Low-temperature spin-glass behaviour in a diluted dipolar Ising system

Juan J. Alonso Departamento de Física Aplicada, Universidad de Málaga, Spain Th.C-P04 Low-temperature spinglass behaviour in a diluted dipolar Ising system. [email protected] Acknowledgments: We thank financial support from Campus Andalucía Tech-Universidad de Málaga (UMA), and MEC FIS2013-43201-P Grant. We thank SCBI at UMA, and to Institute Carlos I at University of Granada for much computer time. Figure 3 Model Results Cross-overlap spikes Equilibration times Introduction Pair correlation function Cumulative distributions References: PRB 86, 140402(R) (2012); PRB 87, 134205 (2013); PRB 91, 09446 (2015). ICM, July 2015, Barcelona 104106108 t 0 0.2 0.4 0.6 0.8 1 q2 , q2 q2 L=10 L=8 103104105106107108 t 0 0.2 0.4 0.6 0.8 1 q2 , q2 q2 L=10 L=8 ~ ~ 10-2 10-1 100101 z 1 Πc 10 8 6 4 10-2 10-1 100101 z 1 Πc 8 6 4 (a) (b) 0.4 0.4 ~ p p ~ ~ -1 -0.5 00.5 1 q 0.1 1 10 (δp)2 4 6 8 10 0.1 1 10 p 4 6 8 10 (a) (b) Diversity in SGs? We consider site-diluted systems of classical Ising spins on SC lattices. All spins point up or down along the z axis of a L3 lattice. At each site a spin is placed with probability x . These spins are coupled by dipolar interactions. We find spiky distributions that are strongly sampledependent. We study sample-to-sample fluctuations. In particular, the deviations of from the average , H=X <i,j> "a(a/rij )3(1 3z2 ij/r2 ij)ij, 0.0 0.2 0.4 0.6 0.8 1.0 x 0 1 2 T AF SG 3D In complex systems, different random distributions of their microscopic constituent parts give rise to diverse values of some macroscopic properties. A paradigmatic example is the SherringtonKirkpatrick (SK) model, where the couplings between any pair of spins are randomly fixed to be FM or AF. SK has both quenched disorder and frustration, the two essential ingredients of spin glasses (SG). SK exhibits RSB: identical replicas of a given sample J (with a given distribution of couplings) may stay trapped in several pure states that are sample-dependent. The distribution of the overlap q between states of a given sample, p_J(q) is like depicted in Fig.(a). In the macroscopic limit, after averaging over J’s, the averaged distribution p(q) is like in Fig.(b). It is said to be non-trivial. Whether the RSB picture describes correctly the behaviour of realistic SGs still an open question. In the droplet picture, the SG phase is described in terms of a unique state with excitations that are compact droplets. According to this scenario, p(q) distributions do not exhibit diversity, as shown in Fig.(c). We obtain equilibrium results by means of Tempered Monte Carlo simulations. This is the phase diagram: We were able to equilibrate systems of several hundreds of dipoles. For strong dilution we observe extremely large relaxation times (we need 108 MCS for x=0.35 and L=10). Here we focus on the character of the SG at very low temperature, well deep in the SG phase. pJ(q) -1 -0.5 00.5 1 q 0 2 4 pJ 0 2 4 6 8 10 pJ (a) (b) q=N1X j (1) j(2) j, We measure the Edwards-Anderson overlap parameter where and are the spins on site j of identical replicas (1) and (2) of a given sample . (1) j (2) j J J For each sample we compute the overlap probability distribution : pJ(q) We focus on spikes situated on . We compute From a pair correlation function gQ, we study the shape and average width of CO spikes: pJ(q) p(q)2=[{pJ(q)p(q)}2]J p(q) We find for with . (This central region stands for overlaps between states of different basins of attraction). This result is in contradiction with Dp. p(q)6=0 q2(Q, Q) Q⇠1/2 On the contrary, we find that ΔQ does not diverge as L increases. This is at odds with RSBp and in sharp contrast with results for the SK model, for which ΔQ diverges as L1/2. Strikingly, do not grow with L, in contradiction with RSBp. For SK one expects many sharp spikes that become δ-like functions as L increases: a diverging diversity. p(q)2 q2(Q, Q) XQ J=ZQ Q pJ(q)dq, Q J=ZQ Q {pJ(q)p(q)}2dq 1/2 We find that XQ does not change with L at low temperatures, in agreement with RSBp. 00.1 0.2 0.3 0.4 T 0.1 0.2 0.3 0.4 0.5 ∆Q L=4 L=6 L=8 L=10 0 0.3 0.6 0.9 1.2 T 0.05 0.1 0.15 0.2 ∆Q/L1/2 L=4 L=6 L=8 (a) (b) DIS SK Conclusions T=0.1 T=0.1 T=0.1 T=0.25 We find gQ (q) curves rather pointed with widths smaller than Q. For DIS, there is no significant size dependence: wQ appear to be size independent. In contrast, wQ values for SK appear to vanish as L-1/2 as L increases. fJ(q1,q 2)⌘pJ(q1)pJ(q2) where . gQ(q) is a conditional probability that q=q2-q1, given that q1,q2∈(0,Q). It is also a sort of average over all CO spikes. gQ J(q)=ZQ 0ZQ 0 dq1dq2(q2q1q)fJ(q1,q 2), DIS SC lattice -0.1 0 0.1 q 1 10 gQ 10 8 6 4 -0.1 0 0.1 q 1 10 gQ 8 6 4 (a) DIS SK T=0.1 T=0.2 DIS SK A suitable width wQ of gQ (q) is , which is a measure of pattern thermal fluctuations. wQ=1/gQ(0) 0 0.1 0.2 0.3 0.4 T 0 0.1 0.2 0.3 0.4 0.5 wQ 4 6 8 10 0 0.3 0.6 0.9 1.2 T 0 0.4 0.8 1.2 L1/2wQ 4 6 8 (a) (b) SK Using Monte Carlo simulations, we study the character of the spin-glass (SG) state of a site-diluted dipolar Ising model. At high dilution, well deep in the SG phase, we find spiky distributions that are strongly sample-dependent. For the system sizes studied, the average width of spikes, and the fraction of samples with spikes higher than a certain threshold does not vary appreciably with system size. These results are at odds with crucial predictions of the droplet and RSB scenarios. Our findings are similar to the ones found for the short-ranged Edwards-Anderson model. DIS We also study the height of CO spikes. For each sample we calculate , the maximum value of in (-Q,Q). MJ pJ(q) MJ<z ⇧ ep c(z) z>0.5 We compute , the fraction of samples having . We find that, at least for , high spikes do proliferate only for the SK model, but not for DIS. Systems of dipoles in crystals have frustration. Put together with spatial-disorder may result in SG behaviour, as observed in some ferroelectrics and in diluted dipolar Ising systems (DIS) as the LiHoxY1-xF4. p(q) q0 q0 q0 q0 q0 q0 q q q (a) (b) (c) pJ(q) p(q) p(q)