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arXiv:1412.3954 · Physical Review B

Infinite volume extrapolation in the one-dimensional bond diluted Levy spin-glass model near its lower critical dimension

Luca Leuzzi, Giorgio Parisi A, Federico Ricci‐Tersenghi, Juan Jesus Ruiz-Lorenzo

Published Feb 5, 2015• 8 citations
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Abstract

We revisited, by means of numerical simulations, the one-dimensional bond diluted Levy Ising spin glasses outside the limit of validity of mean-field theories. In these models the probability that two spins at distance $r$ interact (via disordered interactions, ${J}_{ij}=\ifmmode\pm\else\textpm\fi{}1)$ decays as ${r}^{\ensuremath{-}\ensuremath{\rho}}$. We have estimated, using finite size scaling techniques, the infinite volume correlation length and spin glass susceptibility for $\ensuremath{\rho}=5/3$ and $\ensuremath{\rho}=9/5$. We have obtained strong evidence for divergences of the previous observables at a nonzero critical temperature. We discuss the behavior of the critical exponents, especially when approaching the value $\ensuremath{\rho}=2$, corresponding to a critical threshold beyond which the model has no phase transition. Finally, we numerically study the model right at the threshold value $\ensuremath{\rho}=2$.

DOI
10.1103/physrevb.91.064202

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We revisited, by means of numerical simulations, the one-dimensional bond diluted Levy Ising spin glasses outside the limit of validity of mean-field theories. In these models the probability that two spins at distance rrr interact (via disordered interactions, Jij=\ifmmode±\else\textpm\fi1){J}_{ij}=\ifmmode\pm\else\textpm\fi{}1)Jij​=\ifmmode±\else\textpm\fi1) decays as r\ensuremath−\ensuremathρ{r}^{\ensuremath{-}\ensuremath{\rho}}r\ensuremath−\ensuremathρ. We have estimated, using finite size scaling techniques, the infinite volume correlation length and spin glass susceptibility for \ensuremathρ=5/3\ensuremath{\rho}=5/3\ensuremathρ=5/3 and \ensuremathρ=9/5\ensuremath{\rho}=9/5\ensuremathρ=9/5. We have obtained strong evidence for divergences of the previous observables at a nonzero critical temperature. We discuss the behavior of the critical exponents, especially when approaching the value \ensuremathρ=2\ensuremath{\rho}=2\ensuremathρ=2, corresponding to a critical threshold beyond which the model has no phase transition. Finally, we numerically study the model right at the threshold value \ensuremathρ=2\ensuremath{\rho}=2\ensuremathρ=2.
A spin-glass transition occurs both within and outside of the limits of validity of mean-field theory for a diluted one-dimensional chain of Ising spins where exchange bonds occur with a probability decaying as the inverse power of the distance. Varying the power in this long-range model corresponds, in a one-to-one relationship, to changing the dimension in spin-glass short-range models. Using different finite-size scaling methods, evidence for a spin-glass transition is also found for systems whose equivalent dimension is below the upper critical dimension at zero magnetic field. The application of a new method is discussed, which can be exported to systems in a magnetic field.
The spin-glass transition in a magnetic field is studied both in and out of the limit of validity of mean-field theory on a diluted one dimensional chain of Ising spins where exchange bonds occur with a probability decaying as the inverse power of the distance. Varying the power in this long-range model corresponds, in a one-to-one relationship, to changing the dimension in spin-glass short-range models. Evidence for a spin-glass transition in a magnetic field is found also for systems whose equivalent dimension is below the upper critical dimension in a zero magnetic field.
We investigate zero and finite-temperature properties of the one-dimensional spin-glass model for vector spins in the limit of an infinite number mmm 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 3/43/43/4, corresponding to the lower critical dimension in the ddd-dimensional short-range version of the model. For finite temperatures, the large mmm 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.
We study the de Almeida-Thouless (AT) line in the one-dimensional power-law diluted XY spin-glass model, in which the probability that two spins separated by a distance r interact with each other, decays as 1/r^{2σ}. Tuning the exponent σ is equivalent to changing the space dimension of a short-range model. We develop a heat bath algorithm to equilibrate XY spins; using this in conjunction with the standard parallel tempering and overrelaxation sweeps, we carry out large-scale Monte Carlo simulations. For σ=0.6, which is in the mean-field regime above six dimensions-it is similar to being in 10 dimensions-we find clear evidence for an AT line. For σ=0.75 and σ=0.85, which are in the non-mean-field regime and similar to four and three dimensions, respectively, our data is like that found in previous studies of the Ising and Heisenberg spin glasses when reducing the temperature at fixed field. For σ=0.75, there is evidence from finite-size-scaling studies for an AT transition but for σ=0.85, the evidence for a transition is nonexistent. We have also studied these systems at fixed temperature varying the field and discovered that at both σ=0.75 and at σ=0.85 there is evidence of an AT transition! Confusingly, the correlation length and spin-glass susceptibility as a function of the field are both entirely consistent with the predictions of the droplet picture and hence the nonexistence of an AT line. In the usual finite-size critical point scaling studies used to provide evidence for an AT transition, there is seemingly good evidence for an AT line at σ=0.75 for small values of the system size N, which is strengthening as N is increased, but for N>2048 the trend changes and the evidence then weakens as N is further increased. We have also studied with fewer bond realizations the system at σ=0.70, which is the analog of a system with short-range interactions just below six dimensions, and found that it is similar in its behavior to the system at σ=0.75 but with larger finite-size corrections. The evidence from our simulations points to the complete absence of the AT line in dimensions outside the mean-field region and to the correctness of the droplet picture. Previous simulations which suggested there was an AT line can be attributed to the consequences of studying systems which are just too small. The collapse of our data to the droplet scaling form is poor for σ=0.75 and to some extent also for σ=0.85, when the correlation length becomes of the order of the length of the system, due to the existence of excitations which only cost a free energy of O(1), just as envisaged in the TNT picture of the ordered state of spin glasses. However, for the case of σ=0.85 we can provide evidence that for larger system sizes, droplet scaling will prevail even when the correlation length is comparable to the system size.
Using large-scale Monte Carlo simulations that combine parallel tempering with specialized cluster updates, we show that Ising spin glasses with L\'evy-distributed interactions share the same universality class as Ising spin glasses with Gaussian or bimodal-distributed interactions. Corrections to scaling are large for L\'evy spin glasses. To overcome these and show that the critical exponents agree with the bimodal and Gaussian case, we perform an extended scaling of the two-point finite-size correlation length and the spin-glass susceptibility. Furthermore, we compute the critical temperature and compare its dependence on the disorder distribution width with recent analytical predictions [J. Stat. Mech. (2008) P04006].
We study the three-dimensional (3D) bond-diluted Edwards-Anderson (EA) model with binary interactions at a bond occupation of 45% by Monte Carlo (MC) simulations. Using an efficient cluster MC algorithm we are able to determine the universal finite-size scaling (FSS) functions and the critical exponents with high statistical accuracy. We observe small corrections to scaling for the measured observables. The critical quantities and the FSS functions indicate clearly that the bond-diluted model for dilutions above the critical dilution p∗{p}^{*}p∗, at which a spin glass (SG) phase appears, lies in the same universality class as the 3D undiluted EA model with binary interactions. A comparison with the FSS functions of the 3D site-diluted EA model with Gaussian interactions at a site occupation of 62.5% gives very strong evidence for the universality of the SG transition in the 3D EA model.