Individual eigenvalue distributions for chGSE-chGUE crossover and determination of low-energy constants in two-color QCD+QED

1.7 MB·English

by  Shinsuke M. Nishigaki

Individual eigenvalue distributions for
chGSE-chGUE crossover and determination of
low-energy constants in two-color QCD+QED
5
1
Shinsuke M. Nishigaki∗†
0
2 GraduateSchoolofScienceandEngineering

r ShimaneUniversity,Matsue690-8504,Japan
p
E-mail: [email protected]
A
1 Takuya Yamamoto
GraduateSchoolofScienceandEngineering
]
t ShimaneUniversity,Matsue690-8504,Japan
a
l
-
p We compute statistical distributions of individual low-lying eigenvalues of random matrix en-
e
h semblesinterpolatingchiralGaussiansymplecticandunitaryensembles. Tothisaimweusethe
[ Nyström-type discretization of Fredholm Pfaffians and resolvents of the dynamical Bessel ker-

2 nelcontainingasinglecrossoverparameterρ. Theρ-dependentdistributionsofthefoursmallest
v
eigenvaluesarethenusedtofittheDiracspectraofmodulatedSU(2)latticegaugetheory,inwhich
8
0 therealityofthestaggeredSU(2)DiracoperatorisweaklyviolatedeitherbytheU(1)gaugefield
5
orbyaconstantbackgroundflux.Combineduseofindividualeigenvaluedistributionsiseffective
7
0 in reducing statistical errors in ρ; its linear dependence on the imaginary chemical potential µ
I
.
1 enables precise determination of the pseudo-scalar decay constant F of the SU(2) gauge theory
0
fromasmalllattice. TheU(1)-couplingdependenceofanequivalentofF2µ2intheSU(2)×U(1)
5 I
1 theoryisalsoobtained.
:
v
i
X
r
a
The32ndInternationalSymposiumonLatticeFieldTheory
23-28June,2014
ColumbiaUniversityNewYork,NY
∗Speaker.
†SMNissupportedinpartbyJSPSGrants-in-AidforScientificResearch(KAKENHI)No.25400259.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
IndividualeigenvaluedistributionsforchGSE-chGUEcrossover ShinsukeM.Nishigaki
1. Introduction
Wigner-Dyson universality of the local correlation of energy levels among various stochastic
[1] and quantum-chaotic systems [2] under well-defined conditions was established through the
ten-foldclassificationofsymmetricspacesofspectralσ models[3],towhichtheGutzwillertrace
formulaalsoreduces[4]. Thisuniversalityhasinturnprovidedasolidandsecuregroundonwhich
system-specific information can be decoded by measuring deviations of spectral correlation func-
tionsfromtheiruniversalforms,ortransitionbetweentwouniversalityclasses. Primeexamplesof
theformeraretheweaklocalizationcorrectioninAndersonHamiltonians[5]andthenonuniversal
effectofshortperiodicorbits(smallprimes)inchaoticsystems(intheRiemannζ zeroes[6]).
Study on the latter “universality crossover", initiated by Dyson [7], has also come to encom-
pass a variety of settings, an example being the GUE-GOE transition that appears in a disordered
ring [8] and chaotic systems [9] both under magnetic fields. Recent years saw applications of the
universality crossover in lattice QCD, in an effort to explore the effects of the isospin chemical
potential[10]andofthefinitelatticespacingintheWilsonDiracoperator[11]. Thesestudieshave
revealedthepowerofthespectralapproachindeterminingthepiondecayconstantandtheWilso-
nian chPT constants from relatively small lattices. The aim of this work is to apply this approach
to the determination of low-energy constants in another setting, namely the two-color QCD sub-
jected under the imaginary chemical potential [12] or coupled to QED. Our novelty is to employ
theindividualdistributionsofsmallDiraceigenvalues[13]insteadofn-levelcorrelationfunctions,
infittingthelatticedata. Practicaladvantagesofourmethodwillbemanifestedsubsequently.
2. chGSE-chGUEcrossover
LetAandBbeN/2×N(cid:48)/2quaternionmatrices,representedbycomplexN×N(cid:48) matricesas
3 (cid:16) (cid:17) 3 (cid:16) (cid:17)
A= ∑ A(µ) ⊗e , B= ∑ B(µ) ⊗e (j=1,...,N/2, k=1,...,N(cid:48)/2). (2.1)
jk µ jk µ
µ=0 µ=0
Hereasetoffour2×2matricese =(1 ,−i(cid:126)σ)spansthebasisofthequaternionfieldH. Letthe
µ 2
matrix elements belong to A(µ) ∈R and B(µ) ∈C, so that the matrix A is quaternion-real and B
jk jk
is not (i.e. a generic N×N(cid:48) complex matrix). We consider A(µ), ReB(µ), and ImB(µ) to be inde-
jk jk jk
pendentrandomvariablesdistributedaccordingtotheGaussiandistributionse−12trAA† ande−trBB†,
respectively,andintroduceanensembleof(N+N(cid:48))×(N+N(cid:48))HermitianmatricesH oftheform
(cid:32) (cid:33)
0 C (cid:112)
H = N×N , C=e−τA+ 1−e−2τB. (2.2)
C† 0
N(cid:48)×N(cid:48)
Herearealparameterτ playstheroleoffictitioustimefortheBrownianmotionoftheeigenvalues
[7]. Thisensembleenjoysthechiralsymmetry{H,γ }=0withγ =diag(1 ,−1 ),implyingthat
5 5 N N(cid:48)
thespectrumofH consistsofN ±pairsofnonzeroeigenvaluesandν =|N(cid:48)−N|zeroeigenvalues.
ThepresenceofBviolatesthequaternion-realityofCandtheselfdualityofH,liftingtheKramers
degeneracy of nonzero eigenvalues of H. Accordingly this ensemble interpolates the two limiting
cases,chiralGSEatτ =0andchiralGUEatτ →∞,dependingonasingleparameterτ.
2
IndividualeigenvaluedistributionsforchGSE-chGUEcrossover ShinsukeM.Nishigaki
WeconsiderthecaseinwhichtheKramersdegeneracyisweaklybrokenbyτ (cid:28)1. Thenthe
spectral density of H in the large-N limit is identical to that of the chGSE (τ =0), i.e. Wigner’s
√
semi-circleρ¯(λ)= 4N−λ2/π. Wemagnifythevicinityoftheoriginoftheλ axisbyintroducing
√
unfoldedvariablesx =λ/∆ with∆ =1/ρ¯(0)=π/ 4N. Inordertorealizeanontrivialcrossover
i i
behavior,wetakethetriple-scalinglimitN,N(cid:48)→∞,λ →0,τ →0whilekeepingthecombinations
√ i
ρ = τ/∆,ν =N(cid:48)−N(≥0),andx fixedfinite. Thenthej.p.d.ofN positiveunfoldedeigenvalues
i
P (x ,...,x )isexpressedasaPfaffianofthedynamicalBesselkernelK(x,y)[14],
N 1 N
(cid:34) (cid:35) (cid:34) (cid:35)
(cid:16) (cid:17) S(x,y) I(x,y) 0 1
P (x ,...,x )=Pf Z[K(x,x )]N , K(x,y)= , Z= ⊗1, (2.3)
N 1 N i j i,j=1 D(x,y) S(y,x) −1 0
√ (cid:26)J (πx)yJ (πy)−xJ (πx)J (πy) J (πx)(cid:90) π (cid:27)
S(x,y)=π xy ν ν−1 ν−1 ν − ν dυeρ2(υ2−π2)J (υy) ,
x2−y2 2 ν
0
√
xy(cid:90) π (cid:90) 1
D(x,y)= dυυ dueρ2υ2(1+u2){J (υux)J (υy)−J (υx)J (υuy)},
ν ν ν ν
2
√ 0 0
xy(cid:90) ∞
I(x,y)= dυυ2e−2ρ2υ2{J (υx)yJ (υy)−xJ (υx)J (υy)}.
ν ν−1 ν−1 ν
2
π
(cid:16) (cid:17) (cid:16) (cid:17)
Duetotherecursionrelation(cid:82)∞dx Pf Z[K(x,x )]k =(N−k+1)Pf Z[K(x,x )]k−1 ,cor-
0 k i j i,j=1 i j i,j=1
relationfunctionsofneigenvaluesaregivenby
N! (cid:90) ∞ (cid:16) (cid:17)
R (x ,...,x )= dx ...dx P (x ,...,x )=Pf Z[K(x,x )]n . (2.4)
n 1 n (N−n)! n+1 N N 1 N i j i,j=1
0
3. Individualeigenvaluedistributions
ThePfaffianformsin(2.3)∼(2.4)originatefromquaterniondeterminants(Tdet)composedof
a quaternionic kernel, [K (x,x )] , whose C-number representative is the antisymmetric matrix
i j i,j
Z[K(x,x )] . Accordingly, the probability E (s) for an interval [0,s] to contain exactly k eigen-
i j i,j k
values is also given in terms of the Fredholm Tdet of a quaternionic integral operator Kˆ, i.e. the
s
squarerootofthecorrespondingFredholmdeterminantofKˆ (i.e.FredholmPfaffianofZKˆ ),
s s
1 (cid:12)
E (s)= (−∂ )k Det(1−ξKˆ )1/2(cid:12) . (3.1)
k k! ξ s (cid:12)ξ=1
Here Kˆ denotes an integral operator with the dynamical Bessel kernel K(x,y) (2.3) acting on the
s
spaceoftwo-componentL2-functionsovertheinterval[0,s]. FirstfewE (s)’sareexpressedas
k
T 1 (cid:18)T2 T (cid:19)
E (s)=Det(1−Kˆ )1/2, E (s)=E (s) 1, E (s)=E (s) 1 − 2 , (3.2)
0 s 1 0 2 0
2 2! 4 2
1 (cid:18)T3 3 (cid:19) 1 (cid:18)T4 3 3 (cid:19)
E (s)=E (s) 1 − T T +T , E (s)=E (s) 1 − T2T + T2+2T T −3T ,
3 0 3! 8 4 1 2 3 4 0 4! 16 4 1 2 4 2 1 3 4
where T (s) = Tr(cid:0)Kˆ (I−Kˆ )−1(cid:1)n denote functional traces of the resolvents of Kˆ . Probability
n s s s
distribution pk(s) of the kth smallest positive eigenvalue is then given as pk(s)=−∂s∑(cid:96)k=−01E(cid:96)(s).
An efficient way of numerically evaluating the Fredholm determinant of a trace-class operator Kˆ
s
actingonL2-functionsoveraninterval[0,s]istheNyström-typediscretization[15]
Det(1−Kˆ )(cid:39)det(I−K ), K =(cid:2)K(x,x )√w w (cid:3)m . (3.3)
s s s i j i j i,j=1
3
IndividualeigenvaluedistributionsforchGSE-chGUEcrossover ShinsukeM.Nishigaki
Hereweemployaquadratureruleconsistingofasetofpoints{x}takenfromtheinterval[0,s]and
i
associated weights {wi} such that (cid:82)0s f(x)dx(cid:39)∑mi=1 f(xi)wi. Similarly, the resolvents in (3.2) are
approximatedasT (s)(cid:39)tr(cid:0)K (I−K )−1(cid:1)n.ForapracticalpurposewechoosetheGaussquadra-
n s s
ture rule, i.e. sampling {x} from the nodes of Legendre polynomials normalized to [0,s]. Previ-
i
ously we applied the Nyström-type method to the dynamical Bessel kernels interpolating chGSE-
chGUE (2.3) and chGOE-chGUE and evaluated the smallest eigenvalue distributions p (s) [12].
1
In this work we extend our computation to the first four eigenvalues, aiming to reduce the fitting
errors in determining the low-energy constants. We set the approximation order m to be at least
100,andconfirmedthestabilityoftheresultsforincreasingm(upto200∼400).
Thedistributions p (s),···,p (s)fortheν =0case,computedfrom(2.3)∼(3.3)forρ ≤0.70
1 4
are exhibited in Fig. 1L. A practical advantage of using individual eigenvalue distributions over
the spectral density R1(x)=∑∞k=1pk(x)=S(x,x)(Fig. 1R) for fitting the lattice data is clear from
the figure: the oscillation of the latter immediately becomes structureless and insensitive to the
interpolationparameterρ duetotheoverlappingofmultiplepeaksoftheformer,whereasthequasi-
Gaussian shape of each peak is clearly distinguishable and is extremely sensitive to ρ. Another
advantage specific to the current case originates from the fact that p (s) and p (s) move in
2k−1 2k
oppositedirectionsasρ isincreasedtobreaktheKramersdegeneracy. Bycombiningthetwobest-
fittingvaluesofρ forthesetwodistributions,anyerrorpresentinthemeanlevelspacing∆ ofthe
Diracspectrum,whichwouldresultinshiftingtheunfoldeddataof2k−1thand2ktheigenvaluesto
thesamedirection,isexpectedtobecancelled. Wehaveconfirmedthisbygenerating105 samples
of crossover random matrix ensembles with N = N(cid:48) = 64 and various ρ ≤ 0.50 and by fitting
histograms of first four eigenvalues to the analytic results. Combined values of ρ from these four
fittingshavereproducedthetrueinputvalueswithinafewpermilofsystematicerror(max.0.5%),
anorderofmagnitudeclosertotheinputvaluesthanusinganysingleindividualdistribution. Such
anaccuracycouldneitherbehopedforhadweusedthespectraldensityR (x)forfitting.
1
4. Effectivetheoryandlow-energyconstants
TheDiracoperatorD/ofaQCD-liketheorywithquarksinapseudoreal(real)representation,
1.2
1.5
1.0
0.8
(cid:72)(cid:76)ps1,2,3,4 0.6 (cid:72)(cid:76)Rx1 1.0
0.4 0.5
0.2
0.0 0.0
0 1 2 3 4 5 0 1 2 3 4 5
s x
Figure1:Firstfoureigenvaluedistributions p (s)(left)for0.04≤ρ≤0.70(step0.01,purpletored)and
1∼4
thespectraldensityR (x)for0.01≤ρ≤1.00(step0.01)forthechGSE(black)tochGUE(grey)crossover.
1
4
IndividualeigenvaluedistributionsforchGSE-chGUEcrossover ShinsukeM.Nishigaki
suchasthefundamentalofSp(2N)(SO(N)),possessesanantiunitarysymmetryunlikeQCDwith
quarks in a complex representation [16]: D/ commutes with CZ∗ (C∗), with C being the charge
conjugationand∗thecomplexconjugation. As(CZ∗)2=+1((C∗)2=−1),Dcanbebroughtto
arealsymmetric(quaternionselfdual)matrixbyasimilaritytransformation. Duetothisproperty,
the distinction between left-handed quarks and conjugated right-handed quarks is lost, leading to
the Pauli-Gürsey extension of the flavor symmetry from SU(N ) ×SU(N ) to SU(2N )=:G
F L F R F
anditsvectorsubgroupfromSU(N ) toSp(2N )orSO(2N )=:H. Accordinglyitslow-energy
F V F F
effectivetheorybecomesanonlinearσ modelonanexoticNambu-GoldstonemanifoldG/H.
Since the Dirac operator charged under the U(1) gauge field is complex, coupling QCD-like
theories with electromagnetism or even subjecting them to the constant U(1) background breaks
the antiunitary symmetry of D/ and the Pauli-Gürsey extended flavor symmetry. In the latter case
thatisequivalenttoputtingonaweakimaginarychemicalpotential µ =iµ ,itseffectonthelow-
I
energy Lagrangian is systematically incorporated by the flavor covariantization of the derivatives
[17]. Furthermore,ifthetheoryisinafinitevolumeV =L4 andtheThoulessenergyE (cid:39)F2/ΣL2
c
ismuchlargerthanm,thepathintegralisdominatedbythezero-modeintegration(theε regime),
(cid:90) (cid:18)1 (cid:19)
Z= dU exp VΣmRetrMˆU−Vµ2F2tr(BˆU†BˆU+BˆBˆ) . (4.1)
2 I
SU(2NF)
HereU isanSU(2N )matrix-valuedNambu-Goldstonefield,Bˆ=σ ⊗1 ,Mˆ =iσ ⊗1 (σ ⊗1 )
F 3 NF 2 NF 1 NF
for quarks in a pseudoreal (real) representation. Σ=(cid:104)ψ¯ψ(cid:105)/N denotes the chiral condensate and
F
F thepseudo-scalardecayconstant,bothmeasuredinthechiralandzero-chemicalpotentiallimit.
Note that the above 0D σ model for the case of fermions in a real representation can as well be
derived from the random matrix ensemble (2.2) through the standard procedure: (i) introduce N
F
species of complex Grassmannian (N+N(cid:48))-vectors ψ ,ψ¯ and consider a replicated spectral de-
f f
(cid:68) (cid:69)
terminant (cid:10)det(λ−H)NF(cid:11)= (cid:82) dψdψ¯ e∑fψ¯f(λ−H)ψf , where (cid:104)···(cid:105) denotes averaging over A and
B, (ii) perform Gaussian integrations over A and B, (iii) introduce a 2N ×2N -matrix valued
F F
Hubbard-StratonovichvariableQandopenupthe4-fermiterm,(iv)performGaussianintegrations
overψ andψ¯,(v)taketheaforementionedtriple-scalinglimitanddenotetheangularpartofQ(not
fixedby thelarge-N saddlepoint equation)asU. Then thecoefficients ofthe massand chemical-
potential terms are identified as VΣm = iπx and 2VF2µ2 = π2ρ2. By substituting m → iλ
I Dirac
which turns the QCD partition function into the Dirac spectral determinant, the former equality
provides the definition of unfolded Dirac eigenvalues x=λ /∆ due to the Banks-Casher rela-
Dirac
tionΣ=π/∆V. ThelatterequalityisusedtodetermineF2 fromtheslopeoftheµ -ρ plot.
I
5. FittingDiracspectraofSU(2)×U(1)gaugetheory
Astheaimofthisworkistodemonstratethevalidityandadvantageofthemethodandnotto
approachthecontinuum,chiral,orthermodynamiclimit,wechosethesimplestpossiblesettingon
the lattice side: (i) generate 104 samples of quenched SU(2)=Sp(2) gauge fieldsU (x) on an (in-
µ
tentionally)smalllatticeV =64,withaplaquetteactionatβ =6/g2 =0∼1.75(step.25),
SU(2) SU(2)
usingthestandardheat-bath/overrelaxationalgorithm. (ii-a)multiplytheSU(2)fieldsontemporal
links U0(x) by a constant phase eiµI with µI =0.00524 ∼.05240 (step .00524), or (ii-b) gener-
atequenchednoncompactU(1)gaugefieldsA (x)undertheCoulombgauge-fixingcondition[18]
µ
5
IndividualeigenvaluedistributionsforchGSE-chGUEcrossover ShinsukeM.Nishigaki
andmultiplytheSU(2)fieldsU (x)byexp(ie A (x)),withe =0.0004∼.0024(step.0004),
µ U(1) µ U(1)
(iii)substitutethegaugefieldsintoanunimprovedstaggeredDiracoperatoranddiagonalize.
Due to the absence of the C matrix, the antiunitary symmetries of staggered Dirac opera-
tors are swapped between real and pseudoreal representations [19]. Accordingly, our case with
SU(2)×U(1)fundamentalfermionsindeedcorrespondstothechGSE-chGUEcrossover(2.2). The
low-energyconstantsaredeterminedbythefollowingsteps: (I)fitthehistogramofeachofthetwo
smallest Dirac eigenvalues (i.e. four counting the Kramers degeneracy) of the pure SU(2) case to
therescaledchGSE(ρ =0)prediction p (λ /∆)/∆ byvarying∆,(II)combinetwooptimalvalues
k k
of ∆ and their variances to determine ∆¯ and thus Σ=π/∆¯V, (III) fit the histogram of each of the
four smallest unfolded Dirac eigenvalues x =λ /∆¯ of (a) SU(2)+µ or (b) SU(2)×U(1) case to
k k I
thechGSE-chGUEprediction p (x )byvaryingρ,(IV)combinefouroptimalvaluesofρ andtheir
k k
variancestodetermineρ¯ andthusF2µ2=(π2/2)ρ¯2/V.
I
Wefirstobservethatthefourvaluesof∆ obtainedinthestep(I)aremutuallyconsistent,giving
risetocombinedrelativeerrorsinΣthatareextremelysmall,∼0.1%(Table1,top). One-parameter
fittingsinthesteps(I),(III-a),or(III-b)arequitesatisfactory,withχ2/dof=0.5∼1.5forallrange
of parameters in concern (exemplified in Fig. 2, above). We also confirmed our expectation that
the best-fitting values of ρ for k =1,3 and those for k =2,4 have a tendency to counter-move,
in favor of cancelling the unfolding ambiguity due to a tiny error within ∆. Relative errors in ρ¯
are considerably reduced by the combined use of four individual eigenvalue distributions (Fig. 2
below), and are no larger than ±.018(stat)±.005(sys). Linear response of ρ¯ on µ or e is
I U(1)
confirmedfortheSU(2)+µ case(Fig.3,left),andthepseudo-scalardecayconstantF2 atvarious
I
values of β is obtained from the slopes (Table 1, middle). For the SU(2)×U(1) case, the
SU(2)
coefficients (equivalent of F2µ2) of the trBˆU†BˆU term in (4.1) divided by e2 , extrapolated to
I U(1)
e →0aresummarizedinTable1,bottom. Completelatticeresults,anddetailsofanalyticand
U(1)
numericalcomputationspresentedin§2and§3willbereportedinasubsequentpublication.
β 0 0.25 0.50 0.75 1.00 1.25 1.50 1.75
SU(2)
Σa3 1.310(2) 1.255(2) 1.199(1) 1.139(1) 1.070(1) .987(1) .883(1) .743(1)
F2a2 .284(2) .268(2) .247(2) .226(2) .205(1) .178(1) .153(1) .115(1)
F2µ2a4/e2 220(2) 198(2) 186(2) 163(1) 145(1) 123(1) 99.5(8) 68.0(6)
I U(1)
Table1: ChiralcondensateΣfromquenchedSU(2)[top],pseudo-scalardecayconstantF2 fromSU(2)+µ
I
[middle],andanequivalentofF2µ2(dividedbye2 )fromSU(2)×U(1)[bottom],allinthelatticeunit.
I U(1)
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IndividualeigenvaluedistributionsforchGSE-chGUEcrossover ShinsukeM.Nishigaki
1.2
1.0
0.8 1st 1st 1st
psk0.6 !" 23nrdd 23nrdd 23nrdd
0.4 4th 4th 4th
Full Full Full
0.2
0.0
0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5
s s s
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0.190 0.375
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Ρ 00..118805 !! !! !! !! 00..227705 !! !! !! !! 00..336605 !! !!
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1 2 3 4 1 2 3 4 1 2 3 4
k k k
Figure2: Histogramsofkth unfoldedDiraceigenvaluesforβ =0.25ande =0.0008,.0012,.0016
SU(2) U(1)
[above,lefttoright]andbest-fitting p (s)(k=1∼4)fromthechGSE-chGUEcrossover. Theρ parameter
k
determinedforeachk,theircombinedvalues[dots]andstatisticalerrors[band]areshownbeloweachgraph.
8.20 234
8.15 232
!
8.10 ! 230 !
ΡΜI 8.05! ! ! ! ! ! Ρe 228! ! ! ! ! !
8.00 !
226
7.95 !
224
7.90
0.01 0.02 0.03 0.04 0.05 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025
ΜI e
Figure 3: Ratios ρ¯/µ for SU(2)+µ at β =0.5 [left] and ρ¯/e for SU(2)×U(1) at β =0.25
I I SU(2) U(1) SU(2)
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7