Simulating the scalar field on the fuzzy sphere

0.25 MB·English

by  Fernando Garcia Flores

Simulating the scalar field on the fuzzy sphere
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n
a FernandoGarcíaFlores∗ o
J DublinInstituteofAdvancedStudies,CentrodeInvestigaciónyEstudiosAvanzados
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9 E-mail: [email protected]
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1
DenjoeO’Connor
v L
2 DublinInstituteofAdvancedStudies
1 E-mail: [email protected] A
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T
1 X. Martin
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LMPT,UniversitéF.RabelaisdeTours 2
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0 E-mail: [email protected] 0
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t
a 0
l Thepropertiesofthef 4 scalarfieldtheoryonafuzzyspherearestudiednumerically. Thefuzzy
- 5
p
sphereisadiscretizationofthespherethroughmatricesinwhichthesymmetriesofthespaceare
e )
h preserved. Thismodelpresentsthreedifferentphases: uniformanddisorderedphases,asinthe
: 2
v usualcommutativescalarfieldtheory,andanon-uniformorderedphaserelatedtoUV-IRmixing
i
X like non-commutativeeffects. We have determinedthe coexistence lines between phases, their 6
r triplepointandtheirscaling. 2
a
XXIIIrdInternationalSymposiumonLatticeFieldTheory
25-30July2005
TrinityCollege,Dublin,Ireland
Speaker.
∗
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
Simulatingthescalarfieldonthefuzzysphere FernandoGarcíaFlores
1. Introduction
The fuzzy approximation scheme [1] consists in approximating the algebra of functions on a
manifold withafinitedimensional matrixalgebra instead ofdiscretising theunderlying space asa
latticeapproximation does.
Here we report our results for a hermitian scalar field on the fuzzy sphere. We find the col-
lapsed phasediagramandinparticular wecalculate theuniform ordered/non-uniform ordered line
thatwasabsentin[2].
Thecurrentstudycouldberelativelyeasilyrepeatedforahermitianscalarfieldonotherfuzzy
spaces. ThesimplestextensionwouldbetofuzzyCPN. Somevariantsoftheschemecanbeapplied
to fuzzy versions of S3 and S4[3]. The study reveals that the non-uniform disordered phase lines
should correspond toapurematrixmodeltransition. P
As an approximation scheme, this “fuzzification” is well suited to numerical simulations of o
field theories [4]. As a test run, the first fuzzy approximation to be investigated should be the
S
simplestone,thatofthetwodimensionalsphereCP1=S2. Boththetwo–dimensionalcommutative
(
andMoyalplanescanbeviewedasthelimitsofafuzzysphereofinfiniteradius.
L
2. The two dimensional f 4 Model and itsfuzzy version A
T
Weareinterested inthemodel:
2
S[F ]=Tr aF †[L,[L,F ]]+bF 2+cF 4 , (2.1)
i i 0
(cid:2) (cid:3)
where F is a Hermitian matrix of size N, b and c are mass and coupling parameters respectively. 0
Li is the angular momentum generator in the N dimensional unitary irreducible representation of 5
SU(2). Sincearescaling ofF willallowustoseta=1,theentirephase diagram canbeexplored
)
byranging throughallrealvaluesofbandpositivevaluesofc. Theconventions of[2]area= 4p ,
N 2
b=arR2,c=al R2.
Theinfinite matrix limitofthe action canbe taken andcorresponds toarealscalar fieldf on 6
aroundsphereofradiusRandEuclideanaction 2
S[f ]= d2n f L2f +rR2f 2+l R2f 4 (2.2)
Z
S2 (cid:0) (cid:1)
whereL2=(cid:229) L2 andL aretheusualangularmomentumgenerators.
i=1,3 i i
The eigenvectors of [L,[L, ]] in (2.1) are the polarization tensors Yˆ (normalised so that
i i lm
·
4p Tr(Yˆ†Yˆ ) = 1) and it has eigenvalues l(l+1) with degeneracy 2l+1. This is precisely the
N lm lm
spectrum oftheLaplacianL2 onthecommutativespheretruncated atangular momentumN 1.
−
Thisparticularmodelwaschosenbecauseofitssimplicity. Thediagrammaticexpansionofthe
model (2.2) has only one divergent diagram, the tadpole diagram, is Borel resumable, and defines
thefieldtheoryentirely. Inthefuzzyversion,thetadpolesplitsintoplanarandnon-planartadpoles,
which are also the only diagrams that diverge in the infinite N limit. Their difference is finite and
nonlocal andisresponsible fortheUV/IRmixingphenomenaofthedisordered phase[6].
Even though the scalar field on either commutative or fuzzy spheres cannot have a phase
transition, sincetheyhavefinitevolumeorafinitenumberofdegreesoffreedom,phasetransitions
maybefoundwhenthematrixdimensionortheradiusofthespherebecomeinfinite.
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Simulatingthescalarfieldonthefuzzysphere FernandoGarcíaFlores
Thefuzzyspherecanberecognizedbyintroducingcoordinates(X ,X ,X )proportionaltothe
1 2 3
angular momentumoperators
2R
X = L.
i i
√N2 1
−
Theymustsatisfythealgebra
Q
X2+X2+X2=R21, [X,X ]=i e X
1 2 3 i j R ijk k
whereQ = 2R2 istheparameterofnon-commutativity, Ristheradiusofthesphereand1isthe
√N2 1
unit operator. T−he non-commutativity parameter depends on the matrix size, N, and the radius of
thesphere, R. Bytakingdifferent limitswecanaccessdifferent spaces:
P
N R Q Limit
o
N constant =R 2R2/√N2 1 FuzzySphere
¥ constant =R 0 − CommutativeSphere S
¥ ¥ 0 Commutativeplane (
¥ ¥ constant=q MoyalPlane L
A
2.1 Orderparameters
T
Asuitable setoforderparameters canbeidentifiedfromthecoefficients ofamodedecompo-
2
sitionintermsofthepolarization tensorsbasis[5]
0
F =Tr(F )1 + 12 r L +N(cid:229) −1 (cid:229)+l c Yˆ . (2.3) 0
N N(N2 1) a a lm lm
− l=2m= l 5
−
Wehaveseparatedexplicitlyl=0andl=1fromtheexpansiontoidentifytwoobservableswhose )
expectation values we used to identify the respective phases. The observables are Tr(F ) and
2
| |
3
r 2:=r r = (cid:229) (Tr(L F ))2. ThetotalpowerinallcoefficientsisgivenbyTr(F 2)= 1(Tr(F ))2+ 6
a a a N
a=1
2
N 1 l
12 r 2+ N (cid:229) − (cid:229) c 2 and can beused to estimate the importance of theneglected higher
N(N2 1) 4p | lm|
− l=2m= l
modes. −
2.2 Thephases
This model (2.1) presents three phases. Asageneric illustration of their properties, Fig. 1(a)
andFig.1(b)showthedependenceonthemassparameterboftheprobabilitydistributionsofTr(F )
andr ,respectively, for a=1, c=40, N =4 .
{ }
Disordered: Found for b “small”, the configurations fluctuate close to F =0. This is con-
| |
firmedonthefigure,< Tr(F ) > <r > 0,butalso<Tr(F 2)> 0(notshown).
| | ∼ ∼ ∼
Non-uniform ordered: As b increases, the figure shows multiple symmetric peaks for the
| |
probability distribution of Tr(F ) whose height decreases with increasing Tr(F ), and multiple
| |
peaks not centered near zero for r . Furthermore Tr(F 2) is much larger than both <|Tr(F )|>2 and
N
<r >sothathighermodesactuallydominate. Inparticular,themostprobableconfigurationisnot
rotationallyinvariantandwehavespontaneousbreakdownofrotationalinvariance. Theprobability
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Simulatingthescalarfieldonthefuzzysphere FernandoGarcíaFlores
distributionofTr(F )hasN+1symmetricpeakslocatedapproximatelyat(N 2k) b/2cwhere
− −
k=0,1,...,N, while the probability distributions of r and S[F ] have (N+1)/2 ppeaks for N odd
andN/2+1forN even.
Uniformordered: As b becomeslarge, thefigureshowstwosymmetricpeaksfortheprob-
| |
ability distribution of Tr(F ) corresponding to the outer peaks of the non-uniform ordered phase
and located approximately at Tr(F ) N b/2c, but just one peak near zero for r . Further-
∼± −
more, <Tr(F 2)> <Tr(F )>2/N indicatping thatthepowerinhigher modesisnegligible. This
∼
isgenericandindicates thatF b/2c1andtherotational symmetryisthusrestored.
∼ −
p
N=4 , a= 1, b= -75.0, c= 40 N=4 , a= 1, b= -75.0, c= 40
P
PPrroobbaabbiilliittyy DDiissttrriibbuuttiioonn PPrroobbaabbiilliittyy DDiissttrriibbuuttiioonn
2.5 6 o
2 5
4
1.5 S
3
1 2
0.5 1 (
0 0
L
-100 -100
-90-8b0-70-60-50-40-30-20-10 -5 -4 -3 -2 -1 0 1Tr[ f 2 ] 3 4 5 -90-8b0-70-60-50-40-30-20-10 0 0.5 1 1.5 2 2.5 r 3 3.5 4 4.5 AT2
(a)ProbabilitydistributionofTr(F ) (b)Probabilitydistributionofr . 0
0
Figure1: Figures(a)and(b)showthetypicalbehavioroftheobservablesTr(F )andr inaregionofthe
5
phasediagramwheredecreasingbpassesthesystemthroughthethreephases.
)
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3. Simulationand Results: The specificheat andthe phase diagram 6
2
Weare interested inthephase diagram ofthe model (2.1). Toidentify it, weused the coordi-
nates inparameter space ofthepeaksofthespecific heat,C:=<S2 > <S>2. Therelevant set
−
of parameters is N,b,c whereb and cdepend implicitly inR. Itispossible tofurther reduce by
{ }
one the number ofparameters byfinding ascaling b,c bNqb,cNqc . Ifwefindsuch q b and
{ }→
q ,themodelbecomesindependent ofN andautomatically y(cid:8)ieldsaninfin(cid:9)itematrixlimit.
c
Thesimulations show that inthe non–uniform ordered phase, the fuzzy kinetic term (propor-
tional to a in (2.1)), is negligible compared to the potential term (the other terms). There exists
anexact solution forthecorresponding limitofa=0inthelarge N limitcalled the purepotential
model[7]. Thismodelpredictsathirdorderphasetransitionbetweenadisorderedandnon–uniform
orderedphaseatc=b2/4N. Figure2confirmsnumericallytheconvergenceofthedisordered/non–
uniform orderedtransition towardsthisexactcriticallineofthepurepotential model.
Numerically, it is not difficult to findthe coexistence curve between the uniform ordered and
disorderedphaseswhichexistforlowvaluesofc. Ontheotherhand,thecoexistencecurvebetween
the twoordered phases isdifficult to evaluate because itinvolves ajump inthe fieldconfiguration
andtunnelling overawidepotential barrier.
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Simulatingthescalarfieldonthefuzzysphere FernandoGarcíaFlores
0.28
N=2
N=4
N=6
N=8
0.26 a=0, NN==1 ¥0
0.24
2N
at)/
e
c H 0.22
cifi
e
p
S
(
0.2
P
0.18 c=200000.0
o
S
0.16
-5 -4 -3 -2 -1 0
(
b/(N c)1/2
Figure2: Plotofthespecificheatatthedisordered/non–uniformorderedtransitionforincreasingN andits L
N ¥ limit,theexactpurepotentialmodel. A
→
T
2
3
N=2
N=3 0
N=4
N=6
2.5 N=8 0
N=10
cN-2=(bN-3/2)2/4
5
Disorder phase Non-Uniform Order phase
2 )
2
2N 1.5 6
c/
2
1
0.5 Uniform Order phase
Triple point (0.80 – 0.08,0.15 – 0.05)
0
0 2 4 6 8 10 12 14
-b/N3/2
Figure3: PhasediagramobtainedfromMonte–Carlosimulationsofthemodel(2.1)
The phase diagram obtained by Monte Carlo simulations with the Metropolis algorithm is
shown on figure 3. That plot shows the phase diagram for the model (2.1). The data have been
collapsed usingthescalingformdefinedabovewithq = 3/2andq = 2. Itisremarkablethat
b c
− −
thisscalingalsoworksfortheexactsolutionofthepuremodelpotential.
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Simulatingthescalarfieldonthefuzzysphere FernandoGarcíaFlores
4. Conclusions
The numerical study showed three different phases. In one of those phases, which does not
existinthecommutativeplanarlf 4 theory, therotational symmetryisspontaneously broken. The
othertwophaseshavequalitatively thesamecharacterasthephasesofthislattermodel. Thethree
coexistence curvesintersectatatriplepointgivenby
(b ,c )=( 0.15N3/2,0.8N2). (4.1)
T T −
Thosethreecurvesandthetriplepointcollapseusingthesamescaling functionofN andthus
giveaconsistentN ¥ limit. Thus,allthreephases,andinparticularthenewuniformdisordered
→
phase, and the triple point survive in the limit. We will discuss these issues more completely in
P
Ref. [8].
o
Acknowledgements WewishtothankW.Bietenholz andJ.Medinaforhelpfuldiscussions.
S
(
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