Griffiths-McCoy singularities in the random transverse-field Ising spin chain

PHYSICAL REVIEW B VOLUME 59, NUMBER 17 1 MAY 1999-I Griffiths-McCoy singularitis in th random transvrs-fild Ising spin chain Frnc Iglói Rsarch Institut for Solid Stat Physics and Optics, P.O. Box 49, H-1525 Budapst, Hungary and Institut for Thortical Physics, Szgd Univrsity, H-6720 Szgd, Hungary Róbrt Juhász Institut for Thortical Physics, Szgd Univrsity, H-6720 Szgd, Hungary and Rsarch Institut for Solid Stat Physics and Optics, P.O. Box 49, H-1525 Budapst, Hungary Hiko Rigr Institut für Thortisch Physik, Univrsität zu Köln, 50923 Köln, Grmany and NIC c/o Forschungszntrum Jülich, 52425 Jülich, Grmany Rcivd 30 Novmbr 1998 W considr th paramagntic phas of th random transvrs-fild Ising spin chain and study th dynamical proprtis by numrical mthods and scaling considrations. W xtnd our prvious work Phys. Rv. B 57, 11 404 1998 to nw quantitis, such as th nonlinar suscptibility, highr xcitations, and th nrgydnsity autocorrlation function. W show that in th Griffiths phas all th abov quantitis xhibit powr-law singularitis and th corrsponding critical xponnts, which vary with th distanc from th critical point, can b rlatd to th dynamical xponnt z, th lattr bing th positiv root of (J/h) 1/z av 1. Particularly, whras th avrag spin autocorrlation function in imaginary tim dcays as G av () 1/z, th avrag nrgy-dnsity autocorrlations dcay with anothr xponnt as G av () 21/z. S0163-18299904217-4 I. INTRODUCTION Quantum phas transitions occur at zro tmpratur by varying a paramtr of th Hamiltonian,.g., th strngth of a transvrs fild. Qunchd, i.., tim-indpndnt disordr, has gnrally a profound ffct on th proprtis of th quantum systm not only at th critical point, but also in a whol rgion, which xtnds in both sids of th critical point. In this so-calld Griffiths phas th dynamical proprtis of th random quantum systms ar xcptional: for xampl, th imaginary tim-dpndnt avrag spin-spin corrlations dcay algbraically 1 G av 1/z, 1.1 whr th dynamical xponnt z() is a continuous function of th quantum control paramtr. From hr on w us av to dnot avraging ovr qunchd disordr. Th physical origin of this typ of singular bhavior, as was pointd out by Griffiths 2 for classical systms, is th xistnc of clustrs in th random systm, which ar mor strongly coupld than th avrag. Th spins of such clustrs, bing locally in th ordrd phas, bhav cohrntly as a giant spin and th corrsponding rlaxation tim is vry larg. Thus in an infinit systm thr is no finit tim scal and, as a consqunc, th autocorrlations dcay algbraically, as in Eq. 1.1. 3 Svral physical quantitis, which involv an intgral of th autocorrlation function.g., th static suscptibility ar singular not only at th critical point but also in a finit rgion of th paramagntic phas. This phnomnon was first noticd by McCoy in a two-dimnsional classical modl with corrlatd disordr quivalnt to a on-dimnsional random quantum modl 4 thrfor w call th Griffiths singularitis in quantum systms Griffiths-McCoy singularitis. Many of th thortical studis on random quantum systms ar rlatd to random quantum frromagnts 5 and quantum spin glasss, 6 which also hav xprimntal ralizations. 7 In highr (d2 and d3) dimnsions on gnrally studis th distribution of th linar and nonlinar suscptibilitis, th asymptotic bhavior of thos can b rlatd to th dynamical xponnt z() by scaling considrations. According to numrical studis in agrmnt with ths phnomnological thoris z() is found as a continuous function of th quantum control paramtr, which appars to hav a finit limiting valu at th critical point 0 of spin glasss, 6 whras it is divrging for random frromagnts. 5 Many faturs of Griffiths-McCoy singularitis can alrady b sn in on-dimnsional systms, whr many xact and conjcturd rsults xist. In this papr w considr th prototyp of random quantum systms, th random transvrs-fild Ising modl RTIM in on dimnsion, dfind by th Hamiltonian H l J l x x l l1 h l z l. 1.2 l Hr l x, l z ar Pauli matrics at sit l and th J l xchang couplings and th h l transvrs-filds ar random variabls with distributions (J) and (h), rspctivly. Not that in on dimnsion all th couplings and filds can b takn positiv through a gaug transformation. Th modl in Eq. 1.2 is in th frromagntic paramagntic phas if th couplings 0163-1829/99/5917/113087/$15.00 PRB 59 11 308 1999 Th Amrican Physical Socity

PRB 59 GRIFFITHS-McCOY SINGULARITIES IN THE RANDOM... 11 309 in avrag ar strongr wakr than th transvrs filds. As a convnint quantum control-paramtr on can dfin ln h avln J av varln hvarln J, 1.3 whr varx dnots th varianc and at th critical point 0. Th Hamiltonian in Eq. 1.2 is closly rlatd to th transfr matrix of a classical two-dimnsional layrd Ising modl, which was first introducd and partially solvd by McCoy and Wu. 8 Latr th critical proprtis of th quantum modl was studid by Shankar and Murthy, 9 and in grat dtail by Fishr. 10 Through a rnormalization group RG transformation Fishr has obtaind many nw rsults on static quantitis and qual tim corrlations, which ar claimd to b xact for larg scals, i.., in th vicinity of th critical point. Many of Fishr s rsults hav bn chckd numrically 11 and in addition nw rsults hav bn obtaind about critical dnsity profils, 12 tim-dpndnt critical corrlations, 13 and various probability distributions and scaling functions. 11,14 Latr, using simpl xprssions about th surfac magntization and th nrgy gap svral xact rsults hav bn drivd by making us of a mathmatical analogy with surviving random walks, 14 s also Rf. 15. In th Griffiths phas, whr th RG rsults ar rstrictd to th immdiat vicinity of th critical point, i.., as 0, numrical invstigations both on tmpratur-dpndnt 16 spcific hat, suscptibility and dynamical quantitis spinspin autocorrlations, distribution of th nrgy gap and suscptibility 14,11 hav lad to th conclusion that th bhavior of all ths quantitis is a consqunc of Griffiths- McCoy singularitis and can b charactrizd by a singl varying xponnt z() in Eq. 1.1. Vry rcntly an analytical xprssion for z() has bn drivd 17 by using an xact mapping 18 btwn th Hamiltonian in Eq. 1.2 and th Fokkr-Planck oprator of a random walk in a random nvironmnt. Th dynamical xponnt, which is givn by th positiv root of th quation h J 1/z 1.4 av1, gnrally dpnds both on and on th distributions (J) and (J). Howvr it bcoms univrsal, i.., distribution indpndnt, in th vicinity of th critical point whn z() 1/(2), 1, in accordanc with th RG rsults. 10 Th numrical rsults obtaind about diffrnt singular quantitis in th Griffiths phas ar all in agrmnt with th analytical formula in Eq. 1.4 and th obsrvd small dviations ar attributd to finit-siz corrctions. 16,14 Th singular quantitis studid so far in th Griffiths phas ar all rlatd to th scaling proprtis of th lowstnrgy gap, which xplains th obsrvation that a singl varying xponnt is sufficint to charactriz th singularitis of th diffrnt quantitis. Thr ar, howvr, othr obsrvabls, which ar xpctd to b singular too, but not connctd dirctly to th first gap. For xampl, on could considr th distribution of th scond or som highr gap. For similar rasons as for th first gap ths highr xcitations ar also xpctd to vanish in th thrmodynamic limit and th corrsponding probability distributions ar dscribd by nw xponnts for small valus of th gaps. As anothr xampl w considr th connctd transvrs spin autocorrlation function G l () l z (0) l z (). In th twodimnsional classical vrsion of Eq. 1.2, th McCoy-Wu modl, this function corrsponds to th nrgy-dnsity corrlation function in th dirction whr th disordr is corrlatd. Thrfor w adopt in th following this trminology and call G l () th nrgy-dnsity autocorrlation function. Sinc th invrs tim scal for ths corrlations is, as w shall s, dtrmind by th scond gap, on xpcts that also G av () has an algbraic dcay: G av, 1.5 with an xponnt. Finally on should mntion that th nonlinar suscptibility s distribution is xpctd to b dscribd by a nw varying xponnt. In this papr w xtnd prvious numrical work and study th scaling bhavior of th abovmntiond singular quantitis in th Griffiths phas. W prsnt a phnomnological scaling thory and w confront its prdictions with rsults of numrical calculations, basd on th fr-frmion rprsntation of th Hamiltonian in Eq. 1.2. W show that th physical quantitis w studid ar charactrizd by powr-law singularitis with varying critical xponnts, whos valus ar connctd to th dynamical xponnt through scaling rlations. Throughout th papr w us two typs of random distributions. In th symmtric binary distribution th couplings could tak two valus 1 and 1/ with th sam probability, whil th transvrs-fild is constant: J 1 2 JJ1, hhh 0. 1.6 At th critical point h 0 1, whras in th Griffiths phas, 1h 0, th dynamical xponnt from Eq. 1.4 is dtrmind by th quation h 0 1/z cosh ln z. 1.7 In th uniform distribution both th couplings and th filds hav rctangular distributions: 1 for 0J1, J 0, othrwis, h h 0 1, for 0hh 0, 0, othrwis. 1.8 Th critical point is also at h 0 1, whras th dynamical xponnt is givn by th solution of th quation z ln1z 2 ln h 0, 1.9 whr th Griffiths phas now xtnds to 1h 0. Th structur of th papr is as follows. In Sc. II w prsnt th fr frmion dscription of various dynamical quantitis. Phnomnological and scaling considrations ar

11 310 FERENC IGLÓI, RÓBERT JUHÁSZ, AND HEIKO RIEGER PRB 59 givn in Sc. III and th numrical rsults ar prsntd in Sc. IV. Finally, w clos th papr with a discussion. II. FREE FERMION DESCRIPTION OF DYNAMICAL QUANTITIES W considr th random transvrs-fild Ising modl in Eq. 1.2 on a finit chain of lngth L with fr boundary conditions. Th Hamiltonian in Eq. 1.2 is mappd through a Jordan-Wignr transformation and th following canonical transformation 19 into a fr frmion modl: L H q1 q q q 1 2 2.1 in trms of th q ( q ) frmion cration annihilation oprators. Th nrgy of mods q is obtaind through th solution of an ignvalu problm, which ncssitats th diagonalization of a 2L2L tridiagonal matrix with nonvanishing matrix lmnts T 2i1,2i T 2i,2i1 h i, i 1,2,...,L and T 2i,2i1 T 2i1,2i J i, i1,2,...,l1, and dnot th componnts of th ignvctors V q as V q (2i1) q (i) and V q (2i) q (i), i1,2,...,l, i.., 0 h1 h 1 0 J 1 0 J T 1 0 h 2 h 2 0 J L1 J L1 0 h L h L 0, q q 2 V q q1 q L1 L. 2.2 q q W considr only th q 0 part of th spctrum. 20 Th local suscptibility l at sit l is dfind through th local magntization m l as m l l lim, H l H l 0 2.3 whr H l is th strngth of th local longitudinal fild, which ntrs th Hamiltonian 1.2 via an additional trm H l l x. l can b xprssd as l 2 i i l x 0 2 E i E 0, 2.4 whr 0 and i dnot th ground stat and th ith xcitd stat of H in Eq. 1.2 with nrgis E 0 and E i, rspctivly. For boundary spins on has th simpl xprssion 1 2 q q 1 2 q. 2.5 Similarly, th local nonlinar suscptibility is dfind by and can b xprssd as nl 3 m l l lim 3 H l 0 H l nl l 24 i, 0 x 1 l i i x j,k E i E l j 0 2.6 1 j x 1 E j E l k k x 0 E k E l 0 0 i i l x 2 0 j E i E 0 x l 0 2 j E j E 0. 2.7 It should b notd that it is not th first sum on th right-hand sid RHS of Eq. 2.7 that givs th lading contribution, sinc at last on of th thr nrgy diffrncs most involv a highr xcitation (i l x j0 for i j). For surfac spins l1, Eq. 2.7 simplifis to l nl 24 p,q p 1 2 q 1 2 p q p p p1 p 1 p 1 q 2 q 1 2 q q. 2.8 Nxt w considr th nrgy-dnsity corrlation function at sit l, G l, dfind by G l 0 l z l z 000 l z 00 l z 00 i0 0 l z 0 2 xpe i E 0. In th fr-frmion rprsntation it is givn by 2.9 G l l l l l 2 xp, which can b xprssd for surfac spins as G l h 1 2.10 2 1 l xp. 2.11 Th spin-spin autocorrlation function G l which is dfind as G l in Eq. 2.9 by rplacing l z by l x, is gnrally complicatd and can b xprssd in th form of Pfaffians. 21,14 An xcption is th autocorrlation function for surfac spins, which is simply givn by G 1 q q 1 2 xp q. 2.12

PRB 59 GRIFFITHS-McCOY SINGULARITIES IN THE RANDOM... 11 311 III. PHENOMENOLOGICAL AND SCALING CONSIDERATIONS As dscribd in th Introduction th Griffiths-McCoy singularitis in th paramagntic phas ar connctd to th prsnc of strongly coupld clustrs, which ar locally in th ordrd phas and thrfor th corrsponding xcitation nrgy is vry small. For th RTIM th origin of ths clustrs can b xplaind ithr through th analysis of th RG fixd-point distribution, 10 which works only in th vicinity of th critical point, or by using simpl xplicit xprssions for th xcitation nrgy 22,14 and stimat thos through random walk argumnts. 14 Hr w us a simpl phnomnological approach, 1,6,23 whos rsults ar in agrmnt with th abov microscopic mthods. Considr th quantity P L (N) which masurs th probability that in a chain of L sits thr is a clustr of NL strongly coupld spins. Sinc N conscutiv strong bonds can b found with xponntially small probability xp(an), whras th clustr could b placd at L diffrnt sits w hav P L NL xpan. 3.1 Th xcitation nrgy of this sampl corrsponds to th nrgy ndd to flip all spins in th clustr, which is xponntially small in N: 1 xpbn. 3.2 Combining Eq. 3.1 with Eq. 3.2 w hav for th probability distribution of th first gap P L ln 1 L 1 1/z, 3.3 for 1 0 and 1/zA/B. Hr, from th scaling combination in Eq. 3.3: L 1 1/z 1/z, w can idntify z as th dynamical xponnt. Nxt, w considr th scond gap 2 which is connctd to th xistnc of a scond strongly connctd clustr of NN spins, and its valu corrsponds to th nrgy ndd to flip all th spins in th scond clustr simultanously, consquntly, 2 xpbn. 3.4 Th probability with which a clustr of siz N occurs, providd anothr clustr of siz NN xists, is givn by L P L (N)L xp(an) NN P L (N). For NL or in th infinit systm siz limit L ) this can b stimatd as P L NL 2 xp2an. Thus from Eqs. 3.4 and 3.5 w hav with 1/z2A/B, thus P L ln 2 L 2 2 1/z, zz/2. 3.5 3.6 3.7 Not that th scaling combination on th RHS of Eq. 3.6 is dimnsionlss, as it should b. Rpating th abov argumnt for th third, or gnrally th nth gap th corrsponding distribution is dscribd by an xponnt z (n) z/n, howvr, th finit siz corrctions for ths gaps ar xpctd to incras rapidly with n. Th scaling bhavior of th probability distribution of th suscptibilitis can b obtaind by noticing that both for l and l nl th lading siz dpndnc is connctd with nrgy gaps in th numrators of Eqs. 2.4 and 2.7, rspctivly. Thn for th asymptotic bhavior of th distribution of th local suscptibility w hav lnpln l 1 z ln lconst, 3.8 similar to th invrs gap. For th nonlinar suscptibility th scond trm in th RHS of Eq. 2.7 givs th singular contribution, so that with lnpln nl l 1 z ln nl l nl const, z nl 3z, 3.9 3.10 sinc th asymptotic distribution is th sam as that of th third powr of th invrs gap. W not that th rlation in Eq. 3.10 corrsponds to th phnomnological rsult in Rf. 6. Th scaling bhavior of th avrag spin autocorrlation function is givn by G l av P L 1 M l 2 xp 1 d 1, 3.11 whr th factor with th matrix lmnt is M l 2 1/L, sinc th probability that a low-nrgy clustr is localizd at a givn sit l is invrsly proportional to th lngth of th chain. Thn using Eq. 3.3 on arrivs to th rsult in Eq. 1.1, thus stablishing th rlation btwn th dcay xponnt of th spin autocorrlation function and th dynamical xponnt. For nrgy-dnsity autocorrlations, according to Eqs. 2.10 and 2.11 th charactristic nrgy scal is 2 and th asymptotic bhavior of th avrag nrgy-dnsity autocorrlation function is givn by G l av P L 2 M l 2 xp 2 d 2. 3.12 Now w tak th xampl of th surfac autocorrlation function in Eq. 2.11 to show that th factor with th matrix lmnt M 1 2 is proportional to 2 2. Th rmaining factor in Eq. 2.11 with th first componnts of th ignvctors is xpctd to scal as 1/L du to similar rasons as for th spin autocorrlations, thus M l 2 L 1 2 2 and togthr with Eq. 3.6 on has P L ( 2 )M l 2 L 1/z1 2. Bfor valuating th intgral in Eq. 3.12 w not that for a fixd L th xprssion in Eq. 3.12 stays valid up to L z. Thrfor to obtain th L indpndnt asymptotic bhavior in w should instad vary L, so that according to Eq. 3.6 tak L 1/(2z) 2 and in this way w stay within th bordr of validity of Eq. 3.12 for any. With this modification w

11 312 FERENC IGLÓI, RÓBERT JUHÁSZ, AND HEIKO RIEGER PRB 59 FIG. 2. Th stimats for 1/z and 1/z as a function of h 0 for th uniform distribution. Ths valus and th corrsponding rror bars hav bn obtaind from our analysis of th probability distribution of ln 1 and ln 2, rspctivly, for two systm sizs as xmplifid in Fig. 1. Th full lin for 1/z corrsponds to th analytical rsult 1.4, th brokn lin corrsponds to 2/z, which w prdict to b idntical to 1/z. FIG. 1. Probability distribution of ln 1 and ln 2 for th uniform distribution at h 0 2 top and th binary distribution (4) at h 0 2.5 bottom. Th straight lins ar last squar fits to th data for th largst systm siz, thir slops corrspond to 1/z(h 0 ) and 1/z(h 0 ), rspctivly. Thy follow th prdictd rlation z(h 0 ) z(h 0 )/2. arriv to th rsult in Eq. 1.5 whr th dcay xponnt is rlatd to th dynamical xponnt as 2 1 z, 3.13 whr th rlation in Eq. 3.7 is usd. W xpct that th factor M l 2 has th sam typ of scaling bhavior for any position l, thus th rlation in Eq. 3.13 stays valid both for bulk and surfac spins. W not that th rasoning abov Eq. 3.13 applis also for th spin autocorrlation function, in which cas in Eq. 3.11, howvr, thr is no xplicit L dpndnc. In this way w hav stablishd a phnomnological scaling thory which maks a connction btwn th unconvntional xponnts in Eqs. 3.7, 3.10, and 3.13 and th dynamical xponnt. In th nxt sction w confront ths rlations with numrical rsults. distributions in Fig. 1 w hav stimatd th 1/z and 1/z xponnts for th two largst finit systms L64 and L 128 which ar prsntd in Fig. 2 for diffrnt points of th Griffiths phas for th uniform distribution. As sn in th figur th z xponnt calculatd from th first gap agrs vry wll with th analytical rsults in Eq. 1.9. For th z xponnt, as calculatd from th distribution of th scond gap th scaling rsult in Eq. 3.7 is also wll satisfid, although th rrors of th numrical stimats ar largr than for th first gap. For th third gap, du to th vn strongr finit-siz ffcts, w hav not mad a dtaild invstigation. Extrapolatd rsults at h 0 2 ar found to follow th scaling rsult z (3) z/3. Nxt, w study distribution of th linar and nonlinar local suscptibilitis at th surfac spin. As dmonstratd in Fig. 3 both typs of distributions satisfy th rspctiv asymptotic rlations in Eqs. 3.8 and 3.9, from which th critical xponnts z and z nl ar calculatd. Th stimats ar shown in Fig. 4 at diffrnt points of th Griffiths phas. As sn in th figur th numrical rsults for th dynamical xponnt z ar again in vry good agrmnt with th analytical rsults in Eq. 1.9 and also th xponnt of th nonlinar suscptibility z nl follows th scaling rlation in Eq. 3.10 fairly wll. IV. NUMERICAL RESULTS In th numrical calculations w hav considrd RTIM chains with up to L128 sits and th avrag is prformd ovr svral 10 000 ralizations, typically w considrd 50 000 sampls. For som cass, whr th finit-siz corrctions wr strong, w also mad runs with L256, but with somwhat lss ralizations. W start by prsnting rsults on th distribution of th first and scond gaps. As illustratd in Fig. 1, both for th uniform and th binary distributions, th asymptotic scaling rlations for th distribution of th first two gaps in Eqs. 3.3 and 3.6 ar satisfid. From th asymptotic slops of th FIG. 3. Probability distribution of th linar and nonlinar suscptibility ln 1 and ln 1 nl, rspctivly, for th uniform distribution at h 0 3. Th straight lins ar last squar fits to th data for th largst systm siz, thir slops corrspond to 1/z(h 0 ) and 1/z nl (h 0 ), rspctivly. Thy follow th prdictd rlation z nl (h 0 ) 3z(h 0 ).

PRB 59 GRIFFITHS-McCOY SINGULARITIES IN THE RANDOM... 11 313 FIG. 4. Th stimats for 1/z and 1/z nl as a function of h 0 for th uniform distribution. Ths valus and th corrsponding rror bars hav bn obtaind from our analysis of th probability distribution of ln 1 and ln 1 nl, rspctivly, for two systm sizs as xmplifid in Fig. 3. Th full lin for 1/z corrsponds to th analytical rsult 1.4, th brokn lin corrsponds to 1/3z, which should b idntical with 1/z nl. Finally, w calculat th avrag nrgy-dnsity autocorrlation function. As sn in Fig. 5, G av () displays a linar rgion in a log-log plot, th siz of which is incrasing with L, but its slop, which is just th dcay xponnt has only a wak L dpndnc. Th slop of th curv and thus th corrsponding dcay xponnt has a variation with th paramtr h 0, as illustratd in Fig. 6. Th stimatd xponnts at th critical point, h 0 1, and in th Griffiths phas ar prsntd in Fig. 7. As sn in this figur th variation of is wll dscribd by th form () (0)1/z(). This functional form corrsponds to th scaling rsult in Eq. 3.13, if th critical point corrlations dcay with 02. 4.1 Th numrical calculations with L128 giv a slightly highr valu (0)2.2. 13 Howvr, th finit-siz stimats show a slowly dcrasing (0) with incrasing systm siz. Rpating th calculation with L256 w obtaind (0) 2.1. Thus w can conclud that th scaling rlation in Eq. 3.13 is probably valid and thn Eq. 4.1 is th xact valu of th dcay xponnt of th avrag critical nrgy-dnsity autocorrlations. 24 FIG. 6. Th bulk nrgy-nrgy autocorrlation function G L/2 av () for th binary distribution (4) at diffrnt valus for h 0 for L128 as a function of ln. On obsrvs th variation of th xponnt (h 0 ) idntical to th slop of th straight lin fits with incrasing h 0. V. DISCUSSION In this papr w hav considrd th random transvrsfild Ising spin chain and studid diffrnt consquncs of th Griffiths-McCoy singularitis in th paramagntic phas. Our main conclusion is that all singular quantitis can b charactrizd by powr-law singularitis and th corrsponding varying critical xponnts can b rlatd to th z() dynamical xponnt and, for nrgy-dnsity autocorrlations, to th (0) critical point xponnt. Sinc th xact valu of z() is known in Eq. 1.4 and w xpct that also th rlation in Eq. 4.1 is valid, w hav a complt analytical dscription of th Griffiths phas of th RTIM in on dimnsion. On intrsting fatur of our rsults concrns th distribution of th highr xcitations and th valu of th corrsponding xponnt z (n) z/n. Sinc th dcay of dynamical corrlations of gnral, mor complx oprators ar rlatd to 1/z (n) n/z, w obtain a hirarchy of dcay xponnts which could b simply xprssd by thos of a fw primary oprators. This fatur is rminiscnt of th towrlik structur of anomalous dimnsions in two-dimnsional conformal modls. 25 Our knowldg about th highr xcitations can also b usd to stimat th corrction to scaling contributions. Much of th rasoning of our phnomnological scaling considrations in Sc. III stays valid for othr random quan- FIG. 5. Th bulk nrgy-nrgy autocorrlation function G L/2 av () for th binary distribution (4) at h 0 1.5 for diffrnt systm sizs as a function of ln. Th slop of th straight lin idntifis th xponnt (h 0 ) dscribing th asymptotic dcay of G L/2 av () in th infinit systm siz limit L. Th log-priodic oscillations visibl in th figur ar du to th finit nrgy scal prsnt in th binary distribution. FIG. 7. Th xponnt (h 0 ) for th binary distribution ( 4) as obtaind from th analysis of th asymptotic dcay of th bulk nrgy-nrgy autocorrlation function G L/2 av () in th mannr of Fig. 6. Th full lin is th analytical prdiction (h 0 ) 21/z(h 0 ) with z(h 0 ) givn by th xact formula 1.4 for th binary distribution with 4.

11 314 FERENC IGLÓI, RÓBERT JUHÁSZ, AND HEIKO RIEGER PRB 59 tum systms. Espcially th scaling bhavior of th highr gaps and th corrsponding rlation in Eq. 3.7 should b valid vn for highr dimnsions and th sam is tru for th distribution of th nonlinar suscptibility and th corrsponding rlation in Eq. 3.10. In on dimnsion th univrsality class of th RTIM involvs svral random systms including, among othrs, th random quantum Potts chain. 26 For ths modls on dos not xpct a univrsality of th z() xponnt in th Griffiths phas, although scaling rlations such as th on in Eq. 3.7 ar vry probably valid. It would b intrsting to prform a numrical study on th random quantum Potts modl to chck th xisting conjcturs. Anothr possibl fild whr th prsnt rsults could b applid is th problm of anomalous diffusion in a random nvironmnt. 17,18,27 Making us of th xact corrspondnc 17,18 btwn th Hamiltonian oprator in Eq. 1.2 and th Fokkr-Planck oprator of th on-dimnsional random walk w can us th rlation in Eq. 3.7 to dscrib th distribution of th ignvalus of th corrsponding Fokkr-Planck oprator. On can also dfin an analogous quantity to th nrgy-dnsity autocorrlation function in Eq. 2.10 by considring connctd prsistnc corrlations, whos asymptotic dcay is rlatd to th distribution of th scond gap, as in Eq. 3.12. Rsarch in this fild is in progrss. ACKNOWLEDGMENTS This study was partially prformd during our visits in Köln and Budapst. F.I. s work was supportd by th Hungarian National Rsarch Fund undr Grant Nos. OTKA TO23642 and OTKA MO28418 and by th Ministry of Education undr Grant No. FKFP 0765/1997. H.R. was supportd by th Dutsch Forschungsgminschaft DFG. 1 S H. Rigr and A.P Young, in Complx Bhavior of Glassy Systms, ditd by M. Rubi and C. Prz-Vicnt, Vol. 492 of Lctur Nots in Physics Springr-Vrlag, Hidlbrg, 1997, p. 256. 2 R.B. Griffiths, Phys. Rv. Ltt. 23, 171969. 3 In a classical random systm th singularitis in th corrsponding Griffiths-rgion ar wakr,.g., th spin-spin autocorrlation function has an nhancd powr-law dcay i.., xpc(ln t) with 1 in contrast to th powr-law dpndnc obsrvd in quantum systms. 4 B. McCoy, Phys. Rv. Ltt. 23, 383 1969. 5 H. Rigr and N. Kawashima, Europhys. J. B to b publishd; C. Pich, A.P. Young, H. Rigr, and N. Kawashima, Phys. Rv. Ltt. 81, 5916 1998; T. Ikgami, S. Miyashita, and H. Rigr, J. Phys. Soc. Jpn. 67, 2761 1998. 6 H. Rigr and A.P. Young, Phys. Rv. Ltt. 72, 4141 1994; M. Guo, R.N. Bhatt, and D. Hus, ibid. 72, 4137 1994. 7 W. Wu, B. Ellman, T.F. Rosnbaum, G. Appli, and D.H. Rich, Phys. Rv. Ltt. 67, 2076 1991; W. Wu, D. Bitko, T.F. Rosnbaum, and G. Appli, ibid. 71, 1919 1993. 8 B.M. McCoy and T.T. Wu, Phys. Rv. 176, 631 1968; 188, 982 1969; B.M. McCoy, ibid. 188, 1014 1969. 9 R. Shankar and G. Murthy, Phys. Rv. B 36, 536 1987. 10 D.S. Fishr, Phys. Rv. Ltt. 69, 534 1992; Phys. Rv. B 51, 6411 1995. 11 A.P. Young and H. Rigr, Phys. Rv. B 53, 8486 1996. 12 F. Iglói and H. Rigr, Phys. Rv. Ltt. 78, 2473 1997. 13 H. Rigr and F. Iglói, Europhys. Ltt. 39, 135 1997. 14 F. Iglói and H. Rigr, Phys. Rv. B 57, 11 404 1998. 15 D.S. Fishr and A.P. Young, Phys. Rv. B 58, 9131 1998. 16 A.P. Young, Phys. Rv. B 56, 11 691 1997. 17 F. Iglói and H. Rigr, Phys. Rv. E 58, 4238 1998. 18 F. Iglói, L. Turban, and H. Rigr, Phys. Rv. E 59, 1465 1999. 19 E. Lib, T. Schultz, and D. Mattis, Ann. Phys. N.Y. 16, 407 1961; S. Katsura, Phys. Rv. 127, 1508 1962; P. Pfuty, Ann. Phys. Paris 57, 791970. 20 F. Iglói and L. Turban, Phys. Rv. Ltt. 77, 1206 1996. 21 J. Stolz, A. Nöpprt, and G. Müllr, Phys. Rv. B 52, 4319 1995. 22 F. Iglói, L. Turban, D. Karvski, and F. Szalma, Phys. Rv. B 56, 11 031 1997. 23 M.J. Thill and D.A. Hus, Physica A 15, 321 1995. 24 Numrical stimats for th dcay xponnt of th avrag nrgy-dnsity autocorrlation function for surfac spins at th critical point ar s 2.5 with L128 Rf. 13, which is somwhat largr than for bulk autocorrlations. Discrpancis btwn stimats for z from surfac and bulk quantitis hav bn obsrvd bfor Rf. 14. Thy can b attributd to corrctions to scaling ffcts which ar diffrnt for diffrnt quantitis, s also Fig. 6 in Rf. 16. 25 J.L. Cardy, in Phas Transitions and Critical Phnomna, ditd by C. Domb and J.L. Lbowitz Acadmic, Nw York, 1987, Vol. 11. 26 T. Snthil and S. Majumdar, Phys. Rv. Ltt. 76, 3001 1996. 27 A. Comtt and D. Dan, J. Phys. A 31, 8595 1998; D.S. Fishr, P. L Doussal, and C. Monthus, Phys. Rv. Ltt. 80, 3539 1998.