The fractal dimension of excitations in glassy systems gives information on the critical dimension at which the droplet picture of spin glasses changes to a description based on replica symmetry breaking where the interfaces are space filling. Here, the fractal dimension of domain-wall interfaces is studied using the strong-disorder renormalization group method pioneered by Monthus [Fractals 23, 1550042 (2015)FRACEG0218-348X10.1142/S0218348X15500425] both for the Edwards-Anderson spin-glass model in up to 8 space dimensions, as well as for the one-dimensional long-ranged Ising spin-glass with power-law interactions. Analyzing the fractal dimension of domain walls, we find that replica symmetry is broken in high-enough space dimensions. Because our results for high-dimensional hypercubic lattices are limited by their small size, we have also studied the behavior of the one-dimensional long-range Ising spin-glass with power-law interactions. For the regime where the power of the decay of the spin-spin interactions with their separation distance corresponds to 6 and higher effective space dimensions, we find again the broken replica symmetry result of space filling excitations. This is not the case for smaller effective space dimensions. These results show that the dimensionality of the spin glass determines which theoretical description is appropriate. Our results will also be of relevance to the Gardner transition of structural glasses.
Despite decades of effort, our understanding of low-temperature phase of spin glass models with short-range interactions remains incomplete. Replica symmetry breaking (RSB) theory, based on the solution of the Sherrington-Kirkpatrick mean-field model, predicts many pure states; meanwhile, competing theories of short-range systems, such as the droplet picture, predict a single pair of pure states related by time-reversal symmetry, analogously to the ferromagnet. Since RSB certainly holds for the mean-field (infinite-range) model, it is interesting to study short-range models in high dimensions to observe whether RSB also holds here; however, computer simulations of short-range models in high dimensions are difficult because the number of spins to equilibrate grows so rapidly with the linear size of the system.A relatively recent idea which has been fruitful is to instead study one-dimensional models with long-range (power-law) interactions, which are argued to have the same critical behavior as corresponding short-range models in high dimensions, but for which simulations for a range of sizes (crucial for finite-size scaling analysis) are feasible. For these one-dimensional long-range (1DLR) models, we fill in a previously unexplored region of parameter space where the interactions become sufficiently long-range that they must be rescaled with the system size to maintain the thermodynamic limit. We find strong evidence that detailed behavior of the 1DLR models everywhere in this "nonextensive regime" is identical to that of the Sherrington-Kirkpatrick model, lending support to a recent conjecture.In an attempt to distinguish the RSB and droplet pictures, we study recently-proposed observables based on the statistics of individual disorder samples, rather than simply averaging over the disorder as is most frequently done in previous studies. We compare Monte Carlo results for 1DLR models which are proxies for short-range models in 3, 4, and 10 dimensions with previously-obtained data for the 3D and 4D short-range models and the SK model. For one statistic, which is expected to sharply distinguish between the two pictures in the thermodynamic limit, we find that larger system sizes than those currently feasible to simulate are needed to obtain an unambiguous result. We also find that two other recently-proposed statistics, the median of the cumulative overlap distribution and the "typical" overlap distribution, are not particularly helpful in distinguishing between the RSB and droplet pictures.If there are many pure states in the spin-glass phase, we need to carry out some sort of statistical average over them to obtain the thermodynamics. One such prescription for doing this is called the "metastate." Motivated by similarities between the average over pure states specified by the metastate theory and that presumably generated by the nonequilibrium dynamics, we study a 1DLR model which is a proxy for a short-range model in
d=8 dimensions and measure the evolution of dynamical correlations. We find that the spatial decay of the correlations at distances less than the dynamical correlation length
xi(t) agrees quantitatively with the predictions of the metastate theory, evaluated according to the RSB picture. We also compute the dynamic exponent defined by
xi(t)proptot1/z(T) and find that it is compatible with the mean-field value of the critical dynamical exponent for short-range spin glasses.Finally, we present a unified view of finite-size scaling (FSS) in dimensions
d above the upper critical dimension
du, for both free and periodic boundary conditions. For
d>du, a dangerous irrelevant variable is responsible for both the violation of hyperscaling and the violation of "standard" FSS. We find that the modified hyperscaling proposed to allow for this applies only to
veck=vec0 fluctuations, while standard FSS applies to
veckneqvec0 fluctuations. Hence the exponent
eta describing the power-law decay of correlations at criticality is unambiguously
eta=0. With free boundary conditions, the finite-size "shift" is greater than the rounding. Nonetheless, using
T−TL, where
TL is the finite-size pseudocritical temperature, as the scaling variable, the data do collapse onto a scaling form that includes the behavior both at
TL, where the susceptibility
chi diverges like
Ld/2, and the bulk
Tc, where it diverges like
L2. We support these claims with data from large-scale simulations of the five-dimensional Ising model.
For the one-dimensional long-ranged Ising spin-glass with random couplings decaying with the distance r as J ( r ) ∼ r − σ and distributed with the Lévy symmetric stable distribution of index 1 < μ ≤ 2 (including the usual Gaussian case μ = 2), we consider the region σ > 1/ μ where the energy is extensive. We study two real space renormalization procedures at zero temperature, namely a simple box decimation that leads to explicit calculations, and a strong disorder decimation that can be studied numerically on large sizes. The droplet exponent governing the scaling of the renormalized couplings J L ∝ L θ μ ( σ ) is found to be whenever the long-ranged couplings are relevant θ μ ( σ ) ≥ −1. For the statistics of the ground state energy over disordered samples, we obtain that the droplet exponent θ μ ( σ ) governs the leading correction to extensivity of the averaged value . The characteristic scale of the fluctuations around this average is of order , and the rescaled variable is Gaussian distributed for μ = 2, or displays the negative power-law tail in 1/(− u ) 1+ μ for u → − ∞ in the Lévy case 1 < μ < 2. Finally we apply the zero-temperature renormalization procedure to the related Dyson hierarchical spin-glass model where the same droplet exponent appears.
For the long-range one-dimensional Ising spin-glass with random couplings decaying as J ( r ) ∝ r − σ , the scaling of the effective coupling defined as the difference between the free energies corresponding to periodic and antiperiodic boundary conditions J R ( N ) ≡ F (P) ( N ) − F (AP) ( N ) ∼ N θ ( σ ) defines the droplet exponent θ ( σ ). Here we study numerically the instability of the renormalization flow of the effective coupling J R ( N ) with respect to magnetic, disorder and temperature perturbations respectively, in order to extract the corresponding chaos exponents ζ H ( σ ), ζ J ( σ ) and ζ T ( σ ) as a function of σ . Our results for ζ T ( σ ) are interpreted in terms of the entropy exponent θ S ( σ ) ≃ 1/3 which governs the scaling of the entropy difference S (P) ( N ) − S (AP) ( N ) ∼ N θ S ( σ ) . We also study the instability of the ground state configuration with respect to perturbations, as measured by the spin overlap between the unperturbed and the perturbed ground states, in order to extract the corresponding chaos exponents and .
The one-dimensional long-range Ising spin glass provides useful insights into the properties of finite-dimensional spin glasses with short-range interactions. The defect energy renormalization-group equations derived for it by Kotliar, Anderson, and Stein have been reexamined and a new fixed point has been found. The fixed points which have previously been studied are found to be inappropriate. It is shown that the renormalization-group equations themselves directly imply that that the spin-glass phase is replica symmetric, in agreement with droplet model expectations, when the range exponent $\ensuremath{\sigma}$ lies between the upper and lower critical values.
By using problem-dependent mappings to computer-science problems and by applying sophisticated algorithms, one can numerically study some important problems much better compared to applying standard approaches, such as Monte Carlo simulations. Here, by using calculations of ground states of suitable perturbed systems, droplets are obtained in two-dimensional $\ifmmode\pm\else\textpm\fi{}J$ spin glasses, which are in the focus of a currently very lively debate. By using a sophisticated matching algorithm, exact ground states of large systems up to
L2=2562 spins can be generated. Furthermore, no equilibration or extrapolation to
T=0 is necessary. Three different $\ifmmode\pm\else\textpm\fi{}J$ models are studied here: (a) with free boundary conditions, (b) with fixed boundary conditions, and (c) a diluted system wherein 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 ${E}_{\text{dp}}\ensuremath{\sim}{L}^{{\ensuremath{\theta}}_{\text{dp}}^{\ensuremath{'}}}$ with ${\ensuremath{\theta}}_{\text{dp}}^{\ensuremath{'}}<0$. This is different from the previous studies of domain walls, wherein a convergence to a constant nonzero value $({\ensuremath{\theta}}_{\text{DW}}=0)$ was found. After correcting for the noncompactness of the droplets, the results are likely to be compatible with ${\ensuremath{\theta}}_{\text{dp}}\ensuremath{\approx}\ensuremath{-}0.29$ for all three models. This is in accordance with the Gaussian system wherein ${\ensuremath{\theta}}_{\text{dp}}=\ensuremath{-}0.287(4)$ ($\ensuremath{\nu}\ensuremath{\approx}3.5$ via $\ensuremath{\nu}=\ensuremath{-}1/{\ensuremath{\theta}}_{\text{dp}}$). Nevertheless, the disorder-averaged spin-spin correlation exponent $\ensuremath{\eta}$ is determined here via the probability to have a nonzero-energy droplet, and $\ensuremath{\eta}\ensuremath{\approx}0.22$ is found for all three models, in contrast to the model with the Gaussian interactions, where $\ensuremath{\eta}=0$ is exact.