Volume69A, number3 PHYSICSLETTERS 11 December1978

UNIVERSALITY CLASSES OF ISING MODELS

P. SHUKLA andN.M. ~VRAKH~DepartmentofPhysicsandMaterials ResearchLaboratory, Universityof Illinois at Urbana-Champaign,Urbana,IL 61801,USA

Received5 July 1978

We discussa transformationof Isingspinswhich mapsa d-dimensionalIsingprobleminto a seriesof differentproblemsin thesameuniversalityclass.

In this note we presenta (nontrivial) transforma- ZN = 2Ncosh(KS1,S2,).cosh(KS2S3,)tion which carriesa d-dirnensionalhypercubiclattice (4)of Ising spinsinto anotherd-dimensionalhypercubiclattice of Ising spins.Severalinterestingresultswhich X coSh(KS(N_1)SN’)= 2N(cothK)N~1,follow from this transformationarebriefly discussed. wherewehaveusedthe identity cosh(KS1’S1’) coshK,

Considera set of Nd Ising spins{S1}, S1 = ±1,on a for S~,S1 = ±1.The summationover Si—SNclearlyd-dimensionalhypercubiclattice.We first showthat doesnot generateanynewinteraction,nor doesitthepartition function of this systemwith 2d~spininter- modify theexisting interactionsin theremainingrows.actionamongstthegroupof 2d spinson a unit cell of The completepartition functionfor theN X N latticethelattice canbe evaluatedexactlyfor any integerval- becomesthe productof the partition functionof Nue of d. rows eachequalto the right-hand-sideof eq. (4):

For conveniencewe begin with anNX N squarelat- ZNXN = 2N2(coshK)(~1)2 (5)

tice. LetS1,~2’ -- ~N denotethespinson aboundary

of the square,Si’, S2, ..., S~the spinsalongthe row Similarly theNX N X N simplecubicsystemwith 8-adjacentto the boundaryand so on (refer to fig. 1). spininteraction(amongstthegroupof 8 spinssituatedThepartitionfunction for this systemis on thecornersof a unit cell cube)canbe factoredinto

ZNXN = Tr {s~}exp11NXN’ (1) theproductof N planarpartition functionsZNX N•

Thisprocesscanbe generalizedto anyintegerdimen-where sion providedthespinsinteractvia the2’1-spin interac-

tion of the typeenvisagedabove.Thefinal resultforHNX N = K~ S

1S1S~S~,. (2) thed-dimensionallatticeis

The summationin eq.(2) is overgroupsof 4 spinscor- ZNX .~XN = 2Nd(coshK)(N1)d. (6)respondingto eachdistinctplaquetteof thelattice. Eq. (6)is thepartition functionof Nd — (N —

Thepartitionfunction in eq.(1) canbe evaluated — 1 free spinsand(N — + 1 spinsin a one-dimen-by summingoversuccessiverows of spinsstarting from sionalIsing chainwithnearest-neighborinteractionKtheboundaryS1—SN.Considerthe expression [1]. Thus thed-dimensionalsystemswith the special

interactionsconsideredabovehavethe samecriticalZN = Tr{S1 i = 1,N} expK(S1’S2’S1S2 behavioras theone-dimensionalIsing model.Though

(3) thesesystemsdo not exhibit a phasetransitionat a fi-+ S2’S3’S2S3+ ... + SN~1YSN~S(N_l)SN). nite temperature,it isinterestingto notethat a series

Summationover~5N gives of systemswith differentdimensionalityandinterac-

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Volume69A, number3 PHYSICSLETTERS 11 December1978

Si Si’ versality classasthe8-spin interaction(amongst8spinsat theend-facesof two adjoiningunit cubes)

s2~ ~IIIII~problemon theS-lattice.Thus thetransformationfrom theS-spinsto the p-

spinsgeneratesa seriesof systemswith somewhatUn-

S3 S3 usual interactionsbutwhich are equivalentto the usualnearest-neighbor-interactionIsing systems.Thoughthis

I equivalenceis interestingin its own right we arenotI I

I I awareif theunusualinteractionsencounteredhererep-resentany realisticphysicalsystems.Our transforma-

S(Nltion may be useful,however,in other instances.Forexample,in determiningthecritical behaviorof asys-0tern in an approximatetheorysuchasthe latticerenor-Smalizationtheorydevelopedby Niemeijerand

Fig. 1. N X N planesquarelattice with 4-spin interaction van Leeuwen[3] , it may be moreconvenientsome-(groupsof interactingspins areshownconnectedby circles), timesto choosean alternatesystemfrom the sameuni-

versalityclass.In a systemwith bothferromagnetictionhavethe samecritical behavior,i.e.,they lie in andantiferromagnet~cnearest-neighborinteractions,the sameuniversalityclass [2]. the bondscanbe characterizedby Ising variables~

Next we define ±1. In this casethevariablesp~canbeviewedas the

2d local “frustration functions”of Toulouse[41usedin

= [1 s~, (7) connectionwith the spin-glassproblem.Forsimplicityi=1 considera spinsystemwith randombondsat zerotem-

perature.Following Toulouse,the groundstateof thiswherethe productis over the 2°’spinswhich lie on a systemwould havelong-rangemagneticorderif theunit cell of the lattice. Clearly ji~is an Ising variable, siteswith ji = —1 on theti-lattice do not form an infin-i.e., ji~ ±1,andthe variables{p~}canbe placedon a ite cluster.A detailedunderstandingof the S—ji trans-lattice equivalentto the51-lattice. formationcanproviderelationshipbetweenthesite

ForN°’S-spins,eq.(7) defines(N — i)d t-spins. percolationproblemandthe frustrationproblemofWith suitablechoiceof boundaryconditionswhich Toulouse.fix N’

1 — (N— i)d S-spins,theconfigurationsof(N — I)’1 remainingS-spinscomein one-to-onecorre- We are gratefulto ProfessorG .H. Wannierfor dis-spondencewith the configurationsof the f-spins. cussionsand a critical readingof themanuscript.ThisThereforethespins {p~}canbe consideredindepen- researchwas supportedin part by the NationalSciencedentIsing variables.This is also reflectedin thefact FoundationundertheMRL GrantDMR-76-01058.that theright-hand-sideof eq.(6) is formally thepar-tition function of Nd noninteractingspinswith (N References— i)d spinsplacedin a magneticfield of strengthK.Thereforean Ising problemon theu-latticeis physical- [11See,e.g.,H.E. Stanley,Introductionto phasetransitions

ly equivalentto a correspondingIsing problemon the andcritical phenomena(Oxford U. P., New York, 1971),S-latticeandviceversa.Forexample,a two-dimension- part IV, Cli. 8.

[2] For a discussionof theuniversalityclassesof critical sys-al problemwithnearest-neighborinteractionon the~ tems,see:L.P. Kadanoff,in: Phasetransitionsandcriti-lattice(the Onsagerproblem)mapsinto anequivalent calphenomena,eds.C. Domb andM.S. Green(Academic,problemon theS-latticeinvolving4-spininteractions London,1976),Vol. 5A.

amongstgroupsof 4 spinssuchasS1S1~S3S3~(seefig. [3JTh. NiemeijerandJ.M.J.van Leeuwen,in: Phasetransi-

1). Similarresultscanbe easilyobtainedin higher di- tionsandcritical phenomena,eds.C. Domb andM.S. Green(Academic,New York, 1977),Vol. 6.

mensions.For examplethe 3-dimensionalnearest- 141 G. Toulouse,Commun.Phys.2 (1977) 115.neighborproblemon the .t-lattice is in the sameuni-

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