Volume 146, number5 PHYSICSLETTERSA 28 May 1990

CORRELATION LENGTH EXPONENT OF THE 2-d Z(4) MODELUSING AN EXACT METHOD

Peter WILLIAMSupercomputerComputationsResearchInstituteandDepartmentofPhysics,Florida StateUniversity,Tallahassee,FL 32306, USA

Received1 March1990; revisedmanuscriptreceived20 March1990;acceptedfor publication21 March 1990Communicatedby A.A. Maradudin

Using arecentlysuggestedexactmethodto determinethepartitionfunctionofa discretemodel,thecorrelationlengthexponentv for the two-dimensionalZ(4) spin model is estimated.This estimationis madefrom a study of thezerosof thepartitionfunctionon finite lattices.

1. Introduction whereSandS’ indicateconfigurationsoftheonedi-mensionalspin systemgiven by

In a recentpaperanewmethodhasbeensuggested S= (s1, s2, ..., s1, ..., s1) , (2)to computethe densityof statesexactlyfor a set of ~ — ( ~ (3)discretemodels[1]. Theidea is an extensionof the — ~‘~‘ 2, , :, , /

methodgivenin anearlierwork [2] to computethe The barovers, just switchesthe spin valueat thatpartition function. Following the lines of ref. [1], locationfrom 0 to 1 andviceversa.With eachaddeddiscreteZ( n) modelsin two dimensionsand,in par- spin, the weight arraysare updateduntil the wholeticular, theZ(4) modelis considered.We giveabrief row is completed.After the row is completed,onediscussionof the methodused. The details canbe justmultipliesall thearraysby theBoltzmannweightfoundin ref. [1]. Consideranexampleof Z( 2) sym- ofthespinsin therow. Thistakescareofall thebondsmetry (the Ising model). Boltzmann weights of up tothe particularlayerbeingpresentlyconsidered.u= e~or u= 1 are assignedto abonddependingon Thewholelattice is thusconstructedlayerby layer.whetherthe spinsarealignedantiparallelor parallel. From this processonecould computethe partitionIn orderto obtainthedensityofstatesonehasto store function of the systemat a particulartemperature.theweightsofthebondsofeachconfiguration,hence However,onewould, in general,bemoreinterestedtwo arraysW0 and W,, are constructed,which con- in computingthe densityof states,andthusavoidtam informationregardingtheweightsof thevertical the problemof performingsimulationsat numerousbondsgeneratedin eachconfiguration.Eachtime a valuesof temperatures.With the choiceof u given

spin is addedthe whole arrayis updated.The spe- abovethe partition function canbe expressedas acific updating procedureis describedbelow. This polynomialof finite degreein u,processmakesthe updateshighly nonlocalbut re- em

sultsin thealgorithmbeingefficient. The arraysare Z(u) = ~ G(k)uk. (4)definedasold andnew,andthebasictransformation k0

of theweight arraywith theadditionof a spinat the Hereem is the maximumpossiblevalueof the en-latticesite labelledi, is given as ergy. If we choosethe valueof u suchthat

Wa(S) = W0(S)+uW0(S’) , (1) u=cVm, (5)

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where c1 and rn are non-negativeintegers,then we ory, andthis limits the methodfrom being usedinobservethat very largevolumes.The memory that is requiredin

d dimensionsis roughly given by

Z(u)= k~O (6) mem~2.Onhd_em. (13)

Thisimpliesthat for multiplication by a factor u, we Another limitation results from the accuracytohave which thecoefficientscanbecomputed.Sinceone is

— basically performing integerarithmetic, one is re-

uZ(u)= ~ Crn(k)U’<41 strictedby the largestintegerthatcanberepresented

k=O on a machinein maximumprecision.Theaccuracy=Cm(0)u+Crn(l)u2+...+Crn(ml)u’~. (7) inbitsisroughlygivenby

Consideringa new setof coefficientsC’ ‘s which are acc l’~’log2n . (14)

themselvesfunctionsof thevariableu, we also have Thegeneralisationfor consideringZ(n) modelsfol-that lows quite naturallywith a few redefinitions.In one

rn—I dimensiontheZ(n) configurationis labelledby theuZ(u) ~ C’m(k)U

1’ phasefactorsn, which takevaluesfrom 0 to n—I.k=o The bond energy,in the caseof evenn, betweena

=C~,n(0)+C~rn(l)u+...+C~rn(rnl)um~. (8) pairof spinslabelledby valuesn, and n1 is given by

Comparingtermsin the two previousformsgiven E(n,, n1) =d1 , ford< ~n,above,we havethe following, — A c A> I—n— I, or .-2n,

C’n(O)ciCm(ml) (9) d — n—n ‘15, j .

C~nU)=Crn(jl), J=1 rn—i. (10) Similar equationsholdin thecaseof oddvaluesof

We thusseehow the importantstepof the updating n with n replacedby n — 1 in the inequalities.Theprocedureas given by (1) transformsin terms of generalisationoftheupdatingproceduregiven by (1)Cm’S. is now given by

One cantreateachof the weight arrays W(S)asa partial partitionfunction since W~(S)=~ exp[ —/3E(n1,n1)] W0(S’) . (16)

ni = 0

Z(u)=~W(S). (11)wherethe statesS and S’ aredefinedby

The rule usedin performingthemultiplication by S= (n1, ..., n~, ...) , (17)u on the wholepartition function canbe appliedto s = (n .. ...) (18)eachof the weight arraysseparately.At eachinter- I, .~

mediatestepto obtain W~,we addup therespectivevaluesof Cm’5 from W0(S)and uW0(S’).Finally,for a particular m we addup the coefficientsto get 2. Scalingof zerosandcritical exponentsthe Cm of the whole partition function.Comparingforms (4) and(6) wededucethatif rn= em+ 1 then Having obtainedthedensityof states,we cannowthe spectralcoefficientsare givenby examinethebehaviourof theLee—Yang[3] zerosofG(k)—C (k) k—0 rn—i (12) the partition functionin the complexplane,to look

— m — for phasetransitions.Usingtheresultsofref. [4J one

Thisprovidesa methodfor obtainingthedensityof canobtain quantitativepredictionsfor the correla-stateswithout having to resort to repeatedsimula- tion length critical exponentv.tions at different temperatures.However,this pro- The zeroclosestto thereal axis scaleslike P oncesshasresultedin a considerableincreaseof mem- anli lattice.Hencetherealandimaginarypartsof

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the zerosobey the following relationsgiven by [4] Approximationscherne.The VBS approximationschemeis describedin the following paragraph.A

u,(1)— u~= 1- l/vf 1(0), (19) table of approximantsis constructedfrom a given

Re[u1(1)] —u~=L”f~’(0) , (20) sequenceof values,to which the limiting valueis to

be obtained.Let the given sequencebe denotedbyIm(u1)=l’~f~(0) . (21)

Here u, refersto the ith root, u,, to the infinite vol- AN [N, 0] . (23)ume critical point andf I is in generala complex Then successivecolumns are generatedby thenumber.Fromtheserelationsonefinds estimatesfor formulathe critical exponentv,

~Imu1(1+l) (1)]~ (22) [N,M+l]—[N,M] + [NM—l]—[NM]V Imu1(1) )[— =log~,, log

In order to obtain the critical exponentv andthe= [N+l,M]—[N,M]

critical temperature,we find estimatesfor the ex-ponent from (22) aboveanduse an extrapolation _________________

schemeto getthe infinite volumevalues.Depending + [N—1, M] — [N, M] (24)on the geometrybeing consideredtwo different ex-trapolationschemeswereused. An auxiliary condition [N, — 1] =cx is also im-

posed.Thevalueof aM chosen,for M= 0, 1, 2, ... isgiven by

3.Results aM=—~[l—(—i)]. (25)

We usedthemethodto studythe Z(4) spin model. Successivecolumnsthengeneratethe requiredlim-For this model oneexpectsa singlephasetransition iting value.The displayedtablesshowtheresultsforfrom resultsobtainedby Elitzur et al. [51 in con- the locationoftheclosestzerosin thecomplexplane,sideringthe clockmodelsfirst introducedby Joseet for the differentlatticesconsidered.Applying theBSal. [6]. More recentlyCatterall [7] has examined extrapolationschemetothe dataobtainedin table 1,the scalingof thefree energyto obtain the exponent we get an infinite volume estimatefor the exponentof the correlationlengthin the caseof strip geome- v to be 1.061(6).Consideringdatain table2, v istries.Forthecaseof symmetriclattices,the Bulirsch 1.029(7).HowevertheVBS approximationschemeandStoer(BS) [8] extrapolationschemewas used producesan asymptoticvalueof 0.989(2).Fromtheand foundto work quite satisfactorily.However,the strip geometrydatagiven in table 3, the VBS ap-infinite volume estimates,usingthe VandenBroeck proximation schemegives the value of v to beand Schwartz (YBS) [9] approximationscheme 1.077(4).usedby Hamerand Barber [10] in their finite sizescalinganalysis,were foundto work betterin thecaseof strip geometries.

Table 1Locationof theclosestzerowith freeboundaryconditions.

L Re(u) Im(u) v

2 0.000000000000000 0.414213562328067 1.3274911903006713 0.201248632385597 0.305193065395396 1.1632362949183174 0.269826888749186 0.238324399472003 1.1326302611030105 0.304402531897320 0.195707107877198 1.1166936291140016 0.325340085360403 0.166226294340994

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Table 2Closestzerowith periodicboundaryconditionsin theL direction.

L Re(u) Im(u) p

2 0.257065866406596 0.529085514236190 0.73510882493072923 0.348326604256138 0.304776453932598 0.88573661638629094 0.372604293780331 0.220254659216910 0.94442072978693115 0.383701687949191 0.173904945415597 0.97271947644039126 0.390063758089077 0.144181649043851

Table 3Closestzerowith periodicboundaryconditionsin theL direction.

L L Re(u) Im(u) p

2 10 0.3226097680521 0.4751100854037 0.36105803551551233 10 0.4235834643656 0.1545557866117 0.53827561444868654 10 0.5400899140669 0.09056826396000 0.70678157950670425 10 0.4796154047344 0.06604828388895 0.92520328376785876 10 0.4493215213282 0.05423491799773

4. Conclusion Villanova, R. Bertramfor helpandencouragement.Thanksaredue to Dr. K.M. Bittar for clarifications

The results obtainedfrom this simulation agree and criticisms regardingthis work. This work waswell with those found previously and this method supportedby the Florida State University Super-providesan efficient methodto estimatethe expo- computerComputationsResearchInstitutewhich isnentwith considerablylesserrequirementson corn- partially fundedby the U.S. Departmentof Energyputertime, thelimitationbeingon thesizeof lattices throughContractNo. DE-FCO5-85ER250000.that could be considereddue to large memory re-quirements.However,onedoesnotreallyrequiretosimulatethe systemat verylargelatticesasthe seal- Referencesing of thezerosis a finite sizeeffectandgoodresults [I] G. Bhanot PreprintFSU-SCRI-89-90(1989).

are alreadyobtainedwith the small latticesconsid- [2] K. Binder,Physica62 (1972)508.

ered.The generationof the partition function took [3] C.N. YangandT.D. Lee,Phys. Rev. 87 (1952) 404.

lessthan 30 minuteson the SCRIVAX 8700for the [4] C. Itzykson,RB. PearsonandB. Zuber,NucI. Phys.B 220

largestsystemconsidered. (1983) 415.[5] S. Elitzur, R.B. PearsonandJ. Shigemitsu, Phys. Rev. 19

(1979) 3698.[6] J.V. Jose,L.P. Kadanoff,S. Kirkpatrick and D.R. Nelson,

Acknowledgement Phys.Rev.B 16(1977)1217.[7] S.M. Catterall, Phys. Lett. B 231 (1989)141.[8] R. Bulirsch and J. Stoer, N. Math. 6 (1964) 413.

I amdeeplyindebtedto Dr. GyanBhanotfor many [9] J.M. Vanden Broeck and L.W. Schwartz, SIAM J. Math.patientexplanationsduring the early stagesof the Anal. 10 (1979) 658.

project. I also thank Dr. D. Duke,Dr. U. Heller, R. [10] C.J. Hamer andM.N. Barber,J. Phys. A 14 (1981) 2009.

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