Aoki Phases in Staggered-Wilson Fermions

0.12 MB·English

by  Tatsuhiro Misumi

Aoki Phases in Staggered-Wilson Fermions
∗
2
1 TatsuhiroMisumi†
0
YukawaInstituteforTheoreticalPhysics,KyotoUniversity
2
E-mail: [email protected]
n
a MichaelCreutz‡
J
BrookhavenNationalLaboratory
7
2 E-mail: [email protected]

] Taro Kimura
t
a DepartmentofBasicScience,UniversityofTokyo
l
- MathematicalPhysicsLaboratory,RIKEN
p
E-mail: [email protected]
e
h
TakashiZ.Nakano
[
DepartmentofPhysics,KyotoUniversity
2
YukawaInstituteforTheoreticalPhysics,KyotoUniversity
v
1 E-mail: [email protected]
3
2 Akira Ohnishi
1 YukawaInstituteforTheoreticalPhysics,KyotoUniversity
.
0 E-mail: [email protected]
1
1
1 Weinvestigatetheparity-brokenphase(Aokiphase)forstaggered-Wilsonfermionsbyusingthe
: Gross-Neveu model and the strong-coupling lattice QCD. In the both cases the gap equations
v
i indicatetheparity-brokenphaseexistsandthepionbecomesmasslessonthephaseboundaries.
X
We also show we can take the chiraland continuumlimit in the Gross-Neveumodelby tuning
r
a
massandgauge-couplingparameters. Thissupportstheideathatthestaggered-Wilsonfermions
canbeappliedtothelatticeQCDsimulationbytakingachirallimit,aswithWilsonfermions.
TheXXIXInternationalSymposiumonLatticeFieldTheory-Lattice2011
July10-16,2011
SquawValley,LakeTahoe,California
YITP-11-86,KUNS-2366,RIKEN-MP-33
∗
†Speaker.
‡Authoredunder contract number DE-AC02-98CH10886 withtheU.S.DepartmentofEnergy. Accordingly, the
U.S.Governmentretainsanon-exclusive,royalty-freelicensetopublishorreproducethepublishedformofthiscontri-
bution,orallowotherstodoso,forU.S.Governmentpurposes.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
AokiPhasesinStaggered-WilsonFermions TatsuhiroMisumi
1. Introduction
Recently staggered-based Wilson fermions were proposed by introducing the taste-splitting
mass or the flavored-mass terms into staggered fermions [1, 2, 3]. They can be applied to lattice
QCD not only as Wilson fermions but also as an overlap kernel. One possible advantage of these
novelfermionscalledstaggered-Wilsonandstaggered-overlapisreductionofthematrixsizesinthe
associatedDiracoperators,whichleadstoreductionofnumericalcostsinlatticeQCDsimulations.
Thustheymaybeabletoovercometheusual naive-fermion-based lattice fermions inlattice QCD
[4]. The purpose of this work is reveal properties of staggered Wilson fermions in terms of the
parity phasestructure (Aokiphase) [5]. TheAokiphaseforthestaggered-Wilson wasfirststudied
inRef.[3]andthepresentpapershowsfurtherinvestigationofthistopic. TheexistenceoftheAoki
phase and thesecond-order phase boundary inWilson-type lattice fermions indicates that one can
apply them tolattice QCDsimulations by tuning amass parameter totake achiral limit. Besides,
the understanding of the parity-broken phase gives practical information for the application of its
overlapanddomain-wallversions.
In this paper we elucidate the parity phase structure for staggered-Wilson fermions in the
frameworkoftheGross-Neveumodelandthehoppingparameterexpansioninthestrong-coupling
lattice QCD. We find the gap equations derived from the both theories show the pion condensate
becomes nonzero in some range of the parameters and the pion becomes massless on the phase
boundaries. It means the Aoki phase exists and the order of the phase transition is second-order.
WealsoshowwecantakethechiralcontinuumlimitintheGross-Neveumodelbytuningthemass
and the gauge-coupling. These results on the staggered-Wilson fermion incidate we can obtain
one-ortwo-flavorfermions bytuningthemassparameterandperform thelatticeQCDsimulation
with these fermions as in the Wilson fermion. We note the results on the Gross-Neveu model is
based on the work by some of the present authors [3, 6] while the results on the strong-coupling
latticeQCDarepartsofaworkinprogress.
2. Staggered Wilsonfermions
We begin with staggered-Wilson fermions in which the flavored-mass terms split the four
degenerate tastesinamannersimilartotheusualWilsonterm. Therearetwopossibletypesofthe
flavored-mass termsforstaggered fermionsas
M(1)=e (cid:229) h h h h C C C C =(1 g )+O(a), (2.1)
f 1 2 3 4 1 2 3 4 ⊗ 5
sym
M(2)= (cid:229) i e mn h m h n (Cm Cn +Cn Cm )=(1 (cid:229) s mn )+O(a), (2.2)
f m >n 2√3 ⊗m >n
whereCm =(Vm +Vm†)/2,(h m )xy=(−1)x1+...+xm−1d x,y,(e )xy=(−1)x1+...+x4d x,y,(e mn )xy=(−1)xm +xn d x,y,
with(Vm )xy =Um ,xd y,x+m . Intherighthandsidesweusethespin-tasterepresentation as1 g5. We
⊗
(1) (2)
refertoM astheAdams-typeandM theHoelbling-type. Theformersplitsthe4tastesintotwo
f f
withpositive(m=+1)andtheothertwowithnegative(m= 1)masswhilethelattersplittheminto
−
one withpositive(m=+2), twowithzero(m=0)and theother onewithnegative mass(m= 2).
−
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AokiPhasesinStaggered-WilsonFermions TatsuhiroMisumi
NowweintroducetheWilsonparameterr=rd andshiftthemassfortheactionsaswithWilson
x,y
fermions. ThentheAdams-typestaggered-Wilson fermionactionisgivenby
SA = (cid:229) c¯x[h m Dm +r(1+M(f1))+M]xyc y, (2.3)
xy
with Dm = 12(Vm −V−m ). Here M stands for the usual taste-singlet mass (M = Md x,y). The
Hoelbling-type staggered-Wilson fermionactionisgivenby
SH = (cid:229) c¯x[h m Dm +r(2+M(f2))+M]xyc y. (2.4)
xy
IntheQCDsimulationwewilltunethemassparameterMtotakeachirallimit. Forsomenegative
values of the mass parameter: 1 <M <0 for Adams-type and 2<M <0 for Hoelbling-type
− −
withr=1,weobtaintwo-flavorandone-flavor overlapfermionsrespectively byusingtheoverlap
formula.
The potential problem in lattice QCD with these fermions is the breaking of some discrete
symmetries as the shift symmetry caused by the flavored-mass terms [1, 2]. There has not yet
been a consensus on whether it does harm to lattice QCD with staggered-Wilson fermions. We
can answer this question partly by studying the Aoki phase since a clear symptom is expected to
appear in the phase structure if the symmetry breaking ruins the essential properties of QCD. In
the following sections we will find the Aoki phase structure in the staggered-Wilson fermion is
qualitatively similartotheoriginalWilsononeandthereisnodisease.
3. Gross-Neveumodel
Wefirstinvestigatetheparityphasediagramforstaggered-Wilsonfermionsbyusingthed=2
Gross-Neveu modelasatoymodelofQCD.Tostudythepioncondensate wegeneralize theusual
staggered Gross-Neveumodeltotheonewiththeg -type4-pointinteraction, whichisgivenby
5
S = 1(cid:229) h m c¯n(c n+m c n m )+(cid:229) c¯n(M+r(1+Mf))c n
2n,m − − n
g2 (cid:229) ((cid:229) c¯2N+Ac 2N+A)2+((cid:229) i( 1)A1+A2c¯2N+Ac 2N+A)2 , (3.1)
−2N −
N A A
h i
wherethetwo-dimensionalcoordinateisdefinedasn=2N +AwithsublatticesA=(A ,A )(A =
1 2 1,2
0,1). In this model c is aN-component one-spinor (c ) (j=1,2,...,N) where cc¯ =(cid:229) N c¯ c .
n n j j=1 j j
(−1)A1+A2 corresponds to G 55 =g5⊗g5 in the spinor-taste expression while h m =(−1)n1+...+nm−1
corresponds to gm . In thisdimension the Adams-type andHoelbling-type flavored-mass terms co-
incide and there isonly one type M =G G 1 g +O(a)with G = ih h (cid:229) C C . This
f 5 55 5 5 1 2 sym 1 2
∼ ⊗ −
mass term assigns thepositive mass (m=+1)to one taste and the negative mass(m= 1)tothe
−
other. Withbosonicauxiliary fieldss N ,p N leadingtos -mesonandp -mesonfields,theactionis
rewrittenas
S = 1(cid:229) h m c¯n(c n+m c n m )+(cid:229) c¯nMfc n
2n,m − − n
+2Ng2(cid:229) ((s N −1−M)2+p N2 )+ (cid:229) c¯2N+A(s N +i(−1)A1+A2p N )c 2N+A, (3.2)
N N,A
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AokiPhasesinStaggered-WilsonFermions TatsuhiroMisumi
-2 -1.5 -1 -0.5 0
Figure1: Aokiphasestructureforthestaggered-WislonfermionintheGross-Neveumodel. Astandsfora
paritysymmetricphaseandBforAokiphase.
where we take r = 1 as the Wilson parameter. After integrating the fermion field, the partition
function andtheeffectiveactionwiththeseauxiliary fields(mesonfields)aregivenby
Z= Ds N Dp N e−NSeff(s ,p ), Seff = 21g2(cid:229) ((s N −1−M)2+p N2 )−TrlogD, (3.3)
Z N
pwairtthitDionn,mfu=nc(tsioNn+isig(i−ve1n)Ab1y+Ath2peNsa)dddnl,emp+oihn2mt(odfnt+hme,mac−tiod nn−ams,mZ)+=(eMSfe)ffn(,sm0.,pI0n) wthiethlatrhgeetNranlismlaittiothne-
−
invariant solutions s , p satisfying the saddle-point equations d Seff(s 0,p0) = d Seff(s 0,p0) =0. After
0 0 ds dp
0 0
somecalculationprocesstoderivethefermiondeterminant[3]weobtaintheconcreteformsofthe
saddle-point equations inthemomentumspace
s 1 M dk2 s (s 2+p 2+s2) c2c2s
0− − =4 0 0 0 − 1 2 0 , (3.4)
g2 (2p )2((s +c c )2+p 2+s2)((s c c )2+p 2+s2)
Z 0 1 2 0 0− 1 2 0
p dk2 p (s 2+p 2+s2)+c2c2p
0 =4 0 0 0 1 2 0 , (3.5)
g2 (2p )2((s +c c )2+p 2+s2)((s c c )2+p 2+s2)
Z 0 1 2 0 0− 1 2 0
with cm =coskm /2 and sm =sinkm /2. Now what weare interested in isthe parity phase diagram
inthistheory. Theparity phaseboundary M (g2)isderived byimposing p =0in(3.4)(3.5) after
c 0
theoverallp beingremovedinthesecondone. Thenthegapequations aregivenby
0
1+M dk2 2c2c2s
c =4 1 2 0 , (3.6)
g2 (2p )2((s +c c )2+p 2+s2)((s c c )2+p 2+s2)
Z 0 1 2 0 0− 1 2 0
1 dk2 s 2+s2+c2c2
=4 0 1 2 . (3.7)
g2 (2p )2((s +c c )2+p 2+s2)((s c c )2+p 2+s2)
Z 0 1 2 0 0− 1 2 0
By removing s in these equations, we derive the phase boundary M (g2). The result is shown
0 c
in Fig. 1. It indicates the parity phase structure in the staggered-Wilson fermion is qualitatively
similar to the usual Wilson case [5] reflecting the mass splitting of tastes given by the flavored
mass. We also check the pion mass becomes zero on the second order phase boundary as m2 (cid:181)
p
Vd 2S˜eff =0whereS =VS˜ withV beingthevolume.
d 2p02 |M=Mc eff eff
Wenextconsiderthechiralandcontinuumlimitofthestaggered-WilsonGross-Neveumodels.
ThestrategyistoexpandthefermiondeterminantintheeffectivepotentialinEq.(3.3)withrespect
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AokiPhasesinStaggered-WilsonFermions TatsuhiroMisumi
to the lattice spacing a. After some calculations (See details in Ref. [3]) we obtain the effective
potential remaining inthelimita 0,
→
M+1/a 2 1 1
S˜ = + C s + C˜ + log4a2 p 2
eff − g2s a 1 0 2g2p − 0 p 0
(cid:16) (cid:17) (cid:16) (cid:17)
1 1 1 s 2+p 2
+ C˜ +2C + log4a2 s 2+ (s 2+p 2)log 0 0. (3.8)
2gs2 − 0 2 p 0 p 0 0 e
(cid:16) (cid:17)
with the three numbers asC˜ =1.177,C = 0.896 andC =0.404. Here taking the chiral limit
0 1 2
−
means restoring the rotational symmetry in s and p by tuning the parameters. In this model we
0 0
need introduce two independent coupling constants g2 and g2 to restore the symmetry although
s p
the necessity oftwocouplings isjust amodel artifact. Thetuned point forthechiral limitwithout
O(a)corrections is
2g2 g2
M=− as C1−1, gp2 = 4C2g2ss +1, (3.9)
Totakethecontinuum limitweintroducetheL -parameteras2aL =exp p C˜ p C p . Then
2 0− 2−4g2s
thecoupling renormalization forthechiralandcontinuum limitisgivenhby i
1 1 1 1 1 1
=C˜ 2C + log , =C˜ + log . (3.10)
2g2s 0− 2 p 4L 2a2 2g2p 0 p 4L 2a2
(cid:18) (cid:19) (cid:18) (cid:19)
wherewekeepL finitewhentakingthecontinuum limita 0. Finallytherenormalized effective
→
potential inthechiralandcontinuum limitisgivenby
1 s 2+p 2
S˜ = (s 2+p 2)log 0 0, (3.11)
eff p 0 0 eL 2
This wine-bottle potential yields the spontaneous breaking of the rotational symmetry. We have
shown that the chirally-symmetric continuum limit can be taken by fine-tuning a mass parameter
and two coupling constants in the staggered-Wilson Gross-Neveu model. Considering that the
necessity of the two coupling constants is just a model artifact, this result indicates we can take a
chiral limitby tuning only the mass parameter as in the Wilson fermion. Our results on the chiral
andcontinuum limitforstaggered-Wilson arequalitatively thesameastheWilsoncase[5].
4. Strong-coupling QCD
In this section we investigate the Aoki phase structure in lattice QCD with staggered-Wilson
fermions in the framework of the hopping parameter expansion (HPE) in the strong-coupling
regime. We can detect a symptom of symmetry breaking from this analysis although we can-
notknowdetailsofthetruevacuumfromthis. ForsimplicityweconcentrateontheHoelbling-type
lattice fermion here, but we can also make the same analysis in a parallel way for the Adams-
type fermion. To perform the HPE for the Hoelbling-type fermion, we rewrite the action (2.4) by
redefining c √2Kc withK=1/[2(M+2r)],
→
S=(cid:229) c¯xc x+2K(cid:229) c¯x(h m Dm )xyc y+2Kr(cid:229) c¯x(Mf)xyc y. (4.1)
x x,y x,y
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AokiPhasesinStaggered-WilsonFermions TatsuhiroMisumi
(x,a) (y,b) hχaxχ¯byi0=−δxyδab µ
= +
(x,a) (x+µˆ,b) Kηµ,x(Uµ,x)ab = + µ 0 x 0 x 0 µˆ x
(x,b) (x µˆ,a) −Kηµ,x(Uµ†,x)ab
((xx,,ba)) xx−+−µµˆˆ ((xx−+µµˆˆ+−ννˆˆ,,ba)) 2−K2Krirηiµηνµ,xν(,xU(µU,xν†U,x+ν,µxˆ+Uµˆµ†),xab)/ab(/2(32√3√3)3) + µ ν +0µ νµˆ+νˆ x
Figure2: (Left):FeynmanrulesfortheHPE.(Center):onepointfunction.(Right):twopointfunction.
In Fig. 2(Left) we write down the Feynman rules in the HPE for this fermion. In this paper we
perform the hopping parameter expansion up to O(K3), which works for a small K. We derive
chiralandpioncondensates fromtheone-point functionofthemesonoperator(M =c¯ c )inthe
x x x
mean-field approximation. The equation for the one-point function up to O(K3) is obtained in a
self-consistent wayasshowninFig. 2(Center),
−S x≡hMxi=hMxi0+2K2(cid:229) S x+mˆS x−2·214(Kr)2 (cid:229) S xS x+mˆ+nˆ, (4.2)
m m =n
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where we drop the link variable since we work in the strong-coupling limit. O(K3) diagrams are
found to vanish due to the cancellation between diagrams. Here we solve this equation for the
condensate S within the mean-field approximation. For our purpose we assume S =s +ie p ,
x x x x
where s and p correspond to the chiral and pion condensates. We substitute this form into Eq.
x x
(4.2)andobtaintheself-consistent equation fors andp as
x x
1
(s +ie p )= 1+2K2 4 s 2+p 2 2 (Kr)2 4 3(s +ie p )2, (4.3)
x x
− − · − ·24 · ·
(cid:0) (cid:1)
whichyields s = 1+16K2p 2and ip = 8K2 2isp . Herewehavesetr=2√2forsimplicity.
− − − − ·
Wehavetwosolutions depending onwhetherp =0orp =0: Forp =0wehaveatrivialsolution
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s =1. Forp =0wehaveanon-trivial solution as
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1 1 1
s = , p = 1 . (4.4)
16K2 ±s16K2 −16K2
(cid:18) (cid:19)
Inthissolutionthepioncondensate isnon-zeroandthe signsindicatespontaneous paritybreak-
±
ing. This parity-broken phase (Aoki phase) appears in the range of the hopping parameter or the
mass parameter as K >1/4 or equivalently 4√2 2<M < 4√2+2. We expect that the
| | − − −
expansion up to O(K3) works to give a meaningful result at least around the critical parameter
K 3=1/64 1.
c
| | ≪
We next discuss the two-point function of the meson operator S(0,x) M M up to the
0 x
≡
sameorderastheone-pointfunction. FromFig. 2(Right)wederivethefollowingequationfortwo
pointfunction. (O(K3)diagramsvanishagain.)
S(0,x) c¯ac ac¯bc b = d N +K2(cid:229) c¯ac ac¯bc b
0 0 x x 0x c mˆ mˆ x x
≡h i − h i
m
±
+ 2Kri 1 2 (cid:229) c¯a c a c¯bc b . (4.5)
(cid:18) 23√3(cid:19) m , n h mˆ+nˆ mˆ+nˆ x xi
±(m =±n )
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AokiPhasesinStaggered-WilsonFermions TatsuhiroMisumi
Thentheself-consistent equation forS isgiveninthemomentumspaceas
S(p)= N + K2(cid:229) e ipm +eipm + 2Kr 1 2 (cid:229) (cid:229) ei( pm pn ) S(p). (4.6)
c − ± ±
− (cid:20)− m (cid:18) 23√3(cid:19) m =n (cid:21)
(cid:0) (cid:1) 6 ±
Wefinallyobtainthemesonpropagator as
2 1
S(p)=Nc 2K2(cid:229) cospm +4 2Kr 1 (cid:229) cospm cospn 1 − . (4.7)
(cid:20)− m (cid:18) 23√3(cid:19) m =n − (cid:21)
6
The pole of S(p) gives the meson mass. Remembering g in the staggered fermion is given by
5
ex=( 1)x1+...+x4 andthepionoperatorisgivenbyp x=c¯xiexc x,itisobviousthatthemomentum
−
ofthe pionshould bemeasured from theshifted origin p=(p ,p ,p ,p ). Thusweset p=(imp a+
1 16K2
p ,p ,p ,p ) for1/S(p)=0in(4.7),whichgivesthepionmassmp ascosh(mp a)=1+ − .
6K2
In this result the pion mass becomes tachyonic in the range K >1/4. It indicates there occurs
| |
a phase transition between parity-symmetric and broken phases at K =1/4, which is consistent
c
| |
withtheresultofcondensates inEq.(4.4).
In this section we investigate parity phase structure for staggered-Wilson fermions by using
thehopping parameter expansion uptoO(K3)inthestrong-coupling regime. Although wecannot
giveastrongargumentjustfromthisapproximate calculation, wenotethatourresultonthephase
structure isqualitatively consistent withthat of HPEfor Wilson fermions [5], which implies exis-
tence of the Aoki phase in staggered-Wilson fermions. The full-order calculation in HPE and the
effectivepotential analysis inprogress [7]willgiveusmoreconclusive evidenceforthistopic.
5. Summary
InthispaperwestudytheGross-Neveumodelandthestrong-coupling latticeQCDwithstag-
gered Wilson fermions with emphasis on the Aoki phase structure. We have shown the parity
broken phase andthesecond orderphase boundary existinthestaggered-Wilson fermions aswith
theWilsonfermion. OurresultsindicatethatwecanapplythestaggeredWilsonfermionstolattice
QCDsimulations bymassparameter tuning. Theseresultsalsoindirectly suggest theapplicability
of the staggered overlap and staggered domain-wall fermions to lattice QCD.We note our results
on the Aoki phase diagram exhibit no diseases due to a discrete symmetry breaking, which is
consistent withtheresultsinthelatticeperturbation in[1,2].
References
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[2] C.Hoelbling,Phys.Lett.B696,422(2011)[arXiv:1009.5362].
[3] M.Creutz,T.KimuraandT.Misumi,Phys.Rev.D83,094506(2011)[arXiv:1101.4239].
[4] P.deForcrand,A.KurkelaandM.Panero,PoSLattice2010(2011)080,[arXiv:1102.1000].
[5] S.Aoki,Phys.Rev.D30,2653(1984);S.AokiandK.Higashijima,Prog.Theor.Phys.76,521.
[6] M.Creutz,T.KimuraandT.Misumi,JHEP1012,041(2010)[arXiv:1011.0761].
[7] T.Z.Nakano,T.Misumi,T.KimuraandA.Ohnishi,workinprogress.
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