Dilute One-Dimensional Spin Glasses with Power Law Decaying Interactions

Abstract

We introduce a diluted version of the one dimensional spin-glass model with interactions decaying in probability as an inverse power of the distance. In this model varying the power corresponds to change the dimension in short-range models. The spin-glass phase is studied in and out of the range of validity of the mean-field approximation in order to discriminate between different theories. Since each variable interacts only with a finite number of others the cost for simulating the model is drastically reduced with respect to the fully connected version and larger sizes can be studied. We find both static and dynamic evidence in favor of the so-called replica symmetry breaking theory.PACS numbers: 75.10. Nr,71.55.Jv,05.70.Fh Mean-field spin glass models are known to have a rather complex low-temperature phase [1], which has not been clearly observed so far in numerical simulations of finite-dimensional models with short range interactions. Theories alternative to the mean-field one have been proposed [2], but short-range systems with quenched disorder are very tough to study analytically [3]. Numerical simulations have been, thus, extensively employed, developing more and more refined algorithms over the years, though with no conclusive indication on the nature of the spin-glass phase in finite dimension.Long-range models are such that their lower critical dimension is lower than the one of the corresponding short range model. In particular, one can have a phase transition even in one dimensional systems, provided the range of interaction is large enough. One dimensional spin glass models with power-law decaying interactions actually allow to explore both long-and short-range regimes by changing the power [4,5,6,7]. These models would be perfect candidates for comparing the spin glass phase in and out of the range of validity of the mean field approximation. Unfortunately, since each variable interacts with all the others, numerical simulations are very computer demanding and it is hard to get a clear numerical evidence supporting a specific spin glass theory [6,7].We, therefore, introduce a diluted version of the model, where the mean coordination number is fixed (see also Ref. 8). In diluting, the run time grows as the size N of the system, rather than proportionally to N 2 . This is a fundamental issue because finite volume effects are strong in these models: previous studies were restricted to N ≤ 512, while we can now thermalize systems up to N = 16384, thus keeping these effects under control.We are interested in analyzing the difference among the predictions on the spin glass phase of the droplet theory [2], the trivial-non-trivial (TNT) scenario [9] and the replica symmetry breaking (RSB) theory [1]. Studying the thermodynamics, we focus on site and link overlaps, providing strong evidence that both fluctuate in the infinite volume limit. From the dynamic behavior we learn that the four-point correlation function goes to zero at large distances when extrapolated at infinite times. In this framework w...

Correspondence between long-range and short-range spin glasses

“…(5) for the magnetic exponent y H (=(d + 2 − η)/2) can be written 18 and we use their values here. In particular, they find η SR (3) = −0.375 (10), which, according to Eq. (8), corresponds to a proxy value σ = 0.896.…”

“…Hence it is desirable to understand critical behavior up to, and just above, d = d u . For the case of spin glasses, 1 where much of what we know has come from numerical simulations, this has been difficult because (i) the value of d u is quite large (d u = 6 as opposed to 4 for conventional systems like ferromagnets) and (ii) the slow dynamics, coming from the complicated "energy landscape," prevents equilibration of systems with more than of order 10 4 spins at and below the transition temperature T c . Since the total number of spins V is related to the linear size L by V = L d , for dimensions around d u (=6) it is then not possible to study a range of values of L, which, however, is necessary to carry out a finite-size scaling 2,3 (FSS) analysis.…”

“…The advantage of the 1D model is that one can study a large range of linear sizes for the whole range of σ . Consequently, there have been several subsequent studies [5][6][7][8][9][10][11][12] on these models.…”

Ultrametricity and Clustering of States in Spin Glasses: A One-Dimensional View

“…We observe that for σ values spanning the infinite-range SK universality class (σ = 0) to the non-mean-field universality class (σ > 2/3) an UM organization and a com- plex clustered landscape seem to emerge for the system sizes studied. To check if these results persist at larger length scales, it would be of interest to study even larger systems [27]. This important since the crossover to any putative UM behavior presumably might depend on the system size.…”

Chaos properties of the one-dimensional long-range Ising spin-glass

“…Since it is much simpler to study numerically than hypercubic lattices as a function of the dimension d, this model has attracted a lot of interest recently [35][36][37][38][39][40][41][42][43][44][45][46][47][48][49] (here we will not consider the diluted version of the model [50]). …”