Finite-Volume Scaling of the Wilson-Dirac Operator Spectrum

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by  P. H. Damgaard

Finite-Volume Scaling of the Wilson-Dirac Operator Spectrum
P.H. Damgaard
Niels Bohr International Academy and Discovery Center, Niels Bohr Institute,
University of Copenhagen, Blegdamsvej 17, DK-2100, Copenhagen Ø, Denmark
U.M. Heller
American Physical Society, One Research Road, Ridge, NY 11961, USA
K. Splittorff
Discovery Center, Niels Bohr Institute, University of Copenhagen,
Blegdamsvej 17, DK-2100, Copenhagen Ø, Denmark
(Dated: January 10, 2012)
ThemicroscopicspectraldensityoftheHermitianWilson-Diracoperatoriscomputednumerically
in quenched lattice QCD. We demonstrate that the results given for fixed index of the Wilson-
Diracoperator can bematched bythepredictionsfrom Wilson chiral perturbation theory. Wetest
successfully the finite volume and the mass scaling predicted by Wilson chiral perturbation theory
at fixed lattice spacing.
2
1
0 I. INTRODUCTION [8, 9]. Here we will test these recent predictions against
2 quenched lattice data.

n The microscopicdescriptionworks,atleadingorderin
For a long time, the Wilson construction of Dirac
a the chiral counting rule of the finite-volume ǫ-regime, at
J fermions on the lattice was somewhat overshadowed by two equivalent levels: (i) a Wilson chiral random matrix
9 other formulations with either explicit chiral symmetry theory(WRMT)and(ii)the zero-modesectorofWCPT
orremnantsthereof. Recently,withthehelpofimproved to order a2. Beyond leading order only the approach
] simulationtechniques andmorepowerfulcomputers,nu-
t based on WCPT survives.
a mericalsimulationswithWilsonfermionshavepickedup
The γ -Hermiticity of the Wilson-Dirac operator,
l momentum again. The initial concernabout the explicit 5
-
p breaking of chiral symmetry in this formulation can be D† =γ D γ , (1)
e dealt with in precisely the way originally envisaged: by W 5 W 5
h
simply pushing simulations into the regime of very light together with the explicit form of D tells us that the
[ W
fermions,evento the physicalpointofalmostmasslessu eigenvaluesofD come in complexconjugate pairs[10],
W
2 and d quarks. This requires tight control on the small- a feeble remnant of chiral symmetry. In addition, γ -
5
v
est eigenvalues of the associated Wilson-Dirac operator Hermiticity of D ensures that the Hermitian Wilson-
1 W
5 DW. If the lattice spacing is not small enough, the Dirac operator,
gapassociatedwiththeHermitianWilson-Diracoperator
8
2 D5 ≡ γ5(DW +m), where m is the quark mass in con- D5 ≡γ5(DW +m), (2)
. tinuum language, shrinks due to small eigenvalues pop-
0 as the name suggests, is Hermitian. While the eigenval-
ulating the region between −m and m. This can cause
1
ues of D can be computed by, say, the Arnoldi algo-
1 severe numerical instabilities in the simulations. Studies W
rithm, it is clearly advantageous to focus instead on the
1 of the spectral gap and its potential disappearance have
HermitianWilson-DiracoperatorD ,forwhichfastalgo-
: indicatedthatsimulationsmaybecontinuedtothephys- 5
v rithmsexistthatprovideanascendingsequenceoflowest
icalpoint evenatrealisticlattice spacingsofpresent-day
i
X computational power [1]. eigenvalues.
Here we test the predictions for the quenched micro-
r
a Sparked by this renewed interest in simulations with scopicspectraldensityofD thatwerepresentedin[4,5]
5
Wilson fermions on the lattice, also work on analyti- againstlatticedata. Allresultsof[4,5]aregivenforfixed
cal approachesto understanding the chirallimit of these index of the Wilson-Dirac operator. This index ν is de-
fermions has intensified. As is always the case, it is ad- fined for a given gauge field configuration by
vantageoustophrasethediscussionintermsoftheperti-
nent low-energy effective field theory, Wilson chiral per- ′
ν ≡ sign(hk|γ |ki). (3)
5
turbation theory (WCPT) [2]. Within this framework
k
Sharpe[3]wasthefirsttoconsidertheeffectsoffinitelat- X
ticespacing,a,onthespectrumoftheHermitianWilson- Here,|kidenotesthek’theigenstateoftheWilson-Dirac
Dirac operator. More recently, a complete description operator. Only those eigenvectors which correspond to
of the microscopic details of the Wilson-Dirac operator a real eigenvalue of the Wilson-Dirac operator have a
spectrum up to and including a2 corrections has been nonvanishingexpectationvalue ofγ [10]andhence con-
5
presentedfor both quenched[4–7] andunquenched cases tribute to the index.
2
In the microscopic limit the real eigenvalues of the are considered in the microscopic limit where V → ∞
Wilson-Dirac operator are predicted to be located in re- while
gions with width of order 1/V [4, 5, 7], where V is the
four volume, and we choose only to include the physi- mV, λ5V, a2V (6)
k
cal/leftmost of these regions in the above definition of
the index. This is indicated by the prime on the sum in are kept fixed.
Eq. (3). The microscopicspectral density follows from WCPT.
The predictions [4, 5] for the quenched microscopic ToO(a2)accuracythiseffectivetheorywaswrittendown
spectral density of D are based on the ǫ-regime of Wil- in [2], expanding in all operators that contribute to this
5
son chiral perturbation theory. At leading order in the order in the Szymanzik scheme. To order a2 the zero-
ǫ-regimeofWCPTthreeadditionallowenergyconstants, modepartitionfunctionatfixedindexν canbeobtained
W , W and W parametrize the leading correctionsdue [4] by decomposing the momentum zeromode partition
6 7 8
to the nonzero lattice spacing (order a2 corrections) [2]. function according to
There are large-N arguments (N is the number of col-
c c
∞
ors) to the effect that the Wilson low-energy constants
Z (m,θ;a)≡ eiνθZν (m;a). (7)
W6 andW7 maybe small[3]. Forthe caseW6 =W7 =0 Nf Nf
ν=−∞
it was argued in [4, 5] that only W > 0 correctly de- X
8
scribes lattice QCD with a nonnormaland γ -Hermitian
5 In this ǫ-regime, the partition function reduces to a uni-
Dirac operator. In agreement with this we will demon-
tary matrix integral
strateherethatthequenchedmicroscopiceigenvalueden-
sity ofD obtainedonthe lattice canbe matchedby the
5 Zν (m,z;a)= dU detνU eS[U], (8)
prediction from WCPT with the low energy parameters Nf
W =0, W =0 and W >0. ZU(Nf)
6 7 8
We identify the effects of nonzero W6 and W7 in the where the action S[U] for degenerate quark masses is
spectrum of D5, and conclude that they do not change given by
the above conclusion.
m z
Finally, we explicitly compute the real eigenvalues of S = ΣVTr(U +U†)+ ΣVTr(U −U†)
D and compare to the predictions. 2 2
W
−a2VW [Tr U +U† ]2−a2VW [Tr U −U† ]2
Wilsonchiralperturbationtheoryhasalsobeenconsid- 6 7
eredinthefinite-volumescalinglimitwherethea2-terms −a2VW Tr(U2+U†2). (9)
8 (cid:0) (cid:1) (cid:0) (cid:1)
canbe pulled downfromthe actionandtreated as small
perturbations [11]. The analytic constraint on W that In addition to the chiral condensate, Σ, the action con-
8
we discussedaboveis notobviousinsucha scalinglimit. tains the low-energy Wilson constants W , W and W
6 7 8
Our paper is organized as follows. In the section just as unknown parameters [27]. In the continuum the de-
below,we firstbriefly recallthose results fromWCPT in composition (7) into a sum over sectors of index ν is
theǫ-regimethataremostrelevantforthepresentlattice unambiguous [12]. At nonzero lattice spacing the terms
study. Section III gives the definition of Wilson chiral of order a2 in the chiral Lagrangian, however, allow one
random matrix theory. In section IV we give details of to make different decompositions corresponding to dif-
the lattice simulations we have performed. We compare ferent definitions of θ and ν at a 6= 0. As was shown
our lattice data to the predictions from Wilson chiral in Ref. [5], the specific decomposition (7-9) corresponds
perturbationtheoryinsectionV.SectionVIcontainsour exactly to the definition of ν givenin Eq.(3). Moreover,
conclusions. in the microscopic limit the real modes are predicted to
be in well seprated regions each width width of order
a2W /Σ ∼ 1/V. The gauge field configurations which
i
contribute to the individual terms in the sum (7) can
II. PREDICTIONS FOR THE MICROSCOPIC thus be identified on the basis of the spectral flow [13].
SPECTRUM
This provides a firm foundation for choosing this partic-
ular lattice definition of the index.
Here we recall the predictions for the spectral density To simplify our notation,we absorbthe factor ofVW
i
of D , into a2 and the factor VΣ into m, z and λ5
5 i k
a2VW →aˆ2 mVΣ→mˆ
i i
ρ5(λ5,m;a)= δ(λ5k−λ5) . (4) zVΣ→zˆ λ5VΣ→λˆ5. (10)
* +
k
X
The explicit expression for the quenched microscopic
The eigenvalues λ5, defined by spectral density can be derived following the procedure
described in Ref. [14], now extended to the case of a
D ψ =λ5ψ (5) source z for the pseduoscalar density. In addition, much
5 k k k
3
likeatnonzerochemicalpotential[15,16],onemustcare- where the first(n+ν) entriesare +1,andthe remaining
fully take into account the convergence of the noncom- n entries are -1.
pact integrals. The resulting analytical formula was ob- ResultsbasedontheWilsonchiralrandommatrixthe-
tained in [4, 5]: ory are expected to be universal, as in the case of the
usual chiral random matrix theory [18]. For simplicity
ρν(λˆ5,mˆ;aˆ ) = 1Im Gν(−λˆ5,mˆ;aˆ ), (11) one can therefore take the matrix elements to be dis-
5 i π i tributed with Gaussian weight
where P(A,B,W)≡e−N2Tr[A2+B2]−NTr[WW†]. (16)
∞ π dθ
Gν(zˆ,mˆ;aˆi)= ds icos(θ)e(iθ−s)ν The partition function of the Wilson chiral random ma-
2π
Z−∞ Z−π trix theory is thus defined by
×exp[−mˆ sin(θ)−imˆsinh(s)+izˆcos(θ)−i(zˆ−iǫ)cosh(s)
+4aˆ26(−isin(θ)+sinh(s))2+4aˆ27(cos(θ)−cosh(s))2 Nf
+2aˆ2(cos(2θ)−cosh(2s))] Z˜ν = dAdBdW det(D˜ +m˜ +z˜ γ˜ ) P. (17)
8 Nf W f f 5
mˆ mˆ zˆ zˆ Z f=1
× − sin(θ)+i sinh(s)+i cos(θ)+i cosh(s) Y
2 2 2 2
ThematrixintegralsareoverthecomplexHaarmeasure.
(cid:0)−4(aˆ2+aˆ2)(sin2(θ)+sinh2(s))
6 7 In the limit N → ∞ the partition function (17) coin-
1
+2aˆ2(cos(2θ)+cosh(2s)+eiθ+s+e−iθ−s)+ . (12) cides with that of (8) when W =W =0. It is straight-
8 2 6 7
forwardto extend the Wilson chiral random matrix the-
(cid:1)
Existence of the integrals leads to the constraintW8 >0 ory so that also double-trace terms correspondingto W6
(for W6 = W7 = 0) as described in detail in Refs. [4, 5]. andW7areincluded[5,19]. Thecorrespondencebetween
The sign of W is also crucial for understanding if one parameters of Wilson chiral random matrix theory and
8
approaches the so-called Aoki phase of Wilson fermions Wilson chiral perturbation theory is
[17]. A positivesignofW (andW =W =0)indicates
8 6 7
mΣV zΣV Na˜2
that the Aoki phase will appear as the gap in the spec- Nm˜ = , Nz˜= , =a2W V. (18)
8
trumofD closes. TheanalyticpredictionsofWCPTfor 2 2 2
5
the spectral density D allow us to follow in detail how
5 Oncethisidentificationismade,onecanequallywelluse
the gap disappears as m is taken to zero at fixed lattice
the Wilson chiral random matrix theory to derive spec-
spacing a.
traldistributions. Thishasbeendemonstratedexplicitly
in [6, 7]. Moreover, since no compact expressions are
presently known for individual eigenvalue distributions,
III. WILSON CHIRAL RANDOM MATRIX
itisstraightforwardtogeneratesucheigenvaluedistribu-
THEORY
tions numerically using the Gaussian integrals of Wilson
chiral random matrix theory. This will be used below.
As we will make explicit use of it below, we give also
herethecorrespondingexpressionsintheequivalentWil-
son chiral random matrix theory formulation [4, 5]. The
IV. THE LATTICE SETUP
ideaistointroducethemostgenerallarge-N matrixthat
hasconjugationpropertiessimilartoγ -Hermiticity. Let
5 Here we give a few details of the lattice simulations.
this matrix be
For this initial study we work in the quenched approx-
aA iW imation. It is advantageous to use a pure gauge action
D˜W = iW† aB , (13) which gives configurations with a fairly unique topolog-
(cid:18) (cid:19) ical charge, i.e., a pure gauge action that suppresses so-
where calleddislocations. Experiencefromsimulationswithdo-
main wall fermions, for which the dislocations tend to
A=A† and B† =B (14) increase the residual chiral symmetry breaking at finite
fifth dimensional extent L , suggests that the Iwasaki
s
are (n+ν)×(n+ν) and n×n complex matrices, re- gauge action [20] would be a good choice. This action is
spectively, and W a rectangular complex matrix of size fairly well studied and appears to have a rather smooth
(n+ν)×n. As in [5] we use tildes to indicate quantities continuum limit. Our lattice parameters were chosen
in Wilson chiral random matrix theory which are ana- using r0/a values from interpolation formulae given in
logues of those in the field theory. We consider the limit [21, 22] and using r0 =0.5 fm to set the physical scale.
N =2n+ν →∞ with m˜N, z˜N and a˜2N fixed. We useda mixture ofheatbathandoverrelaxationup-
The matrix D˜W is γ˜5-Hermitian with respect to dateswith,asadvocatedinRef.[22],Nor ≈1.5r0/aover-
relaxation sweeps for each heatbath sweep. We saved
γ˜ =diag(1,··· ,1,−1,··· ,−1) , (15) configurations separated by 100 heatbath sweeps for our
5
4
Ensemble βIw size r0/a a [fm] L [fm] # cfgs Ensemble size βIw m0 # configs
A 2.635 164 5.37 0.093 1.5 6500 ν =0,1,−1
B 2.635 204 5.37 0.093 1.9 3000 A 164 2.635 −0.216 1246, 1088, 1045
C 2.79 204 6.70 0.075 1.5 6000 B 204 2.635 −0.216 379, 319, 322
C 204 2.79 −0.178 1172, 990, 988
TABLE I: Parameters of the pure gauge configurations con- C 204 2.79 −0.184 1172, 990, 988
sidered. They are generated with the Iwasaki gauge action.
TABLE II:
measurements. We found no detectable autocorrelations
V. THE SPECTRUM OF D5: LATTICE
in the ensembles generated. Experience indicates that SIMULATIONS
for studies of the ǫ-regime of chiral perturbation theory,
physical lattice sizes of L = 1.5 to 2 fm are most use-
In this section we compare the microscopic spectral
ful. The lattice size is largeenoughso that severalDirac
density of D as predicted by WCPT for the quenched
eigenvalues are governed by the ǫ-regime of CPT, but 5
case to the lattice simulations described above.
it is small enough that sufficient statistics in sectors of
The values of m, λ5 and a given in the figures refer to
given, small topological charge can be obtained. Some
mˆ, λˆ5 and aˆ . For all plots except Figure 8 we have set
relevant parameters of the three ensemble considered in 8
aˆ =aˆ =0.
this study are summarized in Table I. 6 7
A. Volume scaling
Ourfirsttestofthe analyticpredictions forthe micro-
A. Spectral flow
scopic spectrumof D concernsthe finite volume scaling
5
at fixed lattice spacing. That is, the scaling of mˆ, λˆ5
In order for the match to Wilson chiral perturbation and aˆ8 with V for fixed a. To this end we consider the
theory to be valid the assignment of index ν to a given two sets of data with βIw =2.635 and m0 =−0.216,see
gauge field configuration must be obtained from Eq. (3) TableII.Becauseofthelargerstatisticsobtainedwefirst
usingeithera)the(small)realeigenvaluesoftheWilson- match the 164 lattice data with ν = 0 to the prediction
Dirac operator, together with the chiralities of the cor- (12). The result is displayed in the top panel of Figure
responding eigenvectors or equivalently b) by counting 1. We observethatitis possibleto makeareasonablefit
the number of net crossings of zero of the eigenvalues of tothe data. Therangeoverwhichthe predictionsmatch
theHermitianWilson-Diracoperatorasafunctionofthe the data is limited to the lowesttwo eigenvalues,beyond
massparameterm [13]andweightingtheircontribution which the increasing slope of the data indicates that the
0
by the sign of the slope at the crossing. Since we did p-regime effects set in. Given the best values found in
not have an implementation of the Arnoldi algorithm to the fit we then scale mˆ by 204/164 and aˆ8 by 202/162
compute the (complex) eigenvalues of the Wilson-Dirac to obtain the analytic prediction for the spectraldensity
operator,includingtherelevantrealeigenvalues,weused obtained on the 204 lattice at the same value of the cou-
the spectral flow strategy. pling and lattice mass. This parameter free prediction is
plottedagainstthe latticedatainthelowerpanelofFig-
The index determined from the realeigenvalues of the
ure 1. We observe that this finite-volume scaling of the
Wilson-Dirac operator,or equivalently from the spectral analytic prediction is consistent with the 204 data. The
flow,candependontherangeoftherealeigenvaluescon-
limitedstatisticsdoesnotallowustoresolvethedetailed
sidered, or equivalently, the value of −m at which the
cut structure of the analytic prediction.
spectralflow is terminated. With −m being the critical
c
mass, at which the pion formed from Wilson quarks be-
comes massless, m should be sufficiently larger than Likewisefor|ν|=1wecantesttheanalyticprediction
cut
m , but smaller than the value at which the doubler with the parameters fixed from the fit to the 164 and
c
fermions become light. Forourcomputations we use one ν = 0 data. The result is displayed in Figure 2. Note
HYPsmearingofthegaugefields[23]intheconstruction that the analytic prediction for −ν is obtained from the
oftheWilson-Diracoperatortohelpfurthersuppresslat- one for ν by flipping the sign of λˆ5. To make the most
tice artifacts. Then the choice m =1.0 is safely away of our statistics we therefore overlay the data obtained
cut
from both m and the doubler region. With the Iwasaki for ν = 1 and ν = −1 flipping the sign of the eigenval-
c
gaugeactionanditssuppressionofdislocations,crossings ues of the latter. This is done throughout this paper.
in the neighborhood of m are rare, but they do occur We observe that the general structure of the 164 ν = 1
cut
occasionally. Hence a change in m would change the data is reproduced by the analytic prediction. Also the
cut
index on a few configurations. overall features of the 204 ν = 1 data are followed by
5
Ev_2635_16_HYP_Q0_m216 Ev_2635_16_HYP_Q1_Qm1_m216
0.6 m=3.5 a=0.35 ν=0 0.6 m=3.5 a=0.35 ν=1
FIT PREDICTION
0.5 0.5
a) a)
m; 0.4 m, 0.4
5λ, 5λ,
ν=0ρ(50.3 ν=1ρ(50.3
0.2 0.2
0.1 0.1
0 0
-12 -8 -4 0 4 8 12 -12 -8 -4 0 4 8 12
λ5 λ5
Ev_2635_20_HYP_Q0_m216 Ev_2635_20_HYP_Q1_Qm1_m216
0.6 m=8.5 a=0.55 ν=0 0.6 m=8.5 a=0.55 ν=1
PREDICTION PREDICTION
0.5 0.5
a) a)
m; 0.4 m, 0.4
5λ, 5λ,
ν=0ρ(50.3 ν=1ρ(50.3
0.2 0.2
0.1 0.1
0 0
-12 -8 -4 0 4 8 12 -12 -8 -4 0 4 8 12
λ5 λ5
FIG.1: TheeigenvaluedensityofD5 inthesectorwithν =0 FIG.2: TheeigenvaluedensityofD5 inthesectorwithν =1
for V =164 (top) and V =204 (bottom) both with βIw = for V = 164 (top) and V =204 (bottom) both with βIw =
2.635andm0 =−0.216. Thex-axishasbeenrescaledbyΣV, 2.635andm0 =−0.216. Thex-axishasbeenrescaledbyΣV,
where Σ164 = 157.5. In the top plot the smooth red curve where Σ164 = 157.5. The smooth red curves displays the
displays a fit of the analytic prediction for the microscopic analytic prediction for the microscopic density of D5. The
density of D5 to the 164 data. The fit values are mˆ = 3.5 values, mˆ = 3.5 and aˆ8 = 0.35 for 164 and mˆ = 8.5 and
and aˆ8 = 0.35 (aˆ6 = aˆ7 = 0). In the plot below the smooth aˆ8=0.55 areobtainedfromthefitinthetoppanelofFigure
red line shows the prediction for the 204 data (mˆ = 8.5 and 1andbyfinitevolumescaling from 164 to204 (aˆ6 =aˆ7=0).
aˆ8 = 0.55) obtained from finite volume scaling of the 164
values. We observe that the finite volume scaling at fixed
lattice spacing is well respected.
We first consider the m = −0.184 data with ν =
0
0. We make a fit and find the best values of mˆ = 3.5,
the parameter free analytic prediction which in this case ΣV = 216 and aˆ8 = 0.35. (The fact that mˆ and aˆ takes
follows from finite volume scaling. the same value as for the 164 lattice at βIw = 2.635 is
We conclude that W = W = 0 and W > 0 is con- accidental.) For these values the analytic curve matches
6 7 8
sistent with the lattice data at β = 2.635 and that thedatawelloverthelowestfoureigenvalues,seethetop
Iw
the volume scaling at fixed lattice spacing predicted by panel of Figure 3.
Wilson chiral perturbation works well. Given these values of mˆ and aˆ we then test the pre-
8
dicted density for ν = 1 against the data. As can be
observed from the lower panel of Figure 3 this again
B. Mass scaling works well for the lowest four eigenvalues. Finally, we
consider also the higher mass data at the same value of
Oursecondtestoftheanalyticpredictionsforthespec- the coupling. Since the lattice mass has changed from
trum of D concerns the scaling with the quark mass at m = −0.184 to m = −0.178 we have δm = 0.006
5 0 0
fixed lattice spacing. For this we choose a 204 lattice so that δmΣV = 0.006∗216 = 1.3. Hence, we predict
with β = 2.79 and two masses m = −0.178 respec- that the β = 2.79 data with m = −0.178 should be
Iw 0 Iw 0
tively m =−0.184. matched with mˆ = 4.8 and aˆ = 0.35. As can be seen
0
6
Ev_279_20_HYP_Q0_m184 Ev_279_20_HYP_Q0_m178
0.6 m=3.5 a=0.35 ν=0 0.6 m=4.8 a=0.35 ν=0
FIT PREDICTION
0.5 0.5
a) a)
m; 0.4 m; 0.4
5λ, 5λ,
=0( 0.3 =0( 0.3
ν ν
5 5
ρ ρ
0.2 0.2
0.1 0.1
0 0
-12 -8 -4 0 4 8 12 -12 -8 -4 0 4 8 12
λ5 λ5
Ev_279_20_HYP_Q1andQm1_m184 Ev_279_20_HYP_Q1andQm1_m178
0.6 m=3.5 a=0.35 ν=1 0.6 m=4.8 a=0.35 ν=1
PREDICTION PREDICTION
0.5 0.5
a) a)
m; 0.4 m; 0.4
5λ, 5λ,
=1( 0.3 =1( 0.3
ν ν
5 5
ρ ρ
0.2 0.2
0.1 0.1
0 0
-12 -8 -4 0 4 8 12 -12 -8 -4 0 4 8 12
λ5 λ5
FIG.3: TheeigenvaluedensityofD5 forβIw =2.79,V =204 FIG.4: TheeigenvaluedensityofD5 forβIw =2.79,V =204
and m0 =−0.184 top: ν =0 and bottom: |ν|=1. For the and m0 =−0.178 top: ν =0 and bottom: |ν|=1. We use
ν = 0 data we find the best values mˆ = 3.5, aˆ8 = 0.35 and the value ΣV = 216 (found for m0 = −0.184) and δm =
ΣV = 216. With these values we then test the prediction −0.006 to determine the value mˆ = 4.8 for the βIw = 2.79,
against the|ν|=1 data. V =204,m0 =−0.178data. Sincethecouplingisunchanged
also aˆ8 = 0.35 is fixed by the previous fit. The parameter
freepredictionsforν =0and|ν|=1arethenplottedagainst
from Figure 4 this parameter free prediction works well the data. We observe that the mass scaling at fixed lattice
for the ν = 0 as well as for the |ν| = 1 sector. Figure spacing works well.
5 compares the corresponding parameter free prediction
for |ν|=2 against the data.
We conclude that W6 = W7 = 0 and W8 > 0 is con- aˆ8. Here we demonstrate the strong sensitivity of the
sistent with the 204 lattice data with β = 2.79 and analytic predictions for the microscopic eigenvalue den-
Iw
that the mass scaling at fixed lattice spacing implied by sity of D5 with fixed index to the values of mˆ and aˆ8.
WCPT works well. For the influence of W and W see Figure 6 give examples of this strong dependence which
6 7
below. allow for a precise determination of the best values for
the match to the data. Figure 7 compares the best fit to
thecontinuum,a=0,curve. Inthiswayweestimatethe
error on aˆ in our data sets to be at most 0.05 and the
8
error on mˆ to be below 0.5. With increased statistics on
slightly larger lattices at slightly smaller lattice spacing
it should be possible to determine mˆ and aˆ with high
8
accuracy.
C. The mˆ and aˆ8 dependence
Above we have obtained best fits of the analytic pre-
diction, Eq. (12), to the lattice data by varying mˆ and
7
Ev_279_20_HYP_Q2_Qm2_m184 m=4.8 a=0.35 ν=0
0.6 m=3.5 a=0.35 ν=2 0.6 m=4.3 a=0.35 ν=0
PREDICTION m=5.3 a=0.35 ν=0
0.5 0.5
a) a)
m; 0.4 m; 0.4
5λ, 5λ,
=2(50.3 ν=0( 0.3
ν 5
ρ ρ
0.2 0.2
0.1 0.1
0 0
-12 -8 -4 0 4 8 12 -12 -8 -4 0 4 8 12
λ5 λ5
0.7
0.6 Emv=_42.87 9a_=200.3_5H Yν=P2_Q2_Qm2_m178 0.5 mmm===444...888 aaa===000...334500 ννν===000
PREDICTION
0.5
m;a) 0.4 m;a) 0.4
ν=25ρ(λ,50.3 ν=05ρ(λ,500..23
0.2
0.1
0.1
0 -12 -8 -4 λ05 4 8 12 0 -12 -8 -4 λ05 4 8 12
FIG.5: TheeigenvaluedensityofD5 forβIw =2.79,V =204 FIG. 6: The dependence of the eigenvalue density of D5 for
and |ν| =2 top: m0 =−0.184 and bottom: m0 =−0.178. ν =0ontop: thequarkmassmˆ andonbottom: thelattice
The smooth curves are the parameter free predictions from artifacts aˆ8. The central (Red) line in both plots correspond
WCPT (all parameters are fixed from the fit to the ν = 0, tothe(Red)analytic curvewhich isplotted against thedata
m0 =−0.184 data). in the top panel of Figure 4.
E. Individual eigenvalue distributions
D. Effects of W6 and W7
In order to understand in greater detail whether our
lattice data match the predictions from WCPT and
The effect of W < 0 or W < 0 is to smear out the WRMTitisadvantageoustoconsiderthedistributionsof
6 7
analytic predictions so that the bumps in the curves be- individualeigenvalues. Aswedonothaveanysimpleex-
come less pronounced. In Figure 8 we give an example pression available from the chiral Lagrangian approach,
of the effect of aˆ2 and aˆ2. The more oscillating curve in we will simply make use of the relation to Wilson chi-
6 7
both plots corresponds to the curve in the top panel of ralrandommatrixtheory,andgeneratethedistributions
Figure 4 which has aˆ = aˆ = 0. As a less pronounced numerically. See Figure 9.
6 7
structure of the oscillations due to the individual eigen- For the β = 2.635 data on the 164 lattice the first
Iw
values is unfavored by the lattice data we conclude that eigenvalue matches well that generated from WRMT.
|aˆ2| and |aˆ2| appear to be smaller than 0.01. Note that The second eigenvalue however is somewhat broader.
6 7
aˆi only enters as squares in the action(9), that is why it In the βIw = 2.79 data from the 204 lattice the
is natural to give the bound on the square. first eigenvalue again matches well that generated from
WRMT.Thistimethesecondeigenvalueisrepelledfrom
the first in a manner comparable to the one observed in
the simulation of WRMT.
8
2
m=4.8 a=0.0 a=0.0 a=0.35
6 7 8
Ev_279_20_HYP_Q0_m184 m=4.8 a=i0.2 a=0.0 a=0.35
m=3.5 a=0.00 ν=0 0.6 6 7 8
m=3.5 a=0.35 ν=0
1.5
0.5
a) a)i
m; m, 0.4
5λ, 1 5λ,
ν=0ρ(5 ν=0ρ(50.3
0.2
0.5
0.1
0 -12 -8 -4 λ05 4 8 12 0 -12 -8 -4 λ05 4 8 12
FIG. 7: The eigenvalue density of D5 for ν = 0. Shown are
the lattice data and best fit, previously displayed in the top
panel of Figure 3, along with the a = 0 result at the same
value of mˆ. Clearly it is essential to include the order a2 0.6 mm==44..88 aa6==00..00 aa7==0i0.0.2 aa8==00..3355
6 7 8
effectsin WCPTin ordertomatchthedata. Notethatscale
on they-axisis different from that in theother plots. 0.5
a)i
m, 0.4
F. The real eigenvalues of DW =05(λ, 0.3
ν
5
ρ
From the flow of λ5 as a function of m we have also 0.2
determined the real eigenvalues of D as well as their
W
chiralities. This enables us to measure both the density 0.1
of the real modes
0
-12 -9 -6 -3 0 3 6 9 12
λ5
ρν (λW;a)= δ(λW −λW) (19)
real k
* k + FIG. 8: The eigenvalue density of D5 for ν = 0 top: the
X
dependence on aˆ6 and bottom: the dependence on aˆ7. The
as well as the distribution more oscillating (Red) line in both plots has a6 = a7 = 0
andcorresponds tothe(Red)analyticcurvewhich isplotted
against the data in the top panel of Figure 4. Both aˆ26 <
ρνχ(λW;a)= δ(λW −λWk )signhk|γ5|ki . (20) 0 or aˆ27 < 0 tends to smear out the oscillations. Since the
* +
k magnitude of the oscillations in the lattice data presented in
X
Figure 4 is of the same magnitude as those in the analytic
The analytic predictions for both these quantities are prediction with aˆ6 = aˆ7 = 0 we can put an upper bound on
available, see [7] for ρreal and [4, 5] for ρχ. The two these |aˆ26|,|aˆ27|<0.01.
distributions both merge to a ν-fold δ-function at the
origin for aˆ→0.
From the fit to ρ we have already obtained the rel- We observe that the scaling implied by WCPT is re-
5
evant values of ΣV, mˆ and aˆ. This enables a highly markable accurate. After the parameter free shift of the
nontrivial parameter free test of WCPT. In Figure 11 eigenvaluesthecenterofthedatapeakisrightattheori-
the parameterfree analytic predictions areplotted along gin as predicted by WCPT. Moreover, the width of the
with the results from the lattice. Since accidentally the peak in both data sets is exactly of the rightmagnitude.
value aˆ = 0.35 was obtained for both data sets we have We also note that the peak height in the data increases
combinedtheplots. Fortheplotwehaverescaledthelat- towards the analytic prediction when going from 164 to
tice eigenvalues, shown in Figure 10, by ΣV (the value 204.
for which we obtained from the respective fits to ρ ) However, in the data there is an asymmetry of ρ
5 real
and then shifted the eigenvalues by ΣVm −mˆ. For the andρ whichpersist whengoing from164 to 204. More-
0 χ
204 data with m0 = −0.184 and βIw = 2.79 this reads over,theaveragenumberofrealmodesispredictedtobe
ΣVm −mˆ = 216(−0.184)−3.5 = −43.2 while for the 1.017 by WRMT for ν =1 and aˆ =0.35 [7], in the data
0
164 data with βIw = 2.635 and m0 = −0.216 it reads we find 1.237 for the 164 lattice and 1.185 for the 204
ΣVm −mˆ =157.5(−0.216)−3.5=−37.5. lattice. That is, the average number of additional real
0
modes is off by a factor of 10. We note, however, that
thisaveragenumberofadditionalrealmodesdecreaseby
9
50
1st ev Ev_2635_16_HYP_Q0_m216 Ev_real_2635_16_HYP_Q1andQm1
0.6 2nd ev Ev_2635_16_HYP_Q0_m216 Ev_real_279_20_HYP_Q1andQm1
5λ) 20.5 PREDICTION (NUM) a) 40
=0( =0;
ν=05νρ(λ) ρ515000...243 ν=1Wρλ(a ,amreal2300
10
0.1
0
0 4 8 12 0
λ5 0 0.1 0.2 0.3 0.4 a 0λ.5W 0.6 0.7 0.8 0.9 1
FIG. 10: The distribution of the real modes of DW before
subtraction of thebare mass and scaling of theaxis.
1st ev Ev_279_20_HYP_Q0_m184
0.6 2nd ev Ev_279_20_HYP_Q0_m184
5λ) 20.5 PREDICTION (NUM) index of the Wilson-Dirac operator can be matched by
=0( the predictions from Wilson chiral perturbation theory
ν
ρ50.4 with W > 0 and W = W = 0. We have successfully
5λ) 10.3 taesstweedllt8ahsetfihneitmeavsoslus6mcaelinscga7alisngfixaetdfilxaetdticleatstpicaecisnpgacpirneg-
ν=0( 0.2 dicted by Wilson chiral perturbation theory against the
ρ5 lattice data. The spectrum of D in this way gives a
5
0.1 verydirectwayofmeasuringthelow-energyconstantsof
Wilson chiral perturbation theory. Although these con-
00 4 8 12 stants represent lattice artifacts, it is essential to know
λ5 their values in order to extract the physical constants.
Moreover, the detailed predictions can be used with ad-
FIG. 9: The distribution of the first (black) and second
vantagetounderstandhowthespectrumcanbemadeto
(blue) eigenvalue top: in the 164 βIw = 2.635 data set and retainitsgapassimulationsarepushedtowardsphysical
bottom: in the 204 βIw = 2.79 data set. The (red) his- values and towards the continuum limit.
tograms show the first and second eigenvalue distribution in
a WRMT simulation with n=100 and 100000 matrices. Weemphasizethattheassignmentofindexν toagiven
gauge field configuration must be based on the spectral
flowoftheHermitianWilson-Diracinorderforthematch
22% when going from a = 0.93 fm on 164 to a = 0.075 to Wilson chiral perturbation theory to be valid. In the
fm on 204. microscopic limit the analytic results predict that the
The asymmetry of the distribution of the real eigen- real eigenvalues of D are located in well separated re-
W
values is a clear signal that at the present volumes there gions. On the lattice, it should therefore be possible in
are small realeigenvalues of DW which are not captured a clean way to introduce a cut-off such that we consider
by leadingorderWCPT.In[24]it wasobservedthatthe only the physical branch of the spectrum. In support of
eigenvectors corresponding to eigenvalues of DW which this we have explicitly computed the real eigenvalues of
fallto therightofthe mainbandhaveastrongtendency D and found that the bulk of the real eigenvalues fol-
W
to be localized. Obviously such localized modes are not low the predictions from WCPT. However, we observed
described by WCPT. also an asymmetry of the distribution of the real eigen-
values not described by WCPT. For this reason some
configurations that are difficult to classify remain in our
samples. Similar concerns have been voiced already in
the context of the overlap operator [24], where localized
VI. CONCLUSIONS modes that intuitively should play no role in the contin-
uumlimitneverthelesssignificantlydeformthespectrum
The analytical predictions for the quenched micro- of the overlap operator at fixed ν. We stress that such
scopic spectral density of the Hermitian Wilson-Dirac localized modes are not described by Wilson chiral per-
operator have been compared to the low lying eigenval- turbation theory. As such modes are more easily identi-
ues of Wilson fermions on the lattice. In this first study, fiedinthecomplexpartofthespectrumofD acareful
W
we have demonstrated that the results given for fixed and high-statistics study of the microscopic spectrum of
10
theWilson-Diracoperatorwhichaddressthisissuewould
Ev_real_279_20_HYP_Q1andQm1 provideavaluabletestofWilsonchiralperturbationthe-
0.3 Ev_real_2635_16_HYP_Q1andQm1
a=0.35 ory at presently realistic lattice spacings. The analytic
PREDICTION
prediction for the quenched microscopic eigenvalue den-
Wλ;a) 0.2 srietcyenotflyDW[7].inthecomplexplanehasbeenpresentedvery
=1(real
νρ It is also possible to transform all analytical predic-
0.1 tions for the microscopic spectral density of the Hermi-
tian Wilson-Dirac operator into corresponding expres-
sions for the pseudoscalar condensate hψ¯γ ψi, as dis-
5
cussed in Ref. [25]. It would be interesting to use this
0
-10 0 10 20 30 40 50 60 70
λW quantity to extract the Wilson low-energy constants.
Tosummarizewehavedemonstratedherethatthethe-
oretical predictions of Wilson chiral perturbation theory
Ev_real_279_20_HYP_Q1 with W6 = W7 = 0 and W8 > 0 for the microscopic
0.3 Ev_real_2635_16_HYP_Q1 spectral properties of the Wilson-Dirac operator can be
a=0.35
PREDICTION matched to lattice data. It would be most interesting to
extend this quenched study to the physical case of two
W,a) 0.2 light flavors. The detailed theoretical predictions have
=1(λ been given recently [9].
ν
χ
ρ
0.1
End note: We learned at the Lattice 2011 meeting
that Albert Deuzeman, Urs Wenger and Ja¨ır Wuilloud
havebeeninvestigatingthesameissueswehavediscussed
0 here. We thank these authors for coordinating publica-
-10 0 10 20 30 40 50 60 70 tion of results. Their paper is Ref. [26].
λW
FIG. 11: top: The distribution of the real modes. bottom: Acknowledgments: We would like to thank Jac Ver-
the distribution ρχ defined in Eq. (20). All parameters are baarschot,Anna Hasenfratz, Hidenori Fukaya, Christian
fixedfromthefitstoρ5. Theeigenvaluedataarefirstrescaled Lang and Christof Gattringer for discussions. The work
byΣV andthenshiftedbym0ΣV −mˆ wherem0isthelattice of K.S. was supported by the Sapere Aude program of
mass and mˆ is the valueobtained previously for this m0. The Danish Council for Independent Research.
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