IC/80/162 INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS PERCOLATION AND SPIN GLASS TRANSITION Abdullah Sadiq. Raza A. Tahir-Kheli Michael Wortis and INTERNATIONAL ATOMIC ENERGY Naseem A. Bhatti AGENCY UNITED NATIONS EDUCATIONAL SCIENTIFIC AND CULTURAL ORGANIZATION 19R0 MIRAMARE-TRIESTE IC/80/162 International Atomic Energy Agency and United nations Educational Scientific and Cultural Organization INTERHATICNAL CENTRE FOR THEORETICAL PHYSICS PERCOLATION AMD SPIN GLASS TRANSITION * Abdullah Sadiq •• International Centre for Theoretical Physics, Trieste, Italy, Raza A. Tahir-Kheli Physics Department, Temple University, Philadelphia, Pennsylvania, USA, Michael Wortis Department of Physics, University of Illinois, Urbana, Illinois, USA, •and lJa3eem A.. Bhatti Pakistan Institute of Nuclear Science and Technology PINSTECH, Rawalpindi, Pakistan. MIRAMARE - TRIESTE October I960 * Submitted for publication. ** Permanent address: Pakistan Institute of Nuclear Science and Technology, PINSTECH, P.O. Nilc-re, Rawalpindi, Pakistan. Supported by HSF Grant. ••.-i-'-•••?*• .**•.*•• la raoaat y»*ur*, tk* kahavtow of ais«i4arad aacBatla, waars sia flistttsat** kas attract*4 a (raat daal «f attaatloa. Za partiaalar, Ch* qtautloM af aastral latarast ha* •••• ta« *«a«raB«« *f Abstract tte •• «*Uad nmft» glaa*^ jfcisa, «har« «s**l •agswtla loag raaga «r4ar Tha behaviour of clusters of eurvad and normal plmqu«tt« U «BMK« »»t « sin Mttl* aKwiagt Mtwltln t« tto particles In * bond random, ^ J, Ising aodal is studlad la finite •• alb«tt raatfaaOj, ta a«ys«t«4 t* aria* \_1 "\ . square and triangular latticas. Coaputar results for tba concantra- tloa of antifarrooagnetlc bonds when parcolat'ng cluatars first appeara ara found to b« clos* to thoaa raportad for tl« ooenranea and dl6app*arano* of spin glaats phaaas in tha as systsss. aal* t* tdantlfy, at l*aa* la tv* U flalto Uttt**» ta* **a**ie«r *f ta* a*** as) «*U a* awasl. *Xa*a**rt** (1.*, tk*ir a*—itat** fav*i«lM aa4 ta*lr *l««t«iM. N« BMP* Itaal Mat th* AiaappMMtt* *f fW**latia« *laat*M •» a*real *la*wtt* farticlM a*la«tda* «lta ta* a**t "ralU»l*» aatftaat** lav «a* mumammmm *t ta* apta gla** trasaltloa. Taia 1*» BMWVMF* a |Mr*]y g**a*trl*al *ff**%, aai At* r*l*vaac* t* llalta* *trl*tly t* ta* ff • 0 «*** ter ta* Jta—iaa*. tawmrtaAa**, w* tollav* tkat ta* Mta*l*«M* (wlta tk* a«| • 1 lw—lal *r»*rt a* it wn) *f ta» para«Utl*a tar**a*l« *f a«*c a1***j»tta fartlalas **4 ta* astiaataa far ta* apia gt*aa It tkat alta**** tk* *k*l** mt tk* -2- -1- far tka aalrlaas af a«rva4 ylaqvattaa kaltastaf *• dlffaramt elaatara U —* Mattlra toa4a la a»aa aatlraly ntaoaaly, aaajvct. aay, ta tea ta tka Laalataa ai««la wf* ala^aattaa altutaa br froa ottor tka 41atrlk*tl«s fwaatlaa. aluatara. It la tkla aoatrltotlaa. wklek appaara ta glva rlaa ta Qa Baika4 ., aaaartara af tto total alalaal atrlac laagtk varaM tka o«rr«4 pla^aatta •tratiaa froa taa farraaaaatla aafaat aaargy at 0 •.Of la Taalaw- tto ayatlal «latrlto*la* af tto aw-rad (aaraal) plaqaattaa IS aat «meo- rralata*. Alttoajk t&la oarralatlaa cam aa ooaaldarat ta to «Mk, tka Va tova, tkarafara, alaa ate*!** tka arodabllltj, t,<<), »f ••tad. atractvr* «f tkla earralatloa. la, aarartkalaM, far froa alaala. aoaalataly Iaalata4 ala«la taalata« alaatara af twvad pUqaatta partlala*. , it la la • MM* tto arc af tlw «p±a |Un araklM ClaarOy. «,(«) la g&vas ajr tto faUaV.ag axpraaala* If «a aoaaldar ttoaa alacXa pOaa«wtta alaatara wkaaa •>«aa< aa4 tartkar aalgka««ra My ** "9 k* fin* te a«4r«a* thl» 1MM fc] . 7M "«trla«a" »ff Ikm «r« tw* ««»trifc«tio*a t« *M- ««• to thi iatr»-«*utn' tM total atria* l«a«tk d« t* tM latar-elmat«r 1« frofatloaal t* tha ntwaar af placpMttaa la th« almatar. i-C ••a to tka waakawaa af tka Imtair-plaquatta eorralatlasa [%J , th* larga jlaaaartta alaatara ara af lrragalar ahapa. Thla aaaaa tkat aaaa ol«atar B»J aara ta to taka* t«r<n^ tha lat«rr»lnc aoraal pl»qn»tt«a (Wf.1). I*r flalta lattlaaa «•• kav* a<Mpat«4 tto fall atakaklllty ^(a) far Ala* toaa«M af tka «aak a«rr«l»tiona tha av«r«c« «l«at«r tonaiai7 far f aaaglatalf laalata* alaatara af alagl* awrwd ylaaMtt* partlalaa* larat aimatara, la alaa yrofortlanal to tb* aTarkjffet «l«at*r aiaa [h tl . t aaaartara af tto aaafatar raaalta froa tto aarraa glvam Vy aaa. (1) Vkaa tka lattra>al«star WBtriautloa ta th* ntalaal tat«l atrlai l«agt» 1* (5) la aasaMtlv* af TaalMM oto*r*atl«a «M (aoMukat la tka MM toataaiiy af tto wr«ar af tka total inuibcr of ewrraa *laq>«ttaa la tka aylrlt) tmg to aKaata< ta to tka kai*la«*r af aala (laaa CfStaat i.a. fioaantlaaal ta the aarrad ylaqutta fartiela ooMastntlaa V •aajlaaat aoatrlWtloa ta tto aialaal total atrlag lan«tk «oala« -X- -3- On> latar«*t k*r* 1* Ht M mk tk* *a»ly»l» *f f^*) as tk* 3a* altvatlMt 1* a«r* «oa)U«at*a la tk* ta** *f tk* ataflt •swdaatlaa •* the p«r«*l*tla*: •kara*t«ri*ti*a «f tk* •«*¥•* (aaraal) triaaajalar lattlc*. **!• syatwa 1* *ff *r*«lal lat*r*at •*•***• k«r« tk* »laa**tt* partial* *l*at*rs. 0<nMiva*l», £,(*) M; r*fl*«t tt* ayp*ar- f*rr*.aatlf*rr* g*af* i»«P*lo» ayvactry (wUah Is tk* kail aa** *f aat* *f tk*M p*r**latl*f elaat*ra la MM way. Ok* say «*M*1T* tk* ayst*aa *a kyp*r*wM« lattlt**) i**a not *ktala, Avmrdlv^y tk* • « atat* *f tk* Xal*c Mo4*l witt fl.MtwtU| txtkoi* lat*r**tioa (all B*aa* aaU«*rMBja*a*tl*) Is fally rrutr*t*4 (FT), lite tt* tkrt« «lff»r**t tyt*s *f traaaltloas M *a* varies tt* (las* tkla IT *yst«*i kjas a klajkly «*c*a*r«t* gmaa *tat* aaa 1 fin wnwHi) Wat *iai*atwttl»a *. Tk*M ar*t 1) tt* Malt* aatr*wy at I * •, *«as*«ttratXy It «**a a*t kav* aay l*ag traaaltloa UMtltt«4 with tk* *«ia «r*j*riag, 11) tk* *lt* (pl*«Mtt* aai aa vkaaa traaattl** oaewra at ff • • ftJ . Vk* Idmtlty •£ f • • fartlala) p*«Ml>tl«* aaa Ul) «k* ^M« y*r««l*ti9«. tor* •p«clfl<ull7| m wltt tt* saaa gl«*a Is, kmtrw, a«t fklly **ta%Uak*a aai a ^aordratl* Xalag i^ttw tt I • 0 tf f»rr«wi>o*t l»*lly trterac f** • • aa**ti*aa riMla aa t* wkattw tk* IT stat* la tt* trlaafolar Xala« l**lly «Hn«d fiw « • 1, vltk tk* «pl« aatwara la tk* staaaav* afla glass ***** [»J . Mr •< t.9 til ttla •***, t«* w* ass*«lat*a a* la tt* rnn, wttk tk* *f «wr«ratl* Xattl** !>*> tor* t* ajia flaa* traa*lti*a witt tt* at MM TBlW* *f •• *M> n«U «xp*«t *lt* p«wla> traaaltl*a Maa**to« wltk tk* aisan*araaM *f tk* an Ml *la«Mtt* lattait* «l««t«T *f Mml *f aifiii yl«4*a«t* y«rti*lM vltk lalm. Atar* « • tw9* *««»«, tt*r* is aa*tkCT p*r««latl*a traaaltlwi tartl«l*a <l*a»»Mrla« »t *•»» *f • aa« rwppMrlac *&•••*» **sa**t«« witt tk* tn«tnM» *t laflalt* alm*t«r of nml pla^watt* *f •« «f *>>••!?. tar** j**»* partl«l*s. W* lat«nr*t tkla a* aa lailoatloB (ak*lt( «aMrr*b«rat*a *r («hl*k la a M«*mr* •* tk* frutratlaa *a«l la tk* ay*t*a) aaa tt* *yla *tk*r warkars aa *ff ttl* nwit) tkat tk* TW jaa*« Is, la MM HIM •JUaa tr»a*ltl»a MM t* k* r*lat*« t* «a*k *tk*r. «*, tkar*fo»{ pactmlat* 41ff*r*at f**a tt* Sfia gUas paas*. tkat ttm Mall valsM *t • tk* traMltiM froa tk* ft«i*Mca»ti« «aaa* I* tt* *k* alia glaa* yhaM ««««r* at tt* rwttUtloa «kr**k«U aMo*l>t*c with Xk* rasalta *X *w *al*«UtloB* ar* akeva la flfs. U?. Xa flf.t tk* UMpptmn** *f tk* laflalt* eUitw *f Mnal alaqtwtt* parti*!**. At w* *k*w tt* aaalytiMJl (MU« UM) aaa tk* ooapvt.r r**«lt* (palata) for, kl^M* nlw« *f • tk* traultioa firoa tt* apla c^aaa jama* t* 1»« aatlt*r**- #,(•) tk* pntakllltjr *f t*tti>( a «1B«1* la«Ut*a «arv*« (a*rsal> ala<a*- M«Mtl* jjb*— (la th* qoaaratla Uttl**) **lael4*a «itt ta* p*r**UtloB vtt*. aa a fuwtlaa *f tk* tvnl ]tla«m^t* *M**atratl*a. «• sat* tkat tkr*sk*l« asao*lat«4 with th* naniirwi •# tk* laflalt* *lwt«r *f tt* *«9**«r wsaata **part froa tt* aMlrtl*al mwrr* at fl 0.30. Xa flf.5 ffU«ft«tt« partial*** -5- -6- «• g&V* tka resalt* far tka redwed weight average elveter else (HAS) ACKNOWLEDGMEMTS tar tk« trtaagaiar lattice. Ik* ap*a ayanbolfanhcla (for Ot 9.5©.J)) *rar«e f«fa*r tkea One of the authors (A.S.) would like to acknowledge helpful aa* tka dax* (for C > 0.5) are for tha e«rre* discussions with Dr. I. Vilfan. He would also like to thank Professor mla«*etta*. 3a* aarraspoaaiag remits for bath £,<«) aa« WAC (far aai Abdus Salsm, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste. ml**a*ttaa) t*» tka q.«are>atie lattice are shorn In flf.6.