We consider spin glass models in which the number of spin components m is infinite. In the formulation of the problem appropriate for numerical calculations proposed by several authors, we show that the order parameter defined by the long-distance limit of the correlation functions is actually zero and there is only "quasi-long-range order" below the transition temperature. Nonetheless, there can be a finite temperature phase transition where the decay of correlations changes from exponential to power law. We also show that the spin glass transition temperature is zero in three dimensions so power-law behavior only occurs at T=0 in this case. We also argue that the order of limits, m-->infinity and N-->infinity is important, where N is the number of spins.
We consider spin glass models in which the number of spin components m is infinite. In the formulation of the problem appropriate for numerical calculations proposed by several authors, we show that the order parameter defined by the long-distance limit of the correlation functions is actually zero and there is only "quasi-long-range order" below the transition temperature. Nonetheless, there can be a finite temperature phase transition where the decay of correlations changes from exponential to power law. We also show that the spin glass transition temperature is zero in three dimensions so power-law behavior only occurs at T=0 in this case. We also argue that the order of limits, m-->infinity and N-->infinity is important, where N is the number of spins.
We investigate zero and finite-temperature properties of the one-dimensional spin-glass model for vector spins in the limit of an infinite number m of spin components where the interactions decay with a power, $\ensuremath{\sigma}$, of the distance. A diluted version of this model is also studied, but found to deviate significantly from the fully connected model. At zero temperature, defect energies are determined from the difference in ground-state energies between systems with periodic and antiperiodic boundary conditions to determine the dependence of the defect-energy exponent $\ensuremath{\theta}$ on $\ensuremath{\sigma}$. A good fit to this dependence is $\ensuremath{\theta}=3/4\ensuremath{-}\ensuremath{\sigma}$. This implies that the upper critical value of $\ensuremath{\sigma}$ is 3/4, corresponding to the lower critical dimension in the d-dimensional short-range version of the model. For finite temperatures, the large m saddle-point equations are solved self-consistently, which gives access to the correlation function, the order parameter, and the spin-glass susceptibility. Special attention is paid to the different forms of finite-size scaling effects below and above the lower critical value, $\ensuremath{\sigma}=5/8$, which corresponds to the upper critical dimension 8 of the hypercubic short-range model.
The d=1 Ising ferromagnet and spin glass with long-range power-law interactions J r^-a is studied for all interaction range exponents a by a renormalization-group transformation that simultaneously projects local ferromagnetism and antiferromagnetism. In the ferromagnetic case, J>0, a finite-temperature ferromagnetic phase occurs for interaction range 0.742, the phase transition temperature discontinuously drops to zero and for a>2 there is no ordered phase above zero temperature, also as predicted by rigorous results. At the other end, on approaching a=0.74 from above, namely increasing the range of the interaction, the phase transition temperature diverges to infinity, meaning that, at all non-infinite temperatures, the system is ferromagnetically ordered. Thus, the equivalent-neighbor interactions regime is entered before (a > 0) the neighbors become equivalent, namely before the interactions become equal for all separations. The critical exponents alpha,beta, gamma,delta,eta,nu are calculated, from a large recursion matrix, varying as function of a. For antiferromagnetic J<0, all triplets of spins at all ranges have competing interactions and this highly frustrated system does not have an ordered phase. In the spin-glass system, where all couplings for all separations are randomly ferromagnetic or antiferromagnetic (with probability p), a finite-temperatures spin-glass phase is obtained in the absence of antiferromagnetic phase. In the spin-glass phase, the signature chaotic behavior under scale change occurs in a richer version than previously: In the long-range interaction of this system, the interactions at every separation become chaotic, yielding a piecewise chaotic interaction function.
The standard short-range two-dimensional Ising spin glass is numerically well accessible, in particular, because there are polynomial-time ground-state algorithms. On the other hand, in contrast to higher dimensional spin glasses, it does not exhibit a rich behavior, i.e., no ordered phase at finite temperature. Here, we investigate whether long-range correlated bonds change this behavior. This would still keep the model numerically well accessible while exhibiting a more interesting behavior. The bonds are drawn from a Gaussian distribution with a two-point correlation for bonds at distance r that decays as (1+r^{2})^{-a/2}, a≥0. We study numerically with exact algorithms the ground-state and domain-wall excitations. Our results indicate that the inclusion of bond correlations still does not lead to a spin-glass order at any finite temperature. A further analysis reveals that bond correlations have a strong effect at local length scales, inducing ferro- and antiferromagnetic domains into the system. The length scale of ferro- and antiferromagnetic order diverges exponentially as the correlation exponent approaches a critical value, a→a_{crit}=0. Thus, our results suggest that the system becomes a ferro- or antiferromagnet only in the limit a→0.
The nature of the ordering of the one-dimensional Heisenberg spin-glass model with a long-range power-law interaction is studied by extensive Monte Carlo simulations, with particular attention to the issue of the spin-chirality decoupling/coupling. Large system sizes up to L=4096 are studied. With varying the exponent σ describing the power-law interaction, we observe three distinct types of ordering regimes. For smaller σ, the spin and the chirality order at a common finite temperature with a common correlation-length exponent, exhibiting the standard spin-chirality coupling behavior. For intermediate σ, the chirality orders at a temperature higher than the spin, exhibiting the spin-chirality decoupling behavior. For larger σ, both the spin and the chirality order at zero temperature. We construct a phase diagram in the σ versus the temperature plane, and discuss implications of the results. Critical properties associated with both the chiral-glass and the spin-glass transitions are also determined.
We study a p-spin spin-glass model to understand if the finite-temperature glass transition found in the mean-field regime of p-spin models, and used to model the behavior of structural glasses, persists in the nonmean-field regime. By using a three-spin spin-glass model with long-range power-law diluted interactions we are able to continuously tune the (effective) space dimension via the exponent of the interactions. Monte Carlo simulations of the spin-glass susceptibility and the two-point finite-size correlation length show that deep in the nonmean-field regime, the finite-temperature transition is lost whereas this is not the case in the mean-field regime, in agreement with the prediction of Moore and Drossel [Phys. Rev. Lett. 89, 217202 (2002)] that three-spin models are in the same universality class as an Ising spin glass in a magnetic field. However, slightly in the nonmean-field region, we find an apparent transition in the three-spin model, in contrast to results for the Ising spin glass in a field. This may indicate that even larger sizes are needed to probe the asymptotic behavior in this region.
We investigate the balanced M=4, p=4 spin-glass model for a one-dimensional long-range proxy for the finite dimensional short-range p-spin glass model to examine the nature of the glass transition beyond mean-field theory. We perform large-scale Monte Carlo equilibrated simulations for both fully connected and power-law diluted versions of the model. The critical temperatures extracted from the finite-size scaling (FSS) analysis of spin-glass susceptibility are in good agreement with theoretical predictions for σ=0,0.25, and 0.55. For these values of the long-range exponent σ (which is the power of the decrease of the interactions between the spins with their separation), one might have expected that mean-field theory would provide a good description of the system. However, the spin-overlap distribution and the value of the λ-parameter do not provide numerical evidence for a one-step replica symmetry breaking (1RSB) phase transition. Instead, our results indicate a direct transition from the paramagnetic state to a full replica symmetry broken phase, with a renormalized value of λ≡ω2/ω1<1 suggesting a continuous FRSB transition, despite this ratio being equal to 2 at mean-field level. A value of λ>1 is required for the discontinuous 1RSB transition. We argue that strong finite-size effects and closely spaced transition temperatures remove the expected 1RSB transition for the system sizes which we can study. For values of the exponent σ=0.85, which roughly corresponds to a three dimensional system, we find that the renormalized value of λ is again less than 1, with no signs of either the 1RSB transition or the continuous FRSB transition, suggesting that the Kauzmann temperature TK in three dimensions might be zero and the complete absence of phase transitions in structural glasses.