Table Of ContentExact results for two-color QCD at low and high
density∗
1
Takuya Kanazawa
1
0 DepartmentofPhysics,TheUniversityofTokyo,Tokyo113-0033,Japan
2 E-mail: [email protected]
n
a Tilo Wettig†
J
DepartmentofPhysics,UniversityofRegensburg,93040Regensburg,Germany
3
E-mail: [email protected]
]
t Naoki Yamamoto
a
l InstituteforNuclearTheory,UniversityofWashington,Seattle,WA98195-1550,USA
-
p E-mail: [email protected]
e
h
[ Wediscussarandommatrixtheorythatwasoriginallyconstructedtodescribetwo-colorQCDat
1 lowdensityinthephasewithanonzerochiralcondensate.Withaparticularchoiceofaparameter,
v
thesamerandommatrixtheoryalsodescribesthehigh-densityphaseoftwo-colorQCD.Inthis
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8 phaseaBCSsuperfluidofdiquarkpairsisformed, andthepatternofchiralsymmetrybreaking
5
isverydifferentfromthatatlowdensity. Analyticalresultsforthespectraldensityobtainedfrom
0
. thisrandommatrixtheoryallowfortheextractionoftheBCSgapfromlatticedata.
1
0
1
1
:
v
i
X
r
a
TheXXVIIIInternationalSymposiumonLatticeFieldTheory,Lattice2010
June14-19,2010
Villasimius,Italy
∗SupportedbytheGermanResearchFoundation(DFG)andbyJSPS.
†Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
Exactresultsfortwo-colorQCDatlowandhighdensity TiloWettig
1. Introduction
Lattice studies of QCD at nonzero quark chemical potential µ are hindered by the infamous
sign problem, see [1] for a review. Two-color QCD with an even number of pairwise degenerate
quarks does not have a sign problem and can therefore be simulated on the lattice [2]. It shares
many qualitative features, such as confinement and chiral symmetry breaking, with three-color
QCD, but the detailed features of both theories are rather different, such as the pattern of chiral
symmetry breaking, the particle spectrum, or the phase diagram. Nevertheless, two-color QCD is
an interesting theory in its own right. It has been studied in great detail at zero and low density,
see, e.g., [3]. In this contribution, we also address the region of high density in which the pattern
ofchiralsymmetrybreakingisdifferentfromthatatlowdensityandinwhichaBCSsuperfluidof
diquark pairs is expected to be formed because there is an attractive channel between quarks near
theFermisurface. Inearlierwork[4],wehavederivedthelow-energyeffectivechiralLagrangian
for µ (cid:29)Λ , identified the corresponding ε-regime, and derived Leutwyler-Smilga-type sum
SU(2)
rulesfortheeigenvaluesoftheDiracoperator. ThisworkhasbeensummarizedatLattice2009[5].
In the lowest order of the ε-regime, sometimes also called “microscopic domain”, the theory
becomeszero-dimensional. Thiszero-dimensionallimitofthetheorycanalternativelybedescribed
by a random matrix theory (RMT). Many examples of such an exact mapping are known, in par-
ticular for two- and three-color QCD at zero and low density, see [6, 7] for reviews. Therefore
thenaturalquestioniswhatrandommatrixtheorydescribesthemicroscopicdomainoftwo-color
QCDathighdensity. Theanswertothisquestionwasgivenin[8]andwillbereviewedinSec.2.
Theadvantageofhavingarandommatrixtheoryisthatitallowsustocomputealargenumberof
analyticalresults characterizingtheDirac eigenvalues, seeSec.3. Thistask wouldbemuch more
difficultintheeffectivetheory. TheanalyticalresultsathighdensitycontaintheBCSgap∆,which
wascomputedforasymptoticallyhighdensityinaweak-couplingapproachin[9,10],asaparam-
eter. Therefore,∆canbeextractedfromlatticedatafortheDiraceigenvaluesbymatchingthemto
theanalyticalresultsfromrandommatrixtheory. Anotherinterestingfeatureoftwo-colorQCDis
thatitallowsustostudythesignproblem,eitherforanoddnumberN offlavorsbyturningonµ,
f
orforevenN bydetuningthequarkmassesfromtheirdegeneratevalues,seeSec.4.
f
2. Randommatrixtheoryatlowandhighdensity
Thepertinentrandommatrixtheoryfortwo-colorQCDatlowdensityhasbeenformulatedin
[11],withpartitionfunction
(cid:32) (cid:33)
(cid:90) Nf m P+µQ
ZRMT(µ)= dPdQe−12tr(PPT+QQT)∏det −PT +fµQT m , (2.1)
f=1 f
where the m are the quark masses, P and Q are real matrices of dimension N×(N+ν), dP and
f
dQ are Cartesian integration measures, N is assumed to be proportional to the Euclidean space-
time volumeV , and ν can be identified with the topological charge. Note that sometimes other
4
conventions for the width of the Gaussian distribution of P and Q are used in the literature. The
RMTDiracoperatorD(µ)isthematrixin(2.1)withthemasssettozero.
2
Exactresultsfortwo-colorQCDatlowandhighdensity TiloWettig
Itwasshownin[8]thatintheN →∞limitthisRMTpartitionfunctionisidenticaltothepar-
tition function obtained from the static (or zero-dimensional) effective Lagrangian for two-color
QCD at low density1 given in [3]. More precisely, the two partition functions have the same de-
pendenceonthequarkmassesandonthechemicalpotential. Themappingbetweendimensionless
RMTquantitiesandphysicalquantitiesisgivenby
√
Nm=m GV =m2F2V , (2.2a)
phys 4 π 4
1
Nµ2=µ2 F2V , (2.2b)
2 phys 4
whereGandF arelow-energyconstantsintheeffectiveLagrangianinthenotationof[3].
Both the random matrix theory (2.1) and the corresponding effective Lagrangian explicitly
depend on the chemical potential µ. In contrast, the effective Lagrangian at high density derived
in [4] does not explicitly depend on µ. It only depends on the quark masses, which appear in the
combinationm2∆2V . Ahintastowhatthecorrectrandommatrixtheoryathighdensityshouldbe
4
can be obtained by noting that (2.1) is basically symmetric under µ →1/µ (except that real and
imaginary parts are interchanged). Maximum non-Hermiticity, which is expected at high density,
corresponds to µ =1. We therefore conjecture that at high density the random matrix theory is,
afteraredefinitionoftherandommatrices,givenby
(cid:32) (cid:33)
(cid:90) Nf m A
ZRMT= dAdBe−14tr(AAT+BBT)∏det BTf m , (2.3)
f=1 f
where the dimension of A and B is again N×(N+ν).2 In the high-density phase, we restrict
ourselvestoanevennumberofflavors.
Letusfirstcheckthatweobtainthecorrectpatternofchiralsymmetrybreaking. Tothisend,
werewritetheN -flavordeterminantresultingfrom(2.3)inthechirallimitintheform
f
(cid:32) (cid:33) (cid:32) (cid:33) (cid:32) (cid:33)
0 A 0 A 0 B
detNf =detNf/2 detNf/2 . (2.4)
BT 0 −AT 0 −BT 0
ThematricesinthetwofactorsontheRHSofthisequationhavetheformofthechiralorthogonal
ensembleofrandommatrixtheory. Itwasshownin[12]thatthesymmetrybreakingpatterninthat
ensemble with N /2 flavors is U(N )→Sp(N ). Since we have two such factors, (2.3) with N
f f f f
flavors has the symmetry breaking pattern U(N )×U(N )→Sp(N )×Sp(N ). This agrees with
f f f f
thesymmetrybreakingpatternintheeffectivetheoryduetotheformationofadiquarkcondensate,
whichisgivenbySU(N ) ×SU(N ) ×U(1) ×U(1) →Sp(N ) ×Sp(N ) [4].
f L f R B A f L f R
We have also shown [8] that in the N →∞ limit the RMT partition function (2.3) is identical
tothepartitionfunctionofthehigh-densityeffectivetheoryinthezero-dimensionallimit,i.e.,the
twopartitionfunctionshavethesamemassdependence. Themappingbetweenthedimensionless
RMTmassandthephysicalmassisnowquitedifferentfrom(2.2a)andgivenby
√
3 √
m= m ∆ V . (2.5)
phys 4
π
1Bylowdensityweheremeantheregimeofweaknon-Hermiticity,seeSec.3forthedefinitionofthisregime.
2Onlythecaseν=0isphysicallyrelevantsincetopologyisstronglysuppressedathighdensity.
3
Exactresultsfortwo-colorQCDatlowandhighdensity TiloWettig
The arguments presented so far, while giving overwhelming evidence in favor of the equiva-
lence of the random matrix theory (2.3) and the effective theory at high density, do not constitute
a full proof. For such a proof one would have to show that all spectral correlation functions are
identical in both theories, which requires studying the partially quenched version of the theory.
Suchastudyhasnotbeendoneyet,butwehavenodoubtthattheoutcomewouldbepositive.
3. Exactresultsfromrandommatrixtheory
Wecannowproceedtocomputespectralcorrelationfunctionsfromtherandommatrixtheory
in the N → ∞ limit. At µ = 0 the RMT eigenvalues λ are purely imaginary, while at µ (cid:54)= 0
they are either purely real, purely imaginary, or come in complex conjugate pairs. We are mainly
interestedintheso-calledmicroscopicspectraldensityofthesmalleigenvalues,i.e.,werescaleall
eigenvalues by a quantity δ that is, up to a numerical prefactor, equal to the mean level spacing
near zero. This results in complex numbers z=λ/δ of order O(1). To see an effect of the quark
massesonthesmalleigenvaluesweneedtorescaletheminthesameway,resultinginmˆ =m /δ.
f f
Therandommatrixtheorycanbesolvedintwodifferentregimes:
• In the regime of weak non-Hermiticity, the combination µˆ2 =2Nµ2 =4µ2 F2V is kept
phys 4
fixedinthelimitN →∞. Whilethisregimemightappeartobemainlyofacademicinterest
since µ →0inthethermodynamiclimit,ithasanimportantphenomenologicalapplication,
i.e.,theextractionofthelow-energyconstantsGandF fromlatticedata. Inourconventions
√ √ √
wehaveδ =1/2 N inthisregime,i.e.,z=2 Nλ andmˆ =2 Nm.
• Intheregimeofstrongnon-Hermiticity,µ iskeptnonzerointhelimitN→∞. Theanalytical
resultsinthisregimearetheµˆ →∞limitsofthecorrespondingweaknon-Hermiticityresults.
Theirfunctionalformisidenticalforall0<µ≤1,andtheµ-dependenceonlyentersthrough
a rescaling of the eigenvalues. In our conventions we have δ =1 in this regime so that no
N-dependentrescalingoftheeigenvaluesandthemassesisnecessary.
In [13] analytical results for the microscopic spectral “density” (which we put in quotation
markssincethisquantitycanbecomenegativeifthereisasignproblem)wereobtainedinbothof
the above-mentioned regimes in the quenched case, i.e., for N =0 flavors. In the meantime, the
f
generalization to the unquenched case has been worked out [14]. Since the analytical results are
rather cumbersome we will not present them here. Important features of the results are exhibited
inFigs.1through4. Commentsonthesefeaturesaregiveninthefigurecaptions.
4. Thesignproblem
Asagoodmeasureofthesignproblemintwo-colorQCDwedefinethequantity
(cid:68) (cid:69)
sgndet(D+m)∏Nf=f 1|det(D+mf)|
(cid:104)sgndet(D+m)(cid:105) ≡ Nf=0, (4.1)
||Nf|| (cid:68)∏Nf=f 1|det(D+mf)|(cid:69)
Nf=0
see[14]foramoredetaileddiscussion.
4
Exactresultsfortwo-colorQCDatlowandhighdensity TiloWettig
0.08 0.05
ρwIm(y) ρwRe(x)
0.06 0
0.04 mˆ=0.1 −0.05 mˆ=12
mˆ=2 mˆ=15
0.02 mˆ=3 mˆ=20
−0.1
quenched quenched
0
0 2 4 6 8 y 10 0 5 10 15 20 25 x 30
Figure1: Microscopicspectral“density”ofthepurelyimaginary(left)andpurelyreal(right)eigenvalues
intheregimeofweaknon-HermiticityforN =1, µˆ =3,ν =0,anddifferentvaluesofmˆ. Thedensityof
f
therealeigenvaluesgoesthroughzeroforx=mˆ .
f
ρC(z)
w
Imz
ρC(z)
w
Imz
Rez
Rez
Figure2:Microscopicspectral“density”ofthecomplexeigenvaluesintheregimeofweaknon-Hermiticity
for N =1 and ν =0. Left: µˆ =1.8 and mˆ =0. The massless quark causes a depletion of the density
f
neartheorigin. Right: µˆ =6andmˆ =20. Forlarge µˆ thereisanellipticaldomaininwhichthe“density”
oscillatesstrongly.
0.4 0.4
ρs(z) ρs(z)
0.2 0.2
Im Im
Re Re
0 0
0 1 2 3 4 z 5 0 1 2 3 4 z 5
Figure3: Microscopicspectral“density”ofthepurelyimaginary(solid)andpurelyreal(dashed)eigenval-
uesintheregimeofstrongnon-HermiticityforN =2,mˆ =2,mˆ =3,ν =0(left),andν =2(right). The
f 1 2
densityoftherealeigenvaluesgoesthroughzeroforx=mˆ .
f
ρC(z) ρC(z)
s s
Imz Imz
Rez
Rez
Figure4:Microscopicspectral“density”ofthecomplexeigenvaluesintheregimeofstrongnon-Hermiticity
forN =2andν=0.Left:mˆ =mˆ =2.Degeneratemassesgenerateadipinthespectrumatz=mˆ.Right:
f 1 2
mˆ =2,mˆ =8. Unequalmassesresultinadomainofstrongoscillations,indicativeofthesignproblem.
1 2
5
Exactresultsfortwo-colorQCDatlowandhighdensity TiloWettig
1 mˆ=0 1 ν=0
mˆ=6 ν=10
(cid:104)sgn(cid:105) mˆ=15 (cid:104)sgn(cid:105) ν=20
0.5 0.5
0 0
0 1 2 3 4 5 µˆ 0 1 2 3 4 5 µˆ
Figure5: Averagesignatweaknon-HermiticityforN =1asafunctionofµˆ. Left: ν=0iskeptfixedand
f
mˆ isvaried. Right: mˆ =0iskeptfixedandν isvaried.
1 ν=0
ν=1
(cid:104)sgn(cid:105) ν=2
0.5
0
1 2 3 4 5 6 mˆ2
Figure6: Averagesignatstrongnon-HermiticityforN =2,mˆ =1,andν =0,1,2asafunctionofmˆ .
f 1 2
We first consider the regime of weak non-Hermiticity. We choose N =1 and turn on µˆ to
f
study its effect on the average sign, see Fig. 5. It is evident from the plots that the sign problem
(i)increaseswithincreasingµˆ,(ii)decreaseswithincreasingmˆ,and(iii)decreaseswithincreasing
ν (in agreement with [15]). A quantitative analysis [14] reveals that in the thermodynamic limit
(cid:112)
the average sign makes a first-order transition from 1 to 0 at µˆ = mˆ/2, which in physical units
correspondstoacriticalchemicalpotentialµ =m /2.
phys π
Next we consider the regime of strong non-Hermiticity. We choose N =2 and detune the
f
quark masses. The effect on the average sign is shown in Fig. 6. The sign problem is absent for
mˆ =mˆ andincreasesas|mˆ −mˆ |increases. Itagaindecreaseswithincreasingν.
1 2 1 2
5. Conclusions
We have shown that a single random matrix theory describes two-color QCD at low density
in the regime of weak-Hermiticity and at high density in the BCS superfluid phase, depending on
the choice of the RMT parameter µ and on the rescaling factors in (2.2) and (2.5). These two
regimes have very different symmetry breaking patterns. It would be interesting to investigate
theapplicabilityofrandommatrixtheoryintheregionofintermediatedensities,whereintriguing
phenomenasuchasaBEC-BCScrossoverhavebeenconjectured[16].
The analytical RMT results can be used to extract physical parameters such as ∆ from lat-
tice data. Two-color lattice simulations with adjoint staggered fermions, which are in the same
symmetryclassascontinuumfundamentalfermions[17],arecurrentlyunderwaytotestourRMT
predictions.
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Exactresultsfortwo-colorQCDatlowandhighdensity TiloWettig
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