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PHYSICAL REVIEW D, VOLUME 63, 054509
Residual chiral symmetry breaking in domain-wall fermions
Chulwoo Jung,1 Robert G. Edwards,2 Xiangdong Ji,1 and Valeriya Gadiyak11Department of Physics, University of Maryland, College Park, Maryland 20742
2Jefferson Lab, 12000 Jefferson Avenue, MS 12H2, Newport News, Virginia 23606~Received 6 September 2000; published 8 February 2001!
We study the effective quark mass induced by the finite separation of the domain walls in the domain-wallformulation of chiral fermion as the function of the size of the fifth dimension (Ls), the gauge couplingb andthe physical volumeV. We measure the mass by calculating the small eigenvalues of the Hermitian domain-wall Dirac operator@HDWF(m0)# in the topologically-nontrivial quenchedSU(3) gauge configurations. Wefind that the induced quark mass is nearly independent of the physical volume, decays exponentially as afunction ofLs , and has a strong dependence on the size of quantum fluctuations controlled byb. The effect ofthe choice of the lattice gluon action is also studied.
DOI: 10.1103/PhysRevD.63.054509 PACS number~s!: 11.15.Ha, 12.38.Gc
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Simulating massless or near-massless fermions on atice is a serious challenge in numerical quantum field theoThe origin of the difficulty can be traced to the well-knowno-go theorem first shown by Nielsen and Ninomiya, whstates that one cannot write down a 4-dimensional, loHermitian, and chirally symmetric lattice fermion actiowithout the fermion doubling problem@1#. Hence to havechirally symmetric fermions on a lattice, one must use nlocal actions in which the couplings between lattice sitesnot identically vanish even when the separation becomlarge. This implies that the lattice simulation with chiral femions is necessarily more expensive than, e.g., the stanWilson fermion in which only the nearest neighbor coupliis involved.
One of the lattice chiral fermion formalisms that habeen studied extensively in recent years is the domain-fermion, first formulated by Kaplan@2# and later modifiedfor realistic lattice simulation by Shamir@3#. In the domain-wall construction, one introduces an extra fifth dimensioswith a finite extension. After discretization, the fifth directiohas Ls number of lattice sites. If we put the same foudimensional gauge configuration on every four-dimensios slice, the five-dimensional massive theory admits a fodimensional effective theory in which a left-handed chifermion lives near thes50 slice and a right-handed one nes5Ls21. Integrating out the heavy modes@4–7#, one ob-tains an effective four-dimensional chiral theory in the limof Ls→`.
For finite Ls , however, the two chiral modes can coupto produce an effective quark mass. Strong gauge field fltuations can induce rather strong coupling, and hence ralarge quark mass. This quark mass is expected to decrexponentially asLs→` with possible power law correctionsTo gain a quantitative understanding of this induced maColumbia and other groups have considered several diffeways to measure it. One way is to study the behavior of pmass as the function of the wall separationLs @8,9#. Theproblem with this is that the pion mass may not vanishLs→` due to the finite volume effects, and hence the nzero pion mass cannot be entirely attributed to the qumass effect. Another way is to use the Gell-Mann–OakeRenner relation@10,11#. However, this relation assumes
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more complicated form in the domain-wall fermion formaism. The effective mass can be also measured by studthe anomalous contribution in the axial Ward-Takaha~WT! identity @9,12,13#.
In Ref. @14#, we proposed to measure the induced effetive quark mass by considering the eigenvalue of the hermian domain-wall Dirac operator in the presence of aSU(N)gauge background. In theLs→` limit, the lattice version ofthe Atiyah-Singer theorem@15,16# guarantees the existencof exact zero modes. For finiteLs , the lowest eigenvalues othe Dirac operator are not zero. We take the average of thwould-be-zero eigenvalues as the effective quark mass.previous study for 12 crossings from 150 84 gauge configu-rations at b56.0 yields an effective quark mass 0.00760.0007 atLs58, 0.002260.0003 atLs512, and 0.000860.00013 atLs516. The result is qualitatively consistenwith those obtained from other methods@9,12,13#. It shouldbe noted that the probability for the crossings at 1.1,m0,2 is expected to be nonzero, albeit very small~Refs.@17,18#!. Therefore, level crossings close tom051.8 shouldbe observed if we increase the number of configuratioHowever, the effect of those crossings should be dominaby the crossings far fromm051.8 and the effective masshould not be affected, at least for the range ofLs we inves-tigated.
In this paper, we report a more systematic study ofeffective quark masses along this direction. In particular,would like to understand the effects of different lattice sizthe coupling constantb, and the form of gluon actions. Crucial to the size of the effective mass is the near-zero eigvalues of the Hermitian Wilson-Dirac operator at the fixdomain-wall heightm0 . These near-zero eigenvalues reqularge Ls to project out the correct effective Dirac operatoWhile we find that the effective mass is largely independof the lattice size, it is a sensitive function ofb and the formof gluon actions. For a strong coupling~small b! the effec-tive low-energy theory is recovered only at very largeLs .We find that the exponential decay rate for the effective mis well described by the density of zero eigenvalues ofHermitian Wilson-Dirac operator. For an improved gluon ation, the quantum fluctuations are strongly reduced ahence the domain-wall formalism works much more eciently.
©2001 The American Physical Society09-1
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JUNG, EDWARDS, JI, AND GADIYAK PHYSICAL REVIEW D63 054509
The domain-wall fermion was first introduced by Kapla@2#, based on an interesting observation that under cerconditions a five-dimensional massive fermionic theory han effective four-dimensional massless fermion. For practlattice simulations, Shamir considered a five-dimensionaltice having a finite extension in the fifth directions(s50,1, . . .,Ls21). The lattice domain-wall fermion action ia five-dimensional Wilson action atr 51,
SDWF5( cDDWF~m0!c
52 (x,s50
Ls21
cx,sF ~m025!cx,s1PRcx,s21
1PLcx,s1111
2 (m
„~12gm!Ux,mcx1m,s
1~11gm!Ux2m,m† cx2m,s…G , ~1!
wherem0 is the negative of the conventional Wilson maandPR,L51/2(16g5) is the chiral projection operators. Thphysical significance ofm0 is the ‘‘height of the domainwall.’’ Like the inverse of the lattice spacing 1/a ~taken to be1!, it acts as a heavy mass scale of the theory. The gafields live on the four-dimensional sub-lattices and are inpendent ofs. Dirichlet boundary conditions are applied to thfermion fields ats521 andLs . For the sake of simplicity,we omit the Pauli-Villars fields which can be introducedcancel the bulk of the heavy modes of the theory. The inested reader can consult Ref.@3# for details.
As shown in Ref.@3#, the above construction in the absence of gauge fields has a low-energy effective theory wtwo massless chiral fermions nears50 and Ls21 in thelimit of Ls→`. Thus, a single-flavor massless Dirac fieq(x) can be constructed from the domain-wall fermion fiec(x,s) as
q~x!5PLc~x,s50!1PRc~x,s5Ls21!. ~2!
An explicit fermion mass,mf , for the chiral modes can bintroduced through the modified boundary conditions,
PLcx,Ls52mf PLcx,0 , PRcx2152mf PRcx,Ls21 .
~3!
Substituting this into Eq.~1!, we obtain a generic fermionmass term.
To study the low-energy aspects of the domain-wall fmion action, we introduce the Hermitian domain-wall Diroperator, HDWF(m0)5Rsg5DDWF(m0), where Rs denotesthe reflection in the fifth direction. Because of the lowenergy property mentioned above, we expect that the llying spectrum ofHDWF(m0) resembles that of a massleDirac particle and contributes dominantly to low-enerphysical observables.
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A more direct way of studying the low-energy physicsthe domain-wall fermion is to integrate out all heavy modThis has been done by a number of authors@4–7#. The resulthas a very simple form
Seff5(x,y
q~x!DLs
eff~a5 ,m0!q~y!, ~4!
where we have made the lattice spacing in the fifth directa5 explicit and@7#
DLs
eff~a5 ,m0!5
11g5 tanhS Ls
2H~a5 ,m0! D
12g5 tanhS Ls
2H~a5 ,m0! D ,
H~a5 ,m0!51
a5lnS 11a5PRHW~m0!
12a5PLHW~m0! D . ~5!
HW(m0)5g5DW(m0) is the Hermitian Wilson-Dirac operator. The four-dimensional effective Dirac operatDLs
eff(a5,m0) can be used to calculate the propagator of
chiral fermion modes. If an observable involves the fermideterminant, we must take into account the contribution frthe Pauli-Villars particles as well. The effective Dirac opertor in the fermion determinant is
DLs~a5 ,m0!5
1
2 F11g5 tanhS Ls
2H~a5 ,m0! D G . ~6!
In the limit of a continuous fifth dimension, i.e.,a5→0, H(a5 ,m0) in the above expression becomes
H~a550,m0!5HW~m0!. ~7!
And if we take the limitLs→` as well,
tanhS Ls
2H D→e~H !5u~H !2u~2H !, ~8!
the Dirac operatorD` ~a550, m0! becomes the well-knownNeuberger overlap operator@4#. On the other hand, if we takeLs→` first, we have
D`~a5 ,m0!51
2@11g5e„H~a5 ,m0!…#. ~9!
The difference from the Neuberger overlap operator@4# isthat H(a5 ,m0) is now a function ofHW(m0) @see Eq.~5!#.
The above discussion leads to a simple and natural wadefining the topology of lattice gauge configurations@18#.The effective domain-wall Dirac operator atLs→`@D`(a5 ,m0)# has the following interesting property. If thHermitian operatorH(a5 ,m0) has unequal numbers of postive and negative eigenvalues, its determinant vanishes;means thatD`(a5 ,m0) has zero-energy eigenvalues. Ispired by the Atiyah-Singer theorem in the continuum spatime, one can regard the appearance of the zero eigenvaas the signal for a nontrivial topology of the lattice config
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RESIDUAL CHIRAL SYMMETRY BREAKING IN . . . PHYSICAL REVIEW D 63 054509
ration. Since the number of positive and negative levelsthe same form0,0, the difference between the numberpositive and negative levels at a positivem0 can be calcu-lated by tracking the spectral flow of all eigenvaluesH(a5 ,m) betweenm50 andm5m0 and locating the levecrossings@16,17,19#. The topological index of a gauge configuration is given by the difference of the number of potive and negative energy levels ofH(a5 ,m0). The effective
domain-wall Dirac operatorD`(a5 ,m0) has the same number of zero eigenvalues as the topological index. Rather tfind the eigenvalues ofH(a5 ,m), a computationally simplermethod can be used form,2 anda551 where the spectraflow of the lowest eigenvalues of the hermitian Wilson-DiroperatorHW(m) are tracked betweenm50 andm5m0,2.The topological index thus produced is the same as tha
the effective domain-wall Dirac operatorD`(a5 ,m0) @7#.
For finite Ls , however, the chiral modes ofDLs(a5 ,m0)
corresponding to the gauge field topology do not haveexact zero eigenvalue. The limit is approached exponentias Ls→`. Since the low-lying eigenvalues of the effectiv
Dirac operator DLs(a5 ,m0) must reproduce those o
HDWF(m0), the latter has the same number of exponentiasmall eigenvalues as the topological index. We definefinite-Ls induced fermion massmeff as an average of thesquasizero eigenvalues.
As shown in Ref.@18# and proven in@20#, the density ofzero eigenvaluesr(l50;m0) for Hw(m0) is nonzero for thetype of gauge actions studied in this work. From Eq.~5!, wesee that whenHw(m0)50 thenH(a5 ,m0)50, and hence thedensity of zero eigenvalues in an ensemble average ofHw(m0) andH(a5 ,m0) are the same. We define the densof zero eigenvalues ofHw(m0) the same as in Ref.@18#;namely, we obtain in the whole ensemble the integral ofspectral density function,*0
lr(l8;m)dl8. A linear fit in lwith some cutoff is made, andr(0;m0) is obtained as thecoefficient of the linear term. For this work, the valuesr(0;m0) were taken from Ref.@18# except for the improvedgauge action where the data from Ref.@18# was used todetermine the value.
In a previous publication@14#, we have calculatedmefffrom 150 configurations on an 84 lattice atb56.0 andm051.8. Here again we showmeff for each of the zero modes iFig. 1 where the horizontal axis is the Wilson massm repre-senting the locations of the level crossings in the eigenvaflow of HW(m), and the vertical axis ismeff . For those con-figurations with multilevel crossings, we relatemeff andm byassuming that the spectrum flow is repulsive or levels docross, and this simplifies the presentation. Indeed, we didobserve any crossing of the spectral flows in any ofMonte Carlo configurations studied here. The resultsthree differentLs are shown with three different symbols.
How does the effective mass change as the coupgrows stronger~smaller b!? To answer this question, whave generated 100SU(3) lattice gauge configurations onlattice size 84, 50 each atb55.85 and 5.7. We measure theigenvalue flow of the Hermitian Wilson-Dirac operator, a
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calculate the eigenvalues ofHDWF(m051.8) correspondingto the nontrivial topology of the gauge configurations.
Both the eigenvalues of the Hermitian Wilson-Dirac oerator and the Hermitian domain-wall Dirac operator are cculated by using either Implicitly Restarted Arnoldi~IRA!method@21# or Ritz method@22#. Up to 10 eigenvalues arecalculated forHw , while up to 8 eigenvalues are calculatefor HDWF. The stopping condition for IRA method is sucthat the sum of relative changes of the measured eigenvais less than 131025. The number of domain-wall Dirac operator multiplication needed for the eigenvalues measuments using IRA method varies from about 5000 per lattfor b56.0 to 10000 per lattice forb55.7. The stoppingcondition for the Ritz method is 1025 for the relative changeof the eigenvalues, and about 7000 domain-wall Dirac opetor applications are needed for theb57.9 improved gaugeaction.~Due to the properties of algorithms used, the numof converged eigenvalues can be less than originallytended, in which case we increased the number of the msured eigenvalues. Since the smallest eigenvalues convfaster, we did not pursue further when the number of cverged eigenvalues are more than sufficient—sufficientrule out more crossings forHW and more than the absolutvalue of the topological change forHDWF.! The smallestnontopological eigenvalues ranges from 531022 for b56.0 and 131022 for b55.7. In the case where the sepration is not clear, we take the smallest eigenvalues withsame sign as the topological charge of the configurationthe topological eigenvalues.
For larger lattice spacing, quantum fluctuations are strger, and some of these fluctuations can be misidentifiedsmall size would-be instantons. It turns out that they cinduce strong couplings between the left- and right-hanchiral modes and are detrimental to the existence of the lenergy effective theory. Indeed, for the sameLs , the effec-tive masses are much larger at the smallerb’s than those at
FIG. 1. Effective quark mass induced by domain walls wdomain wall heightm051.8 for the Monte Carlo configurations ab56.0, 84 lattice. Ls is the number of lattice sites in the fifthdirection.
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h
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JUNG, EDWARDS, JI, AND GADIYAK PHYSICAL REVIEW D63 054509
b56.0. For example, with Ls516, meff is 0.0008060.00013 atb56.0, 0.00860.002 atb55.85, and 0.01860.001 atb55.7.
Moreover, for the same lattice size, physical volumelarger at smallerb, and hence can house more zero modAs shown in Fig. 2 forb55.85 and Fig. 3 forb55.70, thetotal number of zero-mode crossings in the 50 configuratiis now 32 and 96, respectively. The level crossings happelargerm0 compared with Fig. 1. For.b55.7, there are several crossings very close tom051.8. This is in contrast towhat has been observed atb56.0, where the crossings occumostly aroundm51.0 @14,17#. The equal spacing betweemeff at differentLs is a clear signal for the exponential deca
FIG. 2. Effective quark mass induced by domain walls wdomain wall heightm051.8 for the Monte Carlo configurations ab55.85, 84 lattice. Ls is the number of lattice sites in the fiftdirection.
FIG. 3. Effective quark mass induced by domain walls wdomain wall heightm051.8 for the Monte Carlo configurations ab55.7, 84 lattice. Ls is the number of lattice sites in the fiftdirection.
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.
However, there is a significant variation in the rate amongthe crossings, as evident in the figures.~Note: After theanalysis for the paper was completed, the recent work ofCP-PACS@12# and RBC@13# Collaborations became avaiable, which shows the signal for varying rate of exponendecay and/or nonvanishing effective mass in theLs→`limit. This behavior is only seen at a much largerLs thanthose studied here. However, it is interesting to note taveraging eigenvalues with varying exponential rate can eily reproduce the largerLs behavior of the effective masobserved in the aforementioned reference.!
To see the volume dependence at a fixedm0 and b, wealso measure the effective mass on a set of 50 configuraton an 83316 lattice atb55.85 ~see Fig. 4!. The total num-ber of zero-mode crossings is now 64, doubling that on84 lattice. We notice there are several crossings very closm051.8. The average effective mass turns out to be estially the same as that on the smaller lattice. A similar coclusion can be drawn by comparingmeff at b56.0 andV584 with that at the sameb and V5163324 in Ref. @9#,although we note that the way of determining the effectmass there is quite different. This weak dependence onsize of the volume may allow us to extrapolate our quanttive results to large lattices necessary for realistic simutions.
We also measure the effective mass on a set of 200 cfigurations generated from the 1-loop Lu¨scher-Weisz gaugeaction @23# with the tadpole improvement.~Crossings fromonly 50 lattices are shown in Fig. 5.! Similar studies usingvarious improved gauge actions, including an RG improvaction @24# are reported in@11,12#. The gauge coupling (b57.9) corresponds to a spacing of;0.16 fm, similar tob55.7 of the Wilson action. The spectral flow and thdomain-wall eigenvalues are studied with the same WilsDirac operator. As shown in Fig. 5, the number of zemodes as well as the distribution of the crossing values
FIG. 4. Effective quark mass induced by domain walls wdomain wall heightm051.8 for the Monte Carlo configurations ab55.85, 83316 lattice.Ls is the number of lattice sites in the fifthdirection.
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RESIDUAL CHIRAL SYMMETRY BREAKING IN . . . PHYSICAL REVIEW D 63 054509
fers significantly from the Wilson action at similar latticspacing. Because of the decrease of small-scale quanfluctuations, the probability of crossings form0.1.2 isheavily suppressed, and the density of small eigenvalueHW(m0) is also much smaller~;2.731024, compared to;1.831023 for b55.7 Wilson action!. Therefore, the effec-tive mass decreases faster as a function ofLs . The averageeffective mass atLs516 is ;631023, compared to 1831023 from the Wilson action atb55.7.
The origin of the induced chiral fermion mass is theniteness ofLs . More specifically, the hyperbolic-tangenfunction at finiteLs is used to approximate thee function@Eq. ~8!#. This approximation is good for large eigenvaluof H(a5 ,m0), but poor for small ones. For smallerb,H(a5 ,m0) has more small eigenvalues and hencehyperbolic-tangent is a worse approximation of te-function. This point is also reflected in the dependencethe exponential decay rate on the density of the zero eigvalues ofHW(m0) @r(0;m0)#. We note that since the gaugfields are replicated along the fifth dimensional slices,relevant length scale in the fifth dimension is the inversethe rate of exponential decay~a! and by simple engineeringdimensions is given qualitatively by the density of zerogenvalues ofr(0;m0)1/3. In Fig. 6, we have plotted 1/a as afunction of r(0;m0). The ratea for each coupling is calculated by fittingmeff5m0 exp@2aLs# at differentLs . The sta-tistical errors are estimated by doing correlated fits to singeliminated jackknife blocks. The inverse decay rates fromthe configurations studied show an approximate linear sing as r(0;m0)1/3. This suggests that the density of smeigenvalues is indeed the dominating factor for the expontial decay rate of domain-wall fermions.
Effective masses thus obtained for the different gaucoupling andLs are plotted in Fig. 7. The data from Ref@9,10# are also included for comparison. For a given gau
FIG. 5. Effective quark mass induced by domain walls wdomain wall heightm051.8 for the 50 Monte Carlo configurationgenerated by the 1-loop, tadpole improved gauge action@23# on ab57.9, 83316 lattice.Ls is the number of lattice sites in the fiftdirection.
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ef
-
-lll-
ln-
e
e
coupling, different volumes and methods of measuremehave little effect on the size of the effective mass as wellthe rate of exponential decay. However, the change of gaaction affects the effective mass significantly. It is quite clefrom Fig. 7 that for a practical simulation of the domain-wafermion, one either chooses a largeb with the conventionalWilson action or an improved action keeping lattice spaclarge.
To summarize, we have studied the residual chiral symetry breaking present in domain-wall fermions by measing the eigenvalues of the Hermitian domain-wall Dirac oerator corresponding to the topology of the lattice gau
FIG. 6. Average coefficient of the exponential decreases afunction of r(0;m0). The extrapolation of the fit to the continuumlimit for the Wilson gauge backgrounds is shown on the left. Valufor r(0;m0) for the Wilson and improved gauge action are froRefs.@17,18# for 83316 lattices.
FIG. 7. Average effective masses from various observablesfunction of Ls . Effective masses fromb55.7, GMOR relation arefrom Ref. @10#. Data forb56.0, mp and the axial WT identity arefrom Refs.@9# and @12#, respectively. LW denotes the 1-loop, tadpole improved gauge action from Ref.@23#.
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JUNG, EDWARDS, JI, AND GADIYAK PHYSICAL REVIEW D63 054509
configurations. Individual eigenvalues for the topologiczero modes show clear exponential behavior inLs . We re-gard these eigenvalues as the induced mass for the suchiral modes at finiteLs separation.
For Ls andb, we see little variation ofmeff as a functionof the volume. This is in some sense expected becausecoupling of the chiral modes between the opposite wallslittle to do with the size of the four-dimensional slice. On tother hand, a strong dependence onb is observed. In particu-lar, the effective mass is much larger atb55.85 or 5.7 thanthat atb56.0. For the improved gauge action, the spuriofluctuations are reduced significantly and theLs needed toobtain a good chiral symmetry is reduced. Since the adtional computation needed for the improved action is ne
on,.
l.
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gible, using improved gluon actions may enable us to simlate domain-wall fermions with larger lattice spacing.
We thank N. Christ and J. Negele for useful discussiorelated to the subject of this paper. The numerical calcutions reported here were performed on the Calico AlpLinux Cluster and the QCDSP at the Jefferson LaboratoVirginia. C.J., X.J., and V.G. are supported in part by funprovided by the U.S. Department of Energy~D.O.E.! undercooperative agreement DOE-FG02-93ER-40762. R.G.E.supported by DOE contract DE-AC05-84ER40150 undwhich the Southeastern Universities Research Associa~SURA! operates the Thomas Jefferson National AccelaraFacility ~TJNAF!.
ys.
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. B
rtn.
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