IC/80/162

INTERNATIONAL CENTRE FOR THEORETICAL

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

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*

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-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

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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«ral«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«

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-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. 0tl«* aaa Ul) «k* ^M« y*r««l*ti9«. tor* •p«clflo*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 >••!?. 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

-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. la tkla oaaa The help and cooperation of the centro di Calcolo, University of Trieste, tar raa»ata *, 4a aat axklbit aay algalfleam* de**rt«re froa the and the Computer Division, PINSTECH,are gratefully acknowledged. amrra, altkaagk aoaw klad of atrwatare la theae reamlta aaar tka ?«re«l*tloa tkraakoU la avlAaat. Finally flg.7 illwatratea tka extra- yalatlaa of tha yeraalatlaa tkraakald froa: flalte lattices to iaflalta lattloe.

far tka aaaara lattlsa wa gat p«rcolatloa thrtehold at o - 1* ^ 2 (•at • • M *. 1) Vhlsa lias katwaaa tke raloes given by Tajutiaessa am* t[* J , artaaataia at^l. f 10J and that of Xirkaatrlk ft)J . Far atsflt trlaagalar lattlaa wa gat paroalatlena thraahold at a • 17 £ 20 aaa«3l»

9a tancl«4«, o«r •tvdy af *orr«l»tad parcolatloK of enrrad tt»a la a r*ado« IsUkj a«4al la twa dlaaaatoma) ahow arldasa* of a pfcaaa tranaltloa at aatlf«rraaiayiatla aoaA eonca«%rfttlona «Mak, for aqnara lattice loliu^da wltk tka Mat ralUbla aatiaatea glvaa bj othar aatkor* for tha fm«n|wtl« to mpLm «Uaa tmnslttw C2,1O.1.ijf. . Ikr tha ai»flt triangular Inttlo* o«r reamlta alao lndioata ttut tka FT pbasa may, la aoa a«ttaa, ba Uffrrwat froai tka apia glaaa phaaa.

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-8- CAfWO Bafaraaaaa flg.1 Cl«atara *f «urr«d pl»qm«ttaa i» • fmatratad lalnc »oi*l «• 1. B.r. Bdwards and P.W. Anderson, J. Phya. 15, 965 (1975) 80 s 80 trlamgular lattlea at Ht }( ajttl»farr««a(»«tia aond aoaeaatrattam. D. Sharrlnctoa and 8. Klrkpatriek, Phyt. Uav. Utt. 2£,1992 (197?) Tlga. 2-3 Hffarakt poaslola wtya af f*ttin« a ewrvad ylaquatta (hori>*B>> 2. a. foalous*, COT. Phys. £, 11? (1977) J. Y*uiMm tri a. Toalo*a«, tal k«t«ka« r«giaaa) with aovwil pU^MttM (Tarttaal Jut«ha« J. 1'kjv. £1£, 1*37 (1977) B. Fradkia a*4 3.*. Muwberatn an* ragloaa) la a, (2) trl»»gal»r «W (J) a^vaura ftm*t»«t*d lalsa; acdal. Th» solid linaa ( '••— i doaata farroa»KMtle e.i. shuktr *hy*. B«T. n8. 4789 (1978). a*4 tha 3. P. B**d AW. Brmy ud H.A. Moor* J. Phya. £11^ 1159 (1978). Ft..* Xha prabaMllty of gattios «» laolatad e«rvad (apaa ajaaola) aad aenal (filial «yaa*la) aU A. Afcarmr a&l K. Bl»a>rt"I«w CoMavtrutloa atrinm, Eipanaion fcr •ljta* a« a« x *O (triaa«laa) u««)i«t (•**•»•) aai 9» s 10 Mlvta •>!« SlMSta at ft«r* X««p«nit«ra", KF*t Juliet (19«O) (elr«l«a) trla«g«lar Iattlaaa« Tha Tarlaaa nrm aia Armra «• t. 0.1. Vaaalar, Fky*. Bar. fl, 397 (1990) galda th* aja.

9* >• i«klek, J.S. WaU:*r and N. tforbia Phy«. K.T. B1&, 2209 (1977). rmha.%111%7 ta ham laal«tad a«rrad pU^wattaa. ^ (flllad aymaala) Hat tha r*d«aad valfhtad aTaraga alvatar alas, •**<» lh«a« anUtwrs atrt^y th« o«iEit«tie flald ajai tsaparatara dapa&daaea of (apaa ajafctla) a* a fasatiaa af «WM« ala^aatta ««>aaatratlaa

ta» Tf trtanpnlar X«jng Eyst*».e. Alexander and P. Flncua, J 0y far lalas apim |lui aa to it to (trlaaai**), <• x Co aa« 80 a 10 (etralaa) a, la tk« aMOytiaal raawlt ta lawaat arday, walla ta«\rok*a llaaa 10. 9. GritaUia, C. J«yayr«a*flk u4 M. Wortle, thj». Ser. 31^, 260 ajr« crai* I* fti4* tk* cy*. (1979). Varaelatlaat thraakftlda 1% mt aatl-farraaagsatla a«mda) far 11. S. Eirkffttrlak Mgr»> *•*• H^, ^30 (-.977). trvatrata4 lalna; aedal aa aqmura (a^vauraa) *•< trlnamlar (trlaaglM) lattlaaa aa a faaotloa af tka i*v«raa Uttlc* alaa.

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l£> o 00 lO - I" CM o U_ o o 1x9 o o CD 3 ' uoijDijuapuoo puoq o o o o - AS - — CUHHEHT ICTF Ttj AMU IDTJilUIAb IC/80/55 A.R. HASSAN: Fhonon-asaiated transitions in crossed electric and magnetio IC/80/2 A.P. BUKHYOSTOV and L.H. LIPATOV: Instanton-anti-instanton interaction in fields. the 0{3) non-linear o-model and an exactly soluble fermion theory. IC/80/56 A.R. HASSAN! Two-photon Indirect traositiona in crossed electric and IC/80/3 RIAZUDDIB: K •* 2ir decays in non-relativistic quark-gluon model. magnetic fields. IC/80A G. SEHATORE and M.P. TOSI; Theory of the surface dipole layer and of IC/8O/5T S. HARISOH: Techniques of dimensional renormalization and application to surface tension in liquids of charged particles. the tvo-point functions of QCD and QED. IC/60/15 S. RAI CHOUDHURY: QCD effects in a model of non-leptonic hyperon decays. IC/80/62 N.S. CRAIGIET Quantum chromodynamics - A theory of the nuclear force. IC/80/21 HAMIK K, PAK: Electric charge as the source of CP violation. IC/8O/6T WITHDRAWN IC/80/22 T. HATTERMAUN ami J. PRZYSTAWA: Locking-in and incommensurability of IC/80/68 A. AYI31SU: Dynamic dislocations in high quartz. the structural transition in BaMnFj,. '• IC/80/69 T. PERSI: Time dispersion relations and small-time behaviour in the decay IC/80/29 E.W. HIELKE: On pseudoparticle solutions in Yang's theory of gravity. IHT.REP.* of an unstable system. IC/80/32 and VICTOR ELIAS: Induced Higgs couplings and spontaneous IC/80/70 M.F. KOTKATA, A.A. EL-ELA, E.A. HAHMOUD and M.K. E1-H0USLY: Electrical symmetry breaking. Tlransport and structural properties of Se-Te semiconductors. IC/60/33 A. AMUSA: Point-triton analysis of exchange free direct tvo-nueleon transfer IC/80/Ti A.Yu. GROSBERG and A.R. KHOKHLOV: Some protleos of the statistical theory of IHT.REP.* reaction cross-sections. IHT.REP.* polymeric lyotropic liquid crystals. IC/80/72 J'.C. PATI and ABDUS SALAM: • Quark-lepton unification and proton decay. IC/80/73 ASHOK DAS; Are massless supersymmetric gauge theories really masslessT IC/80/7't 0. MAIELLA,: Chiral rotations and the fension-boson equivalence in the HIT,REP.• Schwinger model. IC/80/38 WITHDRAW! IC/80/75 J. SPAIEK and K.A. CHAO: Kinetic exchange interaction in a doubly degenerate Ca narrow band and its applications to Fe. x0o Sg and i-J^^-^2' IC/60/39 ABDUS SALAM: The nature of the "ultimate" explanation in physics (Herbert Spencer Lecture - 12 November 1979). IC/8O/76 L.K. SHAYOs The generalized pressures on oscillating cantilever pipe* : IHT.REP.* conveying inviscid fluid. IC/80/77 E. GAVA, H. JENGO and C. OMERO: Finite temperature approach to confinement. IC/9O/U3 V. de ALFARO, S. FUBINI and G. FURLAN: Classical solutions and extended IC/80/78 A. AYENSU: The dependence of mechanical, electrical, thermal and acoustic •wpergravity. IHT.REP.* properties of tropical hardwood on moisture content.

IC/80AT J.C. PATI: Unity behind diversity in Nature. IC/80/79 J.S. NKOMA: Effect of impurities on the tvo-dlmensional electron gas polarisability. IC/9OA8 J. LUKTERSKI: Supersymmetric a models and composite Yang-Mills theory. IC/ao/80 A.M. JAYAHHAWAR and N. KUMAP.; Orbital dlamagnetlBm of a charged Brovnlto IC/B0/U9 H. KROLIKCWSKI: Are lepton sod quark families quantized dynamical systemsT particle undergoing birth-death process. IC/fiO/50 W. DEPPERT: The role of interaction in neutrino statistics. IC/80/81 L.K, SHAYO: On the solution of the Uplace equation in the presence of a INT.REP.* semi-infinite boundary - The case of an oscillating cantilever plate in uniform incompressible flow. IC/80/82 E.R. HISSIMOV: Higher qvtantua conserved current in a new completely IC/80/52 D. KUSKO and H.J. MORAVCSIK: On the problsi of the deuteron shearing IHT.REP.* integrable model. corrections - I: The conventional approach. E.W. MELKE: The eight-fold vay to colour geometrodynamics.

• Internal Reports: United distribution. THESE PREPRINTS ARE AVAILABLE FROM THE PUBLICATIONS OFFICE, ICTP, P.O. BOX 586, IC/60/85 Tmo RUIBAO and Pu FUCHO: A proof of the absence of nodulated phase transition and' spin density wave phase transition in one- and two-dimensional systeas. 1-3*100 TRIESTE, ITALY. IC/80/86 Pu FUCHO: Density of states, Foisson's formula of summation and Walfisz's INT.REP.' relativistic field theories. Err.REP.• romuia.

K/80/87 A.K. BAHDWPAEHYAY, S.K. CKATTEHJEE, S.V. SUBRAMA1TYAM and B.H. BULKA: IHT.REF.* Electrical resistivity study of some organic charge transfer complexes under IC/80/lll A. BREZIHI and 0. OLIVIER: Localization on veakly disordered Cayley tree. pressure. IHT.REP.*

IC/80/86 In-Gvu KOH and Yongduk KIM: Global quantity for dyons with various charge IC/80/112 S.K. OH; The BJorken-Paschos relation in the unified gauge theory. HIT.REP.• distributions. IC/80/113 A.A. FADLALLA: On a boundary value problem in a strongly pseudo-convex IC/8O/S9 K.G. AKDENI7., M. GOODMAN and R. PERCACCI: Moncpoles and twisted sigma models. 1ST. REP. • domain. IC/Bo/90 A.R. HASEAH and F. Abu-ALALLA: Exciton-polarizations by two-photon absorption nrr-REP.'11 In semiconductors. IC/80/91 A.R. HASSAH: Phonon-polaritona by tvo-photon absorption in solids. jjix, REP. * IC/80/92 L. LONGA and J. KOHIOR: Exact results for some 2D-Ising models with IHT.REP.* periodically distributed impurities.

IC/80/117 P. FA2EKAS: Laser induced switching phenomena in amorphous ^ IC/80M ABDUS SALAH and J. STRATHUEE: On Witten's charge formula. IHT.REP.* phase transition model.

2 IC/80/95 ABDUS SALAM and J. STRATHDEE: Dynamical mass generation in (\ti) * UR(1)) . IC/80/118 P. BUDINI: On spinor geometry: A genesis of extended supersymmetry. IHT.REF.* IC/6O/LL9 G. CAMPAGNOLI and E. TOSATTI: AsF5-intercalated graphite: Self-conslaten IC/BO/96 H.8. CRAIGIE, f. BALDRACCHINI, V. ROBERTO and M. SOCOLOVEKY: Study of band structure, optical properties and structural energy. factorization in QCD with polarized beams and A production at large p^. IC/80/120 Soe Yin and E. TOSATTI: Core level shifts in group IV semiconductors and semiiEetalB. lC/80/97 K.K. SINGH: Renormalization group and the ideal Bose gas. lb/80/121 D.K. CHATUSVEDI, G. SENATORS and M.P. TOSI: Structure of liquid alkali, IC/8O/96 NARESH DADHICH: The Penrose process of energy extraction in electrodynamics. INT.REP.* metalB as electron-ion plasma. IHT.REP.* IC/80/126 The Seventh Trieste Conference on , 30 June - U July I9S0 '. IC/Bo/99 G. MUKHOPADHYAY and S. LUNDQVIST: Dynamical polarizability of atoms. IHT.REP.• (Contributions) - Part I. ( IHT.REP.* IC/8O/15T The Seventh Trieste Conference on Particle Physics, 30 June - U July I98O IC/8O/IOO A. QADIR and A.A. MUFTI: Do stars disprove multiplicative creation IHT.REP.* (Contributions)- Part II. IHT.REP.* in Dirac's large number hypothesis? IC/80/126 FARID A. KHWAJA: Temperature dependence of the short-range order parameter IC.80/101 V.V. MOLOTKOV and I.T. T0DOR0V: Gauge dependence of world lines and invariance and the concentration dependence of the order-disorder temperature for Hi-P IHT.REP.* of the S-matrix in relativistic classical mechanics. and Ni-Fe systens in the improved statistical pseudopotential approximation IC/8O/IO2 Kh.I. PUSHKAROV: Solitary excitations in one-dimensional ferromagnet3 at IC/80/130 A. QADIR: Dirac's large number hypothesis and the red shifts of distant IHT.REP.• T J< 0. DPT.REP.* galaxies. IC/80/103 Kh.I. PUSKHAROV and D.I. PUSHKAROV: Solitary clusters in one-dimensional' IC/80/131 S. HOBASZKIEWICZ, R. MICNAS and K.A. CHAO: Ground-8t»te phase diagram of IHT.REP.* ferromagnet. INT.REP.* extended attractive Hubbard model. IC/oG/lQll V.V. MOLOTKOV; Equivalence between representations of conforms! superalgebra from different subgroups. Invariant subspaces. IC/80/133 K.K. SINGH: Landau theory and Cinzburg criterion for Interacting bosons. 1 IC/80/105 MENDEL SACHS: On proton mass doublet from general relativity. IC/80/13 ! R.P. HAZOUME: Reconstruction of the molecular distribution functions from t IHT.REP.* Bite-site diotribution functions in classical molecular fluids at equilibri'• IC/60/106 L.C. PAPALOUCAS: Uncertainty relations and semi-groups in B-algebras. IC/80/135 R.P. HAZOUME: A theory of the nematic liquid crystala. IHT.REP.* IHT.REP.* IC/80/136 K.O. AKDESIZ and M. HORTACSO: Functional determinant for the Thirring mode INT.REP.• yith instanton A