at distances of O(N). This can be shown already in the free case. On theother hand, in practical simulations the typical lattice sizes used are not muchlarger than N and consequently D is not local at the distances probed. Inthis situation, a quark mass renormalization cannot cancel the chirality-breakingeects induced by D.The convergence rate in NFor gauge eld congurations with a restricted plaquette value, the operator AyAhas a spectral gap [22]:0 < u  AyA  v; (16)ensuring the exponential convergence in N of DN . The minimum rate of conver-gence is given by! = mini[!i]; !i  ln 1 +pij1−pij; (17)5where i are the eigenvalues of AyA.However, in realistic simulations the plaquette bound is not satised and it isimportant to study the convergence rate ! for the values of  and m0 at whichlarge-scale numerical simulations are performed nowadays.The eigenvalues , which determine !, can be obtained through the generalized4D eigenvalue equationa2sMyM = (2 + asM)y(2 + asM) : (18)It is either the minimum or maximum eigenvalue of AyA that minimizes !i.These eigenvalues can be obtained by minimizing (maximizing) the generalizedRitz functionalh ja2sM yM j ih j(2 + asM)y(2 + asM)j i (19)using a straightforward generalization of the algorithm described in [24] 1.Eigenvalues above the lowest one can be computed by modifying the operatorM yM in the numerator of eq. (19) in such a way that the already computedeigenvalues are shifted to larger values. For example, this can be achieved bysubstituting M yM by M yM +Pi iMyM j iih ij(2+asM)y(2+asM) with i (1−i)=i and i,  i being the already computed eigenvalues and vectors. Noticethat in this method no inversion of the matrix (2 + asM)y(2 + asM) is needed.We have studied numerically the convergence rate in the quenched approximation.We nd that it is always controlled by the lowest eigenvalue min of AyA. In Figs. 1and 2 we show the inverse convergence rates !−1i corresponding to the ve lowesteigenvalues of AyA at  = 5:85 on an 83  16 lattice, and at  = 6:0 on a 163  32lattice, respectively. In both cases we have set asm0 = am0 = 1:8, which is atypical value used in previous simulations.Figures 1 and 2 give a rather pessimistic view of the convergence of domain-wall fermions to the exact operator: they imply that several hundred or eventhousand slices in the extra dimension would be needed to achieve a reasonableapproximation. Clearly, this would render domain-wall fermions impracticable.Our results are consistent with the ndings in [19].1A more detailed description of the algorithm can be obtained from the authors on request.6Figure 1: Inverse convergence rate !−1i for the ve lowest eigenvalues of AyA(open symbols correspond to the lowest eigenvalue) as a function of Monte Carlotime tMC, at  = 5:85 on an 83  16 lattice.7Figure 2: The same as Fig. 1 at  = 6 on a 163  32 lattice.8It should be noted, however, that a very similar situation was found for Neu-berger’s operator [22]. Also in this case very small eigenvalues of the corre-sponding AyA = M yM occur, turning the computation of its inverse square rootextremely costly.Acceleration of convergenceIn the case of Neuberger’s operator, the bad convergence behaviour resultingfrom the low-lying eigenvalues of M yM could be cured by treating these modesexactly [25, 26]. It is natural to look for a similar trick also for domain-wallfermions, given the similarity of the two constructions. We have found two waysof achieving this. The rst method is equivalent to the one described in [19],so we will not give any details here. The corresponding improved 5D operatordiers from the standard one by boundary terms.We have tested the inversion of this improved operator, Dimpr, by solving thelinear equation DimprX = Y for a given source Y . As a numerical solver we haveused a conjugate gradient method with a stopping criterion krk=kXk < , wherer = DimprX − Y is the residual vector and  was set to  = 10−8. We foundthat when using such a relatively low value of  the conjugate gradient methodbehaves very poorly: for a number of congurations at  = 5:85 on an 83  16lattice, the norm of the residual vector developed a very long tail at rather smallvalues of krk  O(10−5). This resulted in a very large number of iterations inthe conjugate gradient algorithm before it converged to the desired accuracy. Wesuspect that some subtle cancellations occur in the improved operator leading tounexpectedly large rounding error eects.Since this behaviour of the conjugate gradient algorithm was rather unsatisfac-tory, we developed an alternative improvement method. The key observation forthe new improved 5D operator is that the relations in eq. (15) and eq. (14) holdtrue for any choice of M as long asM y = γ5Mγ5 ; det(2 + asM) 6= 0 : (20)This fact may be used to construct an improved M for which the very low eigen-9values of AyA disappear. A possible form of M that achieves this is given byasM^ = asM −rXk,l=1Xklwk ⊗ wyl γ5; (21)whereγ5Avk = kvk; k = 1;    ; r; (vk; vl) = kl : (22)andwk = (2 + asM)γ5vk: (23)Finally(X−1)kl = 2kl(^k − k)−1 + (vk; (2 + asM)γ5vl) : (24)The corresponding 5D operator D^ is given by eq. (1) after substituting M by M^ .Notice that the improved operator diers from the original one also in the bulkand not just at the boundary.After some algebra it can be shown thatA^  −asM^(2 + asM^)−1 = A+rXk=1(^k − k)γ5vk ⊗ vyk : (25)It is now easy to see that γ5A^ has the same eigenvectors as γ5A, but all eigenval-ues k; k = 1;    ; r; are replaced by ^k. The limit N ! 1 of the correspond-ing improved 4D operator D^N is the same as that of the original DN providedsign(^k) = sign(k). However, the approach to this limit is faster for D^N ifthe lowest eigenvalues of A^yA^, ^k  ^2k, are larger than those of AyA, i.e. ifj^kj > jkj.The concrete choice of j^kj has to be taken with some care to optimize the con-vergence of the inverter. For example, taking j^kj = 1 led to a bad convergencebehaviour of the conjugate gradient algorithm. It is our experience that choos-ing j^kj not much higher than jrj, r being the index of the largest eigenvalueprojected out, see eq. (22), leads to a normal behaviour of the conjugate gradientalgorithm.As an example of the eect of the improvement using M^ , eq. (21), we show inFig. 3 the behaviour of the pion propagator at zero distance Γpi(0) at  = 5:85 on10Figure 3: The pion propagator at zero distance as a function of N . The blacksquares are the results for the original 5D domain wall operator. The open sym-bols correspond to the improved operator with three and ten eigenvalues projectedout. The data are obtained on an 83  16 lattice at  = 5:85 and am = 0:1.11an 83 16 lattice and for a quark mass, am = 0:1. A similar behaviour is obtainedfor the pion propagator at other distances.Already the projection of only three low-lying eigenvalues is sucient to acceleratethe convergence substantially: similar approximations to the N ! 1 limit areobtained for N  150 in the unimproved case and N  30 in the improved one. Itwould, of course, be interesting to see the eect also on other physical quantities.ConclusionsIn this note we presented numerical evidence that in practical simulations domainwall fermions need an unacceptably large number of slices in the extra dimensionto ensure that the chiral symmetry-breaking terms are suppressed. The reason isthat very small eigenvalues of a 4D operator appear, which are directly relatedto the convergence rate of the 5D domain-wall operator. These results conrmthe ndings in [19].As in the case of Neuberger’s operator, it is possible, however, to separate anumber of eigenvalues of the 4D operator and treat them exactly or shift themto larger harmless values. We tested two dierent implementations of this idea.The rst has already appeared in [19], the second, which is described in detailabove, is new. We found that numerically the second implementation works muchbetter.It is our overall impression, however, that there is no particular advantage inusing domain-wall fermions instead of Neuberger’s operator. Theoretical consid-erations demonstrate that both approaches to realize a chiral symmetry on thelattice are equivalent, to the extent that they satisfy the Ginsparg{Wilson rela-tion. However, in practical applications, it is our present experience that it iseasier to control chiral symmetry violations with Neuberger’s operator.AcknowledgementsWe would like to thank Giulia de Divitiis for her help at an early stage of thisproject and the computer centres at NIC (Ju¨lich) and CIEMAT (Madrid) forproviding computer time and technical support.12References[1] D.B. Kaplan, Phys. Lett. B288 (1992) 342.[2] T. Blum, Nucl. Phys. (Proc.Suppl.) B73 (1999) 167.[3] Y. Shamir, Nucl. Phys. B406 (1993) 90.[4] V. Furman and Y. Shamir, Nucl. Phys. B439 (1995) 54.[5] H. Neuberger, Phys. Rev. D57 (1998) 5417.[6] Y. Kikukawa and T. Noguchi, hep-lat/9902022.[7] P.H. Ginsparg and K.G. Wilson, Phys. Rev. D25 (1982) 2649.[8] P. Hasenfratz, Nucl. Phys. B (Proc.Suppl.) 63A-C (1998) 53.[9] H. Neuberger, Phys. Lett. B427 (1998) 353.[10] M. Lu¨scher, Phys. Lett. B428 (1998) 342.[11] P. Hasenfratz, V. Laliena and F. Niedermayer, Phys. Lett. B427 (1998) 125.[12] P. Hasenfratz, Nucl. Phys. B525 (1998) 401.[13] S. Chandrasekharan, Phys. Rev. D60 (1999) 074503.[14] H. Neuberger, Phys. Lett. B417 (1998) 141.[15] Y. Shamir, hep-lat/0003024.[16] T. Blum, A. Soni and M. Wingate, Nucl. Phys. B (Proc.Suppl.) 73 (1999)201.[17] T. Blum and A. Soni, Phys. Rev. Lett. 79 (1997) 3595.[18] A. Ali Khan, et al. (CP-PACS Collab.), hep-lat/9909049.[19] R.G. Edwards and U. Heller, hep-lat/0005002.[20] A. Borici, Phys. Lett. B453 (1999) 46.13[21] Y. Kikukawa, hep-lat/9912056.[22] P. Hernandez, K. Jansen and M. Lu¨scher, Nucl. Phys. B552 (1999) 363.[23] M. Lu¨scher, Nucl. Phys. B538 (1999) 515.[24] B. Bunk, K. Jansen, M. Lu¨scher and H. Simma, DESY report (September1994); T. Kalkreuter and H. Simma, Comput. Phys. Commun. 93 (1996) 33.[25] R.G. Edwards, U.M. Heller and R. Narayanan, Phys. Rev. D59 (1999)094510.[26] P. Hernandez, K. Jansen and L. Lellouch, Phys. Lett. B469(1999) 198; hep-lat/0001008.14

A note on the practical feasibility of domain-wall fermions

Abstract

Similar works

Full text

Available Versions

CERN Document Server