Correlation length exponent of the 2-d Z(4) model using an exact method
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Volume 146, number 5 PHYSICS LETTERS A 28 May 1990 CORRELATION LENGTH EXPONENT OF THE 2-d Z(4) MODEL USING AN EXACT METHOD Peter WILLIAM Supercomputer Computations Research Institute and Department of Physics, Florida State University, Tallahassee, FL 32306, USA Received 1 March 1990; revised manuscript received 20 March 1990; accepted for publication 21 March 1990 Communicated by A.A. Maradudin Using a recently suggested exact method to determine the partition function of a discrete model, the correlation length exponent v for the two-dimensional Z(4) spin model is estimated. This estimation is made from a study of the zeros of the partition function on finite lattices. 1. Introduction where S and S’ indicate configurations of the one di- mensional spin system given by In a recent paper a new method has been suggested S= (s 1, s2, ..., s1, ..., s1) , (2) to compute the density of states exactly for a set of ~ — ( ~ (3) discrete models [1]. The idea is an extension of the — ~‘~‘ 2, , :, , / method given in an earlier work [2] to compute the The bar over s, just switches the spin value at that partition function. Following the lines of ref. [1], location from 0 to 1 and vice versa. With each added discrete Z( n) models in two dimensions and, in par- spin, the weight arrays are updated until the whole ticular, the Z( 4) model is considered. We give a brief row is completed. After the row is completed, one discussion of the method used. The details can be just multiplies all the arrays by the Boltzmann weight found in ref. [1]. Consider an example of Z( 2) sym- of the spins in the row. This takes care of all the bonds metry (the Ising model). Boltzmann weights of up to the particular layer being presently considered. u = e~or u = 1 are assigned to a bond depending on The whole lattice is thus constructed layer by layer. whether the spins are aligned antiparallel or parallel. From this process one could compute the partition In order to obtain the density of states one has to store function of the system at a particular temperature. the weights of the bonds of each configuration, hence However, one would, in general, be more interested two arrays W0 and W,, are constructed, which con- in computing the density of states, and thus avoid tam information regarding the weights of the vertical the problem of performing simulations at numerous bonds generated in each configuration. Each time a values of temperatures. With the choice of u given spin is added the whole array is updated. The spe- above the partition function can be expressed as a cific updating procedure is described below. This polynomial of finite degree in u, process makes the updates highly nonlocal but re- em sults in the algorithm being efficient. The arrays are Z(u) = ~ G(k)uk. (4) defined as old and new, and the basic transformation k0 of the weight array with the addition of a spin at the Here em is the maximum possible value of the en- lattice site labelled i, is given as ergy. If we choose the value of u such that Wa(S) = W0(S) +uW0(S’) , (1) u=cVm, (5) 0375-9601/90/S 03.50 © Elsevier Science Publishers B.V. (North-Holland) 261
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Transcript of Correlation length exponent of the 2-d Z(4) model using an exact method
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)
0375-9601/90/S03.50© ElsevierSciencePublishersB.V. (North-Holland) 261
Volume 146, number5 PHYSICSLETTERSA 28 May 1990
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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Volume 146, number5 PHYSICSLETTERSA 28 May 1990
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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Volume 146, number5 PHYSICSLETTERSA 28 May 1990
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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