We consider the Dyson hierarchical version of the quantum spin-glass with random Gaussian couplings characterized by the power-law decaying variance and a uniform transverse field h . The ground state is studied via real-space renormalization to characterize the spin-glass-paramagnetic zero temperature quantum phase transition as a function of the control parameter h . In the spin-glass phase , the typical renormalized coupling grows with the length scale L as the power-law with the classical droplet exponent , where the stiffness modulus vanishes at criticality , whereas the typical renormalized transverse field decays exponentially in terms of the diverging correlation length . At the critical point , the typical renormalized coupling and the typical renormalized transverse field display the same power-law behavior L − z with a finite dynamical exponent z . The RG rules are applied numerically to chains containing L = 2 12 = 4096 spins in order to measure these critical exponents for various values of σ in the region .
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 .
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.
Understanding the low-temperature pure state structure of spin glasses remains an open problem in the field of statistical mechanics of disordered systems. Here we study Monte Carlo dynamics, performing simulations of the growth of correlations following a quench from infinite temperature to a temperature well below the spin-glass transition temperature T_{c} for a one-dimensional Ising spin-glass model with diluted long-range interactions. In this model, the probability P_{ij} that an edge {i,j} has nonvanishing interaction falls as a power law with chord distance, P_{ij}∝1/R_{ij}^{2σ}, and we study a range of values of σ with 1/2<σ<1. We consider a correlation function C_{4}(r,t). A dynamic correlation length that shows power-law growth with time ξ(t)∝t^{1/z} can be identified in the data and, for large time t, C_{4}(r,t) decays as a power law r^{-α_{d}} with distance r when r≪ξ(t). The calculation can be interpreted in terms of the maturation metastate averaged Gibbs state, or MMAS, and the decay exponent α_{d} differentiates between a trivial MMAS (α_{d}=0), as expected in the droplet picture of spin glasses, and a nontrivial MMAS (α_{d}≠0), as in the replica-symmetry-breaking (RSB) or chaotic pairs pictures. We find nonzero α_{d} even in the regime σ>2/3 which corresponds to short-range systems below six dimensions. For σ<2/3, the decay exponent α_{d} follows the RSB prediction for the decay exponent α_{s}=3-4σ of the static metastate, consistent with a conjectured statics-dynamics relation, while it approaches α_{d}=1-σ in the regime 2/3<σ<1; however, it deviates from both lines in the vicinity of σ=2/3.
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.
The one-dimensional Ising spin-glass model with power-law long-range interactions is a useful proxy model for studying spin glasses in higher space dimensions and for finding the dimension at which the spin-glass state changes from having broken replica symmetry to that of droplet behavior. To this end we have calculated the exponent that describes the difference in free energy between periodic and antiperiodic boundary conditions. Numerical work is done to support some of the assumptions made in the calculations and to determine the behavior of the interface free-energy exponent of the power law of the interactions. Our numerical results for the interface free-energy exponent are badly affected by finite-size problems.