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Correlated random fields in dielectric and spin glasses M. Schechter1,2 and P. C. E. Stamp1,3 1 Department of Physics and Astronomy, University of British Columbia - Vancouver, B.C., Canada V6T 1Z1 2 Department of Physics, Ben Gurion University - Beer Sheva 84105, Israel 0 3 Pacific Institute for Theoretical Physics, University of British Columbia - Vancouver B.C., Canada V6T 1Z1 1 0 2 n PACS 64.70.ph–Nonmetallic glasses a PACS 75.10.Nr–Spin-glass and other random models J PACS 75.10.Jm–Quantized spin models 5 Abstract. - Both orientational glasses and dipolar glasses possess an intrinsic random field, ] coming from the volume difference between impurity and host ions. We show this suppresses n the glass transition, causing instead a crossover to the low T phase. Moreover the random field n is correlated with the inter-impurity interactions, and has a broad distribution. This leads to a - s peculiarvariantoftheImry-Mamechanism,with’domains’ofimpuritiesorientedbyafewfrozen i d pairs. These domains are small: predictions of domain size are given for specific systems, and . their possible experimental verification is outlined. In magnetic glasses in zero field the glass t a transitionsurvives,becausetherandomfieldsaredisallowedbytime-reversalsymmetry;applying m a magnetic field then generates random fields, and suppresses the spin glass transition. - d n o c [ Most real solids are either amorphous or disordered posed by symmetry. These RFs are broadly distributed, 2 crystals, and many of these show some sort of ’glassy’ and correlated with the interactions; we find this leads to v behavioratlowtemperatures[1]. Theseglasseshavebeen a peculiar disordering mechanism, intermediate between 3 categorized into different types, including spin [2], struc- simple alignment along the RFs and Imry-Ma [11] behav- 9 tural[3], electron[4], orientational [5], anddipolarglasses ior. 8 [6]. All of these show aging, rejuvenation and memory 0 . effects. However, thermodynamically there is a marked (i) Orientational & Dipolar glasses: For typical 0 difference between systems like spin glasses (SGs), which OGs like KBr:CN, various approaches have been used to 1 9 show a clear phase transition between the paramagnetic describe the interactions [12–15]. If one adds to the stan- 0 and SG phases (with diverging length scale and conse- dard strain-mediated interaction [14,16] another term re- : quent diverging nonlinear susceptibility [7]), and systems flecting the volume difference between host and impurity v i like orientational glasses (OGs) or electric dipole glasses ions, an effective random field (RF) is found to be gener- X (DG) where the transition to the low-T phase is smeared ated by the orientational impurities themselves. The OG r [6,8,9]. degreesoffreedomcanbetreatedformallyinthesameway a At first glance, this difference seems paradoxical - in as the pseudospin degrees of freedom in DG systems like OGs, the rotational degree of freedom behaves like an Li:KCl, allowing one to analyse the interactions of both Ising pseudospin 1/2, similar to that in DGs; and the systems on the same footing [15]. electric- and strain-mediated interactions between these One then finds [15] that over a wide range of temper- pseudospins have a similar form to those between the real atures extending up to the putative glass transition tem- spins in dipolar SGs. The quenched randomness then im- perature TG, both OGs and DGs are described by the plies that OGs and DGs should show the same critical pseudospin effective Hamiltonian behaviorasdipolarSGs,thatoftheshortrangeEdwards- Anderson model [10]. H = Uzzσˆzσˆz+ b σˆz. (1) eff ij i j j j Inthepresentpaperwearguethatthefullanswertothis ij j X X questionisfoundinananalysisoftheeffectiveHamiltoni- ans for these systems - including the form of the random We emphasize that both the Ising interaction Uzz and ij fields(RFs)thataregenerated-andintheconstraintsim- the RFs b are generated by the same defect-phonon j { } p-1 M. Schechter1,2 P. C. E. Stamp1,3 interaction the trivial relations between high and low impurity densi- ties). Now, sinceb U (whereU istheinteraction be- 0 0 0 ∂X ≪ V = (η δαβ +γαβσz) jα (2) tween nearest-neighbour defects/impurities), two regimes dp − o j j ∂xjβ are possible, viz.: (a) b U n and (b) U n b U . Xj Xαβ 0 ≪ 0 0 ≪ 0 ≪ 0 Regime(a)issimple. AllRFsaresmallerthanthetypi- expanded to second order in perturbation. Here Xjα calinteraction. Onecanthenfollowthestandardapplica- represents the displacement operator at point j in di- tionoftheImry-Maargument[11]toSGs[18]. Theenergy rection α. One finds [15] Uizjz = gzzTizjz/Ri3j, and bj ≡ cost to flip a domain is related to the typical interaction, U0z = g 0z/R3 with R = r r , and where and therefore is U¯Lθ, where L is the domain size, and i ij 0z iTij ij ij i− j ∼ g γ2/ρc2, and g (η γ /ρc2). Here ρ is the den- thestiffnessexponentθ 0.2in3D[18,20]. However, the Pzz ∼ o o P 0z ∼ o o l ≈ sity,co(cl)isanaverage(longitudinal)soundvelocity,and effective RF is given by bav = b0√n = b0¯b. Thus, the zz, 0z O(1) are complicated angular averages over number of spins in a domain N , and the related dimen- Tij Tij ∼ d p the tensor components of the interaction. sionless correlation length ξ˜ ξn1/3 N1/3 are both n ≡ ∼ d ThetypicalparametersoftheHamiltonian(1)arethere- dependent, the latter being given by fore given by γ2 η γ ξ˜ U0n (3/21)−θ n0.39 U¯ 0.78. (4) U¯ o n ; ¯b o on (3) ≈ b √n ≈ ¯b ≈ ρc2 ≈ ρc2 (cid:18) 0 (cid:19) (cid:18) (cid:19) o o In comparison to Gaussian-distributed RFs, the domain and we note that the ratio U¯/¯b is independent of n. size here is smaller by a factor n0.39. In principle, one should add to the hamiltonian (1) a Regime (b) (which usually applies in dilute dipolar tunneling term ∆ σˆx, which is crucial for the dynam- j j j glasses) is more complex, because the spread of RFs is ics of the system. However, we are interested in the case larger than the typical interactions, so Imry-Ma argu- where interactioPns dominate, ie., U¯ > ∆¯; the tunneling mentscannotbeuseddirectly. Considerapairofnearest- term is then irrelevant when considering the phase dia- neighbour impurities/defects l,l′ (3-impurity correlations gram. For concentration n = 10−3, typical values are can be neglected when n 1). Since U U¯, the two U¯ 1.5 K and¯b somewhat smaller. Typically we assume ≪ 0 ≫ ∼ impurities will mutually order, ie., lock together. If this n 1 in what follows, but we will also compare with ≪ locking is antiferromagnetic, then the net RF on the pair larger n. comes only from distant impurities - the internally generated RF U0z U0z =0, since U0z =U0z for any pair i,j. (ii) Correlated random fields: Consider now the ll′ − l′l ij ji However, for ferromagnetic ordering the internally gener- RFs b . Thesewerepreviouslyarguedtoleadtoaglassy j { } ated RF is b , and since U n b , this pair will be state [13], or to a state intermediate between a true spin ∼ 0 0 ≪ 0 oriented along this internal RF, and hardly influenced by glass and a pure random-field system [17]. Here we ar- theinteractionswithotherimpurities,whichhavestrength gue that actually the RFs destroy long-range glassy or- U¯ = U n! In fact, this is the case for all ferromagnet- der in the low-T phase [18], which is then reached by ∼ically alig0ned impurity pairs satisfying b /R3 > U¯. In a a crossover and not a phase transition. Furthermore, 0 ij sample of N impurities there are Nb /U such ’frozen since the RFs originate from the impurity lattice inter- ≈ 0 0 pairs’. actions themselves, they are correlated with the interactions (because U0z is correlated with Uzz) and have ConsidernowasamplecontainingN impurities/defects. ij ij The typical RF resulting from the frozen pairs is then a broad distribution. When defects/impurities occupy b N/U U n, while that from all other impurities is nearest-neighbour sites, the field of one on the other is 0 0 0 ∼ b0B≈ecηaouγsoe/(Uρc2o)≫1/¯bR. 3 and is random, U2 1/R6 and tohnelpyfr∼oz√enNpba0inr.s TdohmusindaetsepsittehethteoirtaslcaRrFci.tyU,stihnegRnFowfrtohme ij ∝ h iji ∝ standard argument which compares the energy gain from is effectively short range [10,18]. The situation is thus the RF and the interaction energy cost at domain bound- similar to that analysed by Imry-Ma [11], and its gen- aries,wefindforadiluteDGinregime(b)adimensionless eralisation to glasses by Fisher and Huse [18]. However correlation length given by there is an important difference - we find that in systems like dilute OGs, where the RFs are broadly distributed and are correlated with the interactions, one gets qualita- U¯ (3/21)−θ U¯ 0.39 ξ˜ . (5) tively different results from systems where the RFs are ≈ √U¯¯b ≈ ¯b (cid:18) (cid:19) (cid:18) (cid:19) normally distributed. The largest RF, occuring in the event when defects/impurities occupy nearest-neighbour Thus,inthisregimethecorrelationlengthisdetermined sites, is b . Since n 1, such events are rare; how- neither by the typical RF b n, nor by the average RF 0 0 ever the average RF b≪ (the standard deviation of the b √n, but by the algebraic mean of U¯ and ¯b. This is av 0 distribution) is dominated by these rare events, and is because the relaxation of the RFs occurs in two stages. given by b η γ √n/(ρc2) ¯b (see Ref. [19], noting The rare pairs with b > U¯ relax locally to their frozen av ≈ o o o ≫ i p-2 Correlated random fields in dielectric and spin glasses state. The majority of the spins then order via the Imry- 12 Mamechanism,albeitwithRFsdominatedbytheirinter- action with the rare frozen pairs. Because U¯/¯b is scale- 10 10 invariant N (U¯/¯b)1.17, independent of n. The low d ∼ power in the exponent is dictated by the unique mech- 8 ~ξ anism above, and results in ”quasi-domains”, which in 1 practise contain very few defects/impurities (see the ex- ~ξ 6 amplesbelow). Moreover,ineachsuchquasi-domainthere are only ∼ (U¯/¯b)0.17 ≈ 1 frozen pairs. In this limit one 4 0.11 10 100 cannot of course assume that the random fields behave as E/n (*109 V/m) in the large N limit; however, the mechanism described 2 above gives an upper limit to the size of the domains, since it minimizes the effect of the RFs. A lower limit 0 can be given by the assumption that the total RF from 0 5 10 15 20 the frozen pairs is maximized, leading to N (U¯/¯b). In E/n (*109 V/m) d ∼ practise these two limits are indistinguishable, suggesting that domains are formed in such a way that interactions Fig.1: Thedimensionlesscorrelationlengthξ˜≡ξ·n1/3 ofthe with frozen pairs are maximized. glass order is plotted as function of the normalized external field E/n for KBr:CN (¯b/U¯ = 1/20, d = 0.3 D) with n = 0.5 Thus, in regime (b) we arrive at a picture of very (solid black line) and n = 10−3 (dotted red line), against its small quasi-domains or ’clusters’ of Nd pseudospins, of valuewithzeroeffectiveRF(dashedblueline). Theinsetshows size ξ˜, each containing O(1) frozen pseudospin, whose a log-log plot, emphasizing the n dependence of the deviation ∼ field then orients the rest of them. For n 1 (regime from power low behavior. ≈ (a)) we find N (U/b)2.34, giving large clusters, but d ≈ on strong dilution this crosses over to the much smaller Nd (U/b)1.17. We note that for all dilutions, the finite izes the Ising interaction, so that ξ˜gi∼ves a finite nonlinear susceptibility χ ξ˜2−η, mark- 3 ingacrossoverratherthanatransitiontot∝heglassystate γ2 2 U¯ n o +U2 (6) [21,22], in agreement with experiments [6]. However, the r ≈ s ρc2 e (cid:18) o(cid:19) decrease in N with n means that the peak in the nonlin- d and allows the coupling to an electric field E. The latter ear susceptibility should be much smaller for dilute OGs allowsameasurementoftheeffectiveRFasfunctionofn, andDGs,incomparisonwithsimilarsystemsathighcon- providinganexperimentalchecktoourtheory. ForE >0 centrations. This is a central prediction of this Letter. 1 In real systems, the ratio U¯/¯b varies widely between U¯ (3/2)−θ ξ˜ . (7) different OGs and DGs, and it is useful to look at some E ≈ b2 +(E d)2! experimental examples. Consider first OH− impurities in eff · KCl, where the electric and elastic impurity-impurity in- where d is the dipolpe moment, beff = b0√n in regime (a) teractions are comparable, and U¯/¯b 3 (see Ref. [23]). and b = √¯bU¯ in regime (b). Thus, increasing E causes eff ≈ In Ref. [9] it was argued, based on the behavior of the a crossover from impurity-dominated to field-dominated non-linear dielectric permittivity of dilute KCl:OH, that ξ at E d b . In Fig. 1 we plot ξ˜ for KBr:CN, for eff there is no transition to a glass phase, but rather to a n = 0.5· an≈d for n = 10−3 ¯b/U¯. For E b /d, eff state analogous to a superparamagnet, with roughly 10 one has ξ˜ E−1/(3/2−θ). H≪owever, the mag≫nitude of E impuritiesperdomain. Thescalingapproachweusehere, ξ˜ at E = 0∝, as well as the region where its functional E embodied in Eq. (5), is strictly applicable only for ξ˜ 1, form deviates considerably from the above power low, are ≫ and even then only gives an order of magnitude. Thus strongly n dependent. Their measurement as functions of our result here is consistent with the above experimen- nwouldmeasuretheeffectiveRFinthesystemasfunction tal picture of small ’superparamagnetic’ domains for this of dilution, and thus provide a check to our theory. system. Consider now systems where the spherical and ExtrinsicimpuritiesalsogenerateRFswithabroaddis- dipolar species are of similar volume, so ¯b/U¯ 1. For tribution,andcanbeanalyzedalongthesamelinesasthe example, ¯b/U¯ 1/20 in KBr:CN (see Ref. [2≪4]), which intrinsic impurities. Thus, at E =0, as the system is pu- is an OG for n≈ 0.5. For n 0.5 we obtain ξ˜ 10 and rified, themagnitudeofξ andofthecuspinthenonlinear ≤ ≈ ≈ N (U/b)2.34 1000. However, in the dilute regime permittivity increase, but only to a value in accordance d (b),≈because of th≈e small exponent in Eq. (5), ξ˜ 3, and with Eqs. (4), (5). ≈ N 30 only, a large reduction! d ≈ (iii)Spinglasses: Wehaveseenthattherandomfields As mentioned above, the electric dipole interaction is generated in OGs and DGs prevent a genuine phase tran- typically quite small. For our purposes here, it renormal- sition to a low-T glass state. Why does one then see a p-3 M. Schechter1,2 and P. C. E. Stamp1,3 glass transition in SGs? After all, one could certainly ex- the Ising model is unstable to small perturbations, and pect similar random fields to be generated therein, via, is therefore not realizable (i.e., RFs will always emerge). eg., magnetoacoustic interactions. However, there is an In particular, one cannot realize the transverse field Ising essential difference between electric and magnetic Ising modelineasy-axisdisorderedmagneticsystemsbyapply- systems. If one neglects the volume term in Eq. (2), then ing a transverse magnetic field. OGs share with SGs a symmetry under σz σz. How- In light of these conclusions for SGs, it is interesting ever this is a symmetry under parity Pˆ in→O−Gs (where to recall that models of 2-level systems (TLSs) interact- the pseudospin variables σz are not real spins), but under ing with phonons are commonly used to describe a wide time-reversal Tˆ in SGs (where the σz are real spins, and variety of glasses at low T, not just OGs and DGs. Inso- couple to real magnetic fields). As a result, terms linear far as the models of TLSs are applicable, and the volume in σ which emerge naturally in OGs, are not allowed in term in the interaction with phonons causes a breaking of the absence of a magnetic field in SGs [25]. For example, the σ σ symmetry, then our theory predicts that in z z ↔− in zero field the magnetoacoustic interaction is, to lowest these systems RFs will emerge as well, and there will also order, bilinear in σ: be a crossover rather than a phase transition between the glass-ordered and disordered phases. To unambiguously ∂X test this, one could measure the non-linear dielectric sus- Vsp = − ηδαβ + Aαjβγδσˆjγσˆjδ ∂xjα, (8) ceptibility in these systems - it should not diverge as a jβ Xj Xαβ Xγδ functionoftemperature. Theseconsiderationsapplyboth   for the glass transition, and for the low energy regime where Aαβγδ is the spin-phonon interaction tensor. Thus, of interacting TLSs [30]. For a recent calculation of the j SGs are well described by the Edwards Anderson model strengthoftheTLS-TLSinteractionsinthetworegimesin with no RF, and have a well defined phase transition and orientational glasses, and a discussion of its applicability divergingnonlinearsusceptibilitybetweentheSGandPM in structural glasses, see ref. [33]. phases [7]. WewouldliketothankA.Burin,L.Ferrari,AJLeggett, The fact that it is the time-reversal nature of the Z2 Z.Nussinov,BSeradjeh,andAPYoungforveryusefuldis- symmetry that prevents the emergence of random lon- cussions. This work was supported by NSERC of Canada gitudinal fields is best exemplified in anisotropic dipolar and by PITP and CIFAR. magnets. Recently it was shown that with the applica- tion of a transverse field H , that breaks time reversal ⊥ but by itself keeps the σz σz symmetry, an effective REFERENCES → − RF h emerges via the intrinsic off-diagonal terms of the dipolar interaction [26]. 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