Simulational studies of spin glasses since the early 2010s have focused on the so-called replicon exponent α as a means of determining whether the low-temperature phase of spin glasses is described by the replica symmetry breaking picture of Parisi or by the droplet-scaling picture. On the latter picture, it should be zero, but we shall argue that it will only be zero for systems of linear dimension L>L^{*}. The crossover length L^{*} may be of the order of hundreds of lattice spacings in three dimensions and approach infinity in six dimensions. We use the droplet-scaling picture to show that the apparent nonzero value of α when L<L^{*} should be 2θ, where θ is the domain wall energy scaling exponent. This formula is in reasonable agreement with the reported values of α.
Simulational studies of spin glasses since the early 2010s have focused on the so-called replicon exponent α as a means of determining whether the low-temperature phase of spin glasses is described by the replica symmetry breaking picture of Parisi or by the droplet-scaling picture. On the latter picture, it should be zero, but we shall argue that it will only be zero for systems of linear dimension L>L^{}. The crossover length L^{} may be of the order of hundreds of lattice spacings in three dimensions and approach infinity in six dimensions. We use the droplet-scaling picture to show that the apparent nonzero value of α when L<L^{*} should be 2θ, where θ is the domain wall energy scaling exponent. This formula is in reasonable agreement with the reported values of α.
We have studied the Parisi overlap distribution for the three-dimensional Ising spin glass in the Migdal-Kadanoff approximation. For temperatures $T\ensuremath{\simeq}{0.7T}_{c}$ and system sizes up to L=32, we found a P(q) as expected for full Parisi replica-symmetry breaking, just as was also observed in recent Monte Carlo simulations on a cubic lattice. However, for lower temperatures our data agree with predictions from the droplet or scaling picture. The failure to see droplet model behavior in Monte Carlo simulations is due to the fact that all existing simulations are done at temperatures too close to the transition temperature so that system sizes larger than the correlation length have not been achieved.
We have studied the diluted Heisenberg spin glass model in a three-component random field for the commonly used one-dimensional long-range model where the probability that two spins separated by a distance r interact with one another falls as 1/r^{2σ}, for two values of σ, 0.75 and 0.85. No de Almeida-Thouless line is expected at these σ values. The spin glass correlation length ξ_{SG} varies with the random field as expected from the Imry-Ma argument and the droplet scaling picture of spin glasses. However, when ξ_{SG} becomes comparable to the system size L, there are departures which we attribute to the features deriving from the replica symmetry breaking picture of spin glasses. For the case σ=0.85 these features go away for system sizes with L>L^{*}, where L^{*} is large (≈4000-8000 lattice spacings). In the case of σ=0.75 we have been unable to study large enough systems to determine its value of L^{*}. We sketch a renormalization group scenario to explain how these features could arise. In this scenario finite size effects on the droplet scaling picture in low-dimensional spin glasses produce some aspects of Parisi's replica symmetry breaking theory of the Sherrington-Kirkpatrick model.
In order to overcome the limitations of small system sizes in spin-glass simulations, we investigate the one-dimensional Ising spin chain with power-law interactions.The model has the advantage over traditional higher-dimensional Hamiltonians in that a large range of system sizes can be studied.In addition, the universality class of the model can be changed by tuning the power law exponent, thus allowing us to scan from the mean-field to long-range and short-range universality classes.We illustrate the advantages of this model by studying the nature of the spin glass state where our results hint towards a replica symmetry breaking scenario.We also compute ground-state energy distributions and show that mean-field and non-mean-field models are intrinsically different. * )Recently Krzakala and Martin, as well as Palassini and Young, suggest an intermediate picture in which droplets have a fractal surface, and their energy is finite in the thermodynamic limit.See Ref. 5) and references therein for details.* * ) Note that we explicitly distinguish between θ , the stiffness exponent for droplet excitations, and θ, the exponent for domain walls, because DP predicts θ = θ , which we want to test here.
Using mappings to computer-science problems and by applying sophisticated algorithms, one can study numerically many problems much better compared to applying standard approaches like Monte Carlo simulations. Here, using calculations of ground states of suitable perturbed systems, droplets are obtained in two-dimensional +-J spin glasses, which are in the focus of a currently very lifely debate. Since a sophisticated matching algorithm is applied here, exact ground states of large systems up to L^2=256^2 spins can be generated. Furthermore, no equilibration or extrapolation to T=0 is necessary. Three different +-J models are studied here: a) with open boundary conditions, b) with fixed boundary conditions and c) a diluted system where a fraction p=0.125 of all bonds is zero. For large systems, the droplet energy shows for all three models a power-law behavior E_D L^θ'_D with θ'_D<0. This is different from previous studies of domain walls, where a convergence to a constant non-zero value (θ_dw=0) has been found for such models. After correcting for the non-compactness of the droplets, the results are likely to be compatible with θ_D= -0.29 for all three models. This is in accordance with the Gaussian system where θ_D=-0.287(4) (ν=3.5 via ν=-1/θ_D). Nevertheless, the disorder-averaged spin-spin correlation exponent ηis determined here via the probability to have a non-zero-energy droplet, and η~0.22$ is found for all three models, this being in contrast to the behavior of the model with Gaussian interactions, where exactly η=0.
The study of the low temperature phase of spin glass models by means of Monte Carlo simulations is a challenging task, because of the very slow dynamics and the severe finite-size effects they show. By exploiting at the best the capabilities of standard modern CPUs (especially the streaming single instruction, multiple data extensions), we have been able to simulate the four-dimensional Edwards-Anderson model with Gaussian couplings up to sizes L=70 and for times long enough to accurately measure the asymptotic behavior. By quenching systems of different sizes to the critical temperature and to temperatures in the whole low temperature phase, we have been able to identify the regime where finite-size effects are negligible: ξ(t)≲L/7. Our estimates for the dynamical exponent (z≃1/T) and for the replicon exponent (α≃1.0 and T independent), that controls the decay of the spatial correlation in the zero overlap sector, are consistent with the replica symmetry breaking theory, but the latter differs from the theoretically conjectured value.
We have studied the diluted Heisenberg spin glass model in a 3-component random field for the commonly-used one-dimensional long-range model where the probability that two spins separated by a distance r interact with one another falls as 1/r2σ, for two values of σ, 0.75 and 0.85. No de Almeida-Thouless line is expected at these σ values. The spin glass correlation length ξSG varies with the random field as expected from the Imry-Ma argument and the droplet scaling picture of spin glasses. However, when ξSG becomes comparable to the system size L, there are departures which we attribute to the features deriving from the TNT picture of spin glasses. For the case σ=0.85 these features go away for system sizes with L>L∗, where L∗ is large (≈4000−8000 lattice spacings). In the case of σ=0.75 we have been unable to study large enough systems to determine its value of L∗. We sketch a renormalization group scenario to explain how these features could arise. On this scenario finite size effects on the droplet scaling picture in low-dimensional spin glasses produce TNT features and some aspects of Parisi's replica symmetry breaking theory of the Sherrington-Kirkpatrick model.