Volume82A, number3 PHYSICSLETFERS 16 March1981

LOW-TEMPERATURE SERIES EXPANSIONS FOR LATTICESOF NON-INTEGRAL DIMENSIONS ~

Felix LEE andH.H.CHENInstitute ofPhysics,National ThingHua University,Hsinchu, Taiwan, ROC

Received19 August1980

Low-temperatureseriesexpansionsfor I singmodelson latticesof non-integraldimensionsarestudied.Critical exponents~3for variousdimensionsareextrapolatedfrom seriesexpansionsfor thegeneralizedequivalentneighborlattice.

The dimensionaldependenceof critical exponents theabove-mentionedhypercubicalandhypertriangu-hasbeenan areaof intensivestudiesduring the past larlatticesareintractableto extrapolationof criticaldecade.In particular,the determinationof critical ex- propertiesfor non-integralvaluesof d.ponentsfor non-integrallattice dimensionsis of great Thelow-temperatureseriesexpansionsof thermo-theoreticalinterest, dynamicfunctionsconsistof termsof the general

Whenthe dimensionalityof the lattice is d 4— e, form~(vq2l)/2, whereu = exp(—4J/kT)is theexpan-

exacte-expansionsofcriticalexponentstothird order sionvariable,q is thegeneralizedcoordinationnumberin e were obtainedfrom the renormalizationgroup of thelattice,andv andlarethenumbersofverticesandapproach[1]. Thec-expansionsof critical exponents linesof a lineargraphwhich contributesto theseriesaregoodapproximationsfor theexponentswhen expansions,respectively.Forinstance,thespontaneouse ~ 1. For generalvaluesof d critical exponentswere magnetizationseriesM

0 for the Ising modelhas theobtainedonly from thehigh-temperatureseriesex- form [6]:pansionmethod[2—5].

Latticemodelssuitablefor investigatingcriticalex- --= 1 2uq/2 — ~

2~(p~v))~u/u_~l (1)ponentsfor arbitrarydimensionsare thehypercubical Nm v2 10 a la

latticeintroducedby FisherandGaunt[2], thehyper-triangularlattice suggestedby VanDyke andCamp wherepr’) is the low-temperaturelattice constantof[3], andthe generalizedequivalentneighbormodel theathlineargraphhavingu verticesand1 lines.proposedby ChenandLee [4,5]. High-temperature Formostregularlatticesq areevennumbers,andseriesexpansionsof thermodynamicfunctionshave eq.(1) is a powerseriesin u. Forthehoneycombandbeenderivedfor theselattices.The critical exponents the hydrogenperoxidelatticesq = 3; then eq.(1) is ay for generalvaluesof d were first determinedby Van powerseriesin u

112 = exp(—2J/kT).For a generalDyke andCamp [3]. More preciseestimatesof ‘y were non-integralvalueof q low-temperatureseriesexpan-thengivenby ChenandLee [4]. sionsare not integer-powerseriesin u (or u~/2,etc.).

Low-temperatureseriesexpansions,andcritical In this casemethodsof analysingtheseseriesareasexponentsfor T -~ T~,for latticesof non-integral yet unavallable.The coordinationnumbersareq = 2ddimensions,however,havenotbeenreported.As will andq d(d + 1) for thehypercubicalandthehyper-beseenbelow,low-temperatureseriesexpansionsfor triangularlattices,respectively.Theyare not integers

for non-integralvaluesof d. Therefore,we cannotob-‘~‘Work supportedby the NationalScienceCouncil of the tam critical propertiesof theselattices from analysis

Republicof China. of the low-temperatureseries.

140 0 031—9163/81/0000—0000/S02.50© North-HollandPublishingCompany

Volume 82A, number 3 PHYSICS LETrERS 16 March1981

Forthegeneralizedequivalentneighbormodelq increasingfunctionsof q for afixed valueofd, anda= (2R + 1 )‘~— 1, whereR is therangeof the inter- cutt-off of theoscillatingtailsin thehigh-temperatureaction.We canfind valuesof R suchthatq are even lattice constantsis required[5]. Thelow-temperaturenumbersfor anynon-integralvaluesof d. Thegeneral- latticeconstants,however,maynotbe increasingfunc-izedequivalentneighbormodelis theonly model, tionsof q. No cut-offprocessis requiredwhenthesetknownsofar, in which critical propertiesfornon- of cut-offhigh-temperaturelattice constantsis trans-integraldimensionscanbe extrapolatedfrom low- formedinto theset of low-temperaturelattice con-temperatureseriesexpansions. stants.

Fora given lattice thequantitiesto bedetermined Fromthelow-temperaturelatticeconstantswe ob-in eq.(1) are the low-temperaturelatticeconstants tam the low-temperatureseriesexpansionsfor thespe-p~)Since theseconstantsarelinearly relatedto the cific heatatzerofield, thespontaneousmagnetization,high-temperaturelattice constants,we canobtain the andtheinitial isothermalsusceptibilityfor the Isingset of low-temperaturelattice constantsfrom the set model.Thelengthsof the seriesexpansionswe ob-ofhigh-temperatureones.Forthegeneralizedequiva- tameddependstronglyonq andweakly on d. Forex-lentneighbormodelthederivationof thehigh-tempera- ample,whenq = 26 thespontaneousmagnetizationture latticeconstantshasbeendescribedin detail by seriesis exactto the 76thorder in u for generalvaluestheauthors[5]. Oneof themain proceduresin de- of d, andwhenq = 14 theseriesis exactto the 27thrivingtheseconstantsis theevaluationof thecoin- orderinu ford 1 andtohigher-ordertermsfor largercidableembeddings((gnx;~my))-The matrix ((g~~ valuesof d.

gmy))transformsthesetofcoincidableoccurrencefac- As in previousstudieson regularlattices [9], ourtors into the set ofhigh-temperaturelattice constants. analysesof the initial isothermalsusceptibilityseries

In ref. [5] thematrix elements((g,~; amy))for and thezero-fieldspecificheatseriesfor the general-starswithn ~ 8 wereobtainedby visualinspection. izedequivalentneighbormodel areunsuccessful.WeWe havecomputerizedtheproceduresfor deriving havenotobtainedmuchfrom analysesof theseseries,theseelements.The main proceduresare to find all andthustheywill not be reportedhere.thereducedgraphsof ~ , andto determinehowmany Our analysisof thespontaneousmagnetizationse-of themareisomorphicwith amy.Previouslywhether rieshasbeenquite successful.In thecriticalregionthetwo graphsareisomorphicornot wasdeterminedby spontaneousmagnetizationis assumedtohaveasim-comparingtheir canonicalmatrices.We havedevel- ple power-lawsingularityfor all lattice dimensions;opedanewmethodtoidentifygraphs[7] ~ . Thekey thenpointis thateachgraphis representedby a modified —

• . . . . (d/du)ln(M0/Nm)~j3/(u—u), for u-~u . (2)incidencematnx.Two graphsare isomorphicwhen c cthedeterminantsof their modified incidencematrices Two methodsof Padéanalysis [10] havebeenper-areequal.Thematrixwhich transformsthe setof high- formed:(a) Fora Padéapproximantto (d/du)ln(M0/temperaturelattice constantsinto theset of low- Nm) the appropriatepole givesuc, andtheresidueattemperatureoneshaselements(g~; g~,).Theseele- thispolegives/3. (b)If agoodestimateof uc isknown,mentsare theweakembeddingsofg~ing~,[8]. /3 is representedby Padéapproximantsto (u — uc)(d/Theyare calculatedby a methodsimilar to that for du)ln(M0/Nm)evaluatedat u~.

((gnx;gmy)). Method (a) predictsbothu~and/3 simultaneously.Wehaveobtainedthelow-temperaturelattice con- In thismethod,however,thevariousapproximants

stantsasfunctionsofd andq for all graphswithv~ 6, are ratherdiverseand theestimatesareless accurate.andfor graphswith u = 7 and 21 ~‘ I ~‘ 15. It will be In method(b) variousapproximantsare moreregular,notedthat thehigh-temperaturelatticeconstantsare but theestimateof /3 is sensitiveto the precisechoice

of u~.If the choiceof u~isincreased,i.e.,higherT~* The diagonal elementsof the incidencematrcesarezeros, is chosen,theestimateof (3 will increase.Note that

while in a modifiedincidencematrix the valuesof the • • . - -

diagonalelementsdependon thenumbers of lines con- m thehigh-temperaturesenesexpansionsif thechoicenectedto the vertices. For defmitions of incidence matrix of T~is increased,then thecorrespondingestimateofand other terminology, seepp. 4—14 of ref. [10]. ~ will decrease.

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Volume82A, number3 PHYSICSLEUERS 16March 1981

As it is well known thatmostreliableestimatesof Table 1

Tc are thoseobtainedfrom analysisof the high-tem- Critical exponentP for theIsing modelon thegeneralizedequivalentlatticeof dimensionalityd.

peraturesusceptibilityseries,we haveadoptedthe ___________________________ ______

criticaltemperaturesw~= tanh(J/kT,~),obtainedfrom d d p

previousanalysisof thehi~h-temperaturesusceptibility —

series[4,5], andsetuc = ~ln [(1 +w~)/(1— we)] lii 1 0 2.7 0.268 ±0.003

method(b).Therangeofqwehavestudiedis8~q~20.1.5 0.024±0.004 2.8 0.283 ±0.003Forsmallvaluesof q our low-temperatureseriesare 1.6 0.042 ±0.004 2.9 0.299±0.004

1.7 0.060±0.004 3.0 0.313 ±0.004tooshort,andfor largervaluesof q theestimatesof 1.8 0.081 ±0.004 3.1 0.328±0.005

w~arelessaccurate.We find that optimumvaluesof 1.9 0.103 ±0.004 3.2 0.342±0.005

q for whichthe seriesarebetterbehavedfor the de- 2.0 0.127 ±0.004 3.3 0.355±0.008terminationofj3 areq = 10, 12 and14. 2.1 0.149±0.003 3.4 0.370±0.008

2.2 0.172 ±0.003 3.5 0.385 ±0.010Within theerrorlimits our results,asexpected,are 2.3 0.194 ±0.003 3.6 0.400±0.010

consistentwith thehypothesisthat /3 isindependent 2.4 0.215 ±0.003 3.7 0.415 ±0.010of q. Ouroverall estimatesof$(d)are givenin table1. 2.5 0.234 ±0.003 3.8 0.430±0.015

Ourestimatesfor 2 ‘~d ~ 3 arequite accurate.The 2.6 0.252 ±0.003 3.9 0.445±0.015

uncertaintiesin theestimatesford< 2 arealso small, 4.0 0.455±0.015

becausevariousPadéapproximantsare regularand —~

lesssensitiveto thechoiceofu~in theregion. The Referencesuncertaintiesin our estimatesaremuchlargerfor d> 3.Not only arevariousPadéapproximantsveryirregular, [1] KG. Wilsonand J. Kogut, Phys.Rep.12C (1974)75.

buttheyare alsosensitiveto thechoiceof u~. [2] M.E. FisherandD.S. Gaunt,Phys.Rev. 133 (1964)

Ford= 4 our estimateof /3 doesnotagreewith the A224.exactvalue (0.5).This is expectedbecausethe loga- [3] J.P.VanDykeandW.J.Camp,AIPC0nI.Proc.29 (1976)

502.rithmic factor in thecritical amplitude [11] wasnot [4] H.H. Chenand F.Lee,Phys.Lett. 70A (1979)77.

takeninto accountin our analysis.We also expect [5] H.H. Chenand F. Lee,J. Phys.C13 (1980)2817;and

thatourestimatesneard = 4 (3.4<d ~ 4) aresmaller to bepublished.

thancorrectvaluessincehigher-ordercorrectionsto [6] N.W. DaltonandD.W. Wood,J. Math. Phys.10 (1969)1271.

thedominantscalingtermare importantin this re- [7] H.H. ChenandF. Lee,to bepublished.

ion [3]. [8] M.F. Sykes,J.W.Essam,B.R. Heapand B.J. Hiley, i.Fromthevaluesof(3 givenin table1 andthevaluesof Math. Phys.7 (1966)1557.

~previouslyobtainedfromthehigh-temperaturesus- [9] D.S. Gauntand M.F.Sykes,J. Phys.A6 (1973)1517.ceptibility series[4,5], otherexponentscanbedeter- [10] D.S. Gauntand A.J.Guttmann,Phasetransitionsand

criticalphenomena,Vol.3, eds.C. DombandM.S. Greenminedby scalinglaws [12]. Forexample~ = 2 — (AcademicPress,New York, 1974)p. 202.

(‘y + 2/3),~= (/3+7)/fl, etc. [11] F..!. WignerandE.K. Riedel,Phys.Rev. 87 (1973)248.

[12] HE. Stanley,Introductionto phasetransitionsand

criticaiphenomena(Oxford U.P., Oxford,1971)p. 185.

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