Volume152, number1,2 PHYSICSLETFERSA 7 January1991

Conformalanomalyof some2-dZ( n) models

PeterWilliamSupercomputerComputationsResearchInstitute,andDepartmentof Physics,Florida StateUniversity,Tallahassee,FL 32306,USA

Received26September1990; revisedmanuscriptreceived29 October1990;acceptedfor publication29October1990Communicatedby A.A. Maradudin

Wedescribeanumericalcalculationoftheconformalanomalyin thecaseof sometwo-dimensionalstatisticalmodelsundergo-ing a second-orderphasetransition,utilizing a recentlydevelopedmethodto computethepartition functionexactly.This com-putationis carriedout on a massivelyparallelCM2 machine,usingthefinite sizescalingbehaviourofthefreeenergy.

1. Introduction ing thecritical system.The interestingquantitiestocomputeare thecorrelationfunctions,with thevac-

Conformal invariance has beenknown to be an uumexpectationvalueof the two point correlationimportantprinciplein the studyof two-dimensional function taking theformstatisticalsystems [1]. Critical systemshavebeen <OIØ(z)Ø(O)10>=r2(~~ e

210~’~ (4)shownto admitordinaryscaleinvarianceasa specialcaseof a moregeneralinvarianceunderthe confor- wherez= relO, andthereforeone identifiesthe scal-ma! group. In recentyearsthe studyof critical sys- ingdimensionof the field 0 as h + kand the spin astemshasreceivedcloserattentionin view of thefact h — £ BPZ haveshown thateveryprimary operatorthata classificationof statisticalmechanicalmodels in the theorytransformsas a representationof thecould be possible,in principle by studyingthe rep- correspondingalgebra called the Virasaro algebraresentationtheory of the conformalgroup (BPZ) with its quantumextensioninvolving the conformal[2]. In two dimensionsthisgroupis infinite dimen- anomaly.Thecommutationrelationssatisfiedby thesionalsinceit is isomorphicto thegroupof analytic operatorsare givenbyfunctions. If one considersthe complexcoordinate F~ ~ 1— ( ~, + ~ 2 i

— . . . L pn, ~~_~m—n, ~

1~cm~m— , rn—nz and its conjugatez, an infinitesimal conformaltransformationis given by wherec is the centralchargeor the conformalanom-

aly, and m, n are integers.On imposingconstraintsZ~Z+ �Z”~

1 (1) due to unitarity [3,4] it was foundthatc couldtake

with a correspondingequationfor Z. Thegenerators only certainrationalvalues,given byof such transformationsi~,i, obey the following 6commutationrules, C 1— m(m+ 1) m=2, 3, 4 (6)

[la, 0] =z’~ +h(n+ 1 )z~Ø (2) And additionallywhenc is restrictedto thosevaluesthe scalingdimensionsof the field x=h+kare givenby

[T~,Ø]=Z’—~+h(n+l)Z~Ø. (3) 2

c9Z hp.q(C) [(m+l)p—mq] —l (7)The generatorsI~and T~are independentof each 4m( m + 1)

other.The Ø(z, ~) are the conformal fields describ- wherep=1, 2, ..., m— 1 andq= 1, 2, ..., p. Henceone

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Volume 152, number1,2 PHYSICSLETTERSA 7 January1991

candeterminethe critical exponentsof two dimen- ..., N— 1, this updatingprocedureis given bysionalmodelsfrom a studyof thehighestweight rep-resentationsof the Virasaroalgebra. W~(S)= ~ u~’~”~W,(S’) , (11)

Furthermorefrom anexaminationof thefinite sizescaling behaviourof the free energy at the critical wherestatesS andS’ differ from eachother, in thatpoint [5,6] a correction term involving the confor- the spin n, is replacedby the spin n1, E(n,, n,) is thema! anomalyhasbeenobtained.In thecaseof a sys- bond energy,and u=e~,J9 beingthe inversetem-tern Ix 1’, with periodicboundaryconditionsin the perature.ReplacingN by N— 1 for the caseof odd/ direction,the leadingordercorrectionterm isgiven valuesof N, definesthebond energyby thefollowingby equation,

~Z_lf+~+O(l/I2) (8) E(n~,n1)=d1, ford1<~N, (12)

=N—d1, ford1~~N, (13)whereZ is thepartition functionof the system,with

the free energy (in units of kT) being definedas = I fl, — fl~I . (14)

— ln Z. (9) In viewof thetransformationintroducedin ref. [7],

Thereforethecorrectionterm involvingthe free en- U=C{”” , c1,m integers, (15)ergy per unit spin is now given by to accountfor a multiplication by a factor u implies

f irc thatspectralweights C’~.undergoa right shift by a= — +0(1/1) . (10) singleelementandthe zerothelementis multiplied

by a factorc1 as in the following,With this onecan obtain thevalueof the conformalanomalyc in variousmodelsby examiningthe seal- C’rn(O) C1 Crn(m 1) , (16)ing of the free energy with the 1’ direction suffi- c~U) = c ~— 1) ~= 1 ... m— 1 . (17)ciently long, 1’ beingconsideredinfinite. On accountof the fact that the whole partition function is ob- It canbe shownthat theC’m areidenticalto theden-tamedexactlyby essentiallyenumeratingthe possi- sity of statesfor a particularchoice of m= em+ 1.ble statesof the system,it is quiteeasyto obtainthe Hereemdenotesthe valueof themaximumpossiblefree energyat thecritical temperaturein eachof the energy in the system.The whole partition functionmodelsconsidered,asit is givenby eq. (10) above, is thus obtainedby accountingfor the bondsgen-

erated, which translates to multiplications byE ( n,, n~)(which are powersof u) with the addition

2. Implementationof the algorithm of eachspin on the lattice. On a massivelyparallelmachinelike the CM2, the elementsof the weight

A parallelimplementation,of an algorithmdueto arraysare storedacrossthe processorsand the effi-Bhanot [7] to generatethe partition function of a ciency is highly dependenton the numberof arraydiscretespinsystem,on theCM2 is describedbriefly, elementsbeingoperatedon in parallel.First themul-As statedearlier [8,9] two arraysW0and W,, (whose tiplying factors for eachof the n’ elementsof therespectivedimensionsare n’x m) containing the weight arrayarepre-calculateddependingon thesiteBoltzmannweights of the bondsgeneratedin each at which the spin isbeingcurrentlyadded.Sincetheconfiguration are constructed,and the subsequent multiplying factors are different for eachelementaddition of a spin at a particularsite resultsin an being updated,the numberof independentupdatesupdateof thesearrays.Thepartitionfunctionis con- possiblearefoundandperformedsimultaneously.Instructedlayerby layerby accountingfor all thebonds an li’ systema totalnumberof n (I updatesaregeneratedwith the additionof a spin at a particular independent,andhenceto updateall elementsofthesite. In thecaseof Z(N) spin configurationswhere weight array, to take into accountthe additionof aeachindividual spinis allowedto takea valuen,=0, spinat a particularsite,onehasto repeatthe process

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Volume152, number1,2 PHYSICSLETTERSA 7 January1991

n times.Continuing theprocessfor everysiteon the thescalingof the freeenergyin the caseof theZ( N)lattice amountsto constructingthe whole partition spin systemsfor N= 2, 3, 4 where a second-orderfunction.As the CM2 hasa hugememoryavailable, phasetransitionis expectedat the critical point. Init overcomesoneof the inherentlimitationsof this thecaseof the Z( 4) model the clockrepresentationmethod.The other limitation is in maintainingthe is consideredas it is equivalentto the actiongivenfull accuracyof thepartitionfunctionasbasicallyin- in eqs. (12)—(14). The free energyat the criticaltegerarithmeticisbeingperformed.Howeverfor the point wasobtainedin the caseof largely asymmet-largerasymmetricalsystemsconsidered,it isnotquite rical lattices,andfrom the scalingformgiven in eq.necessaryto maintaincompleteintegeraccuracyas (10) the correspondingvalue of the conformalone is moreinterestedin obtainingthe free energy. anomalyis estimated.In the caseof all the models

consideredtheagreementwith predictedvalues,andpreviousnumericalsimulation using Monte Carlo

3. Results methods[10] is foundtobe good.In tables1—3 thevaluesofthe freeenergyperunit spin forvariouslat-

Theimplementationgivenabovewasusedto study ticesare tabulatedin thecaseof Z( 2), Z( 3), Z( 4)spin systemsrespectively,for variouslatticesthatareconsidered.Thisfree energyhasbeenobtainedfrom

Table 1 . . . thepartitionfunction,which is exactto machineac-Z(2) free energyfor the lattice sizes lx1’. Periodic boundary

curacy,andthereforehasno statisticaluncertainty.conditionshavebeenimplementedin the1 direction.__________________________________________________Thelastcolumnof eachof thesetablesgivesthevalue

1 1’ /2 —f/Il’ c ofc obtainedfrom eq.(10) abovefor successiveval-uesof!. Forexample,the first entry for the lastcol-

4 60 0.0625 0.069918362459 0.5438589 . .

5 60 0.04 0.063511177347 0.5290366 umn in tableus the valueof c obtainedby consid-6 60 0.02777 0.060123431952 0.5221923 ering the data for !=4 and 1=5. Thus obtaining7 60 0.020408 0.058110522022 0.5181502 successivevaluesof c allowsone to makean extrap-8 60 0.015625 0.056812877240 0.5130711 olationfor slopeof the line in the infinite volume9 60 0.012345 0.055931727558 0.5128042 limiting case.It is assumedthattheasymptoticvalue

10 60 0.010000 0.055302087528 of c correspondsto the theoreticalvalueof the con-

Table 2Z( 3) freeenergyfor thelatticesizestx!. Periodicboundaryconditionshavebeenimplementedin the 1 direction.

I I’ l_2 —f/Il’ c

4 40 6.2500000000000000E—02 9.8605372994996599E—02 0.87953165 40 4.0000000000000000E—02 8.8243630796801508E—02 0.85186866 40 2.7777777777777778E—02 8.27920683l6927l48E—02 0.8362263

7 40 2.0408l63265306l23E—02 7.9565299723606826E—02 0.82644078 40 l.5625000000000000E—02 7.7495515670373952E—02

Table 3Z( 4) free energy for the lattice sizes 1xl’. Periodic boundary conditionshavebeenimplementedin the1 direction.

1 1’ 12 —f/Il’ c

3 25 0.llllllllllllllll 0.1948137725217248 1.1480234 25 6.2500000000000000E—02 0.1655934548717464 1.0888905 25 4.0000000000000000E—02 0.1527652702131197 1.0527016 25 2.7777777777777778E—02 0.1460284694328458 1.060982

7 25 2.0408163265306123E—02 0.1419344358952235

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Volume 152,number1,2 PHYSICSLETTERSA 7 January1991

formalchargein view of higherordercorrectionterm cersof the machinefor a totaltimeof approximatelypresentin eq. (10). For the purposesof extrapola- 15 hours.tion, the VandenBroeckandSchwartz(VBS) [111approximationscheme,was used. From data ob-tainedin table 1 the infinite volumevalueof c in the Acknowledgementcaseof Z(2) was found to be c=0.51(2) in com-parisonwith the predictedvalueof c= ~. In the case I wish to thankDr. DennisDuke,Dr. Urs Heller,of theZ( 3) modelfromvaluesin table2, theinfinite Dr. K.M. Bitar foruseful suggestionsto improvethevolumevalueis givenby c=0.8l(4), whilst the pre- manuscript.Thanksareduealso to Terry Dontje,anddictedvalueis c=~.Fromtable3, theasymptoticand PaulOppenheimerof ThinkingMachines,andDr.predictedvalues are c= 1.05(3) and c= 1 respec- Jim Hudgensfor help with use of the Connectiontively. The uncertaintiesquotedarisefrom the ap- Machine.This work was supportedby the Floridaproximationschemethat hasbeenused.In all the StateUniversity SupercomputerComputationsRe-abovecasesthe predictedvaluesof c havebeenob- searchInstitutewhich is partially fundedby theU.S.tamedby matchingtheknown scalingdimensionsof Departmentof Energy through Contract No. DE-theoperators,which arelinearcombinationsof h and FCO5-85ER250000.hintroducedin eq. (2) andeq. (3) with theallowedvaluesof c correspondingto thosevaluesof h andh [3]. References

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to the limits of machineaccuracythe methodused mathematicsandphysics(Springer,Berlin, 1984)p. 419.here has provento be very useful in estimatingc. [5] H.W. Blote, J.L. Cardy andM.P. Nightingale, Phys.Rev.

Furthermoreit hastheadvantagein thatonedoesn’t Leit. 56 (1986)742.

haveto resortto Monte Carlo simulationsinvolving [6] I. Affleck, Phys.Rev.Lett. 56 (1986)746.[71G. Bhanot,preprintFSU-SCRI-89-90(1989).

a largeamountof time in evaluatingthefree energy. [81G.Bhanot, preprint FSU-SCRI-89-165 (1990).

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