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arXiv:1207.7014 · Physical Review B

Correspondence between long-range and short-range spin glasses

R. A. Baños, L. A. Fernández, Víctor Martín-Mayor, A. P. Young

Published Oct 17, 2012• 50 citations
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Abstract

We compare the critical behavior of the short-range Ising spin glass with a spin glass with long-range interactions which fall off as a power $\ensuremath{\sigma}$ of the distance. We show that there is a value of $\ensuremath{\sigma}$ of the long-range model for which the critical behavior is very similar to that of the short-range model in four dimensions. We also study a value of $\ensuremath{\sigma}$ for which we find the critical behavior to be compatible with that of the three-dimensional model, although we have much less precision than in the four-dimensional case.

DOI
10.1103/physrevb.86.134416

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We compare the critical behavior of the short-range Ising spin glass with a spin glass with long-range interactions which fall off as a power \ensuremathσ\ensuremath{\sigma}\ensuremathσ of the distance. We show that there is a value of \ensuremathσ\ensuremath{\sigma}\ensuremathσ of the long-range model for which the critical behavior is very similar to that of the short-range model in four dimensions. We also study a value of \ensuremathσ\ensuremath{\sigma}\ensuremathσ for which we find the critical behavior to be compatible with that of the three-dimensional model, although we have much less precision than in the four-dimensional case.
Abstract We present results of numerical simulations on a one-dimensional Ising spin glass with long-range interactions. Parameters of the model are chosen such that it is a proxy for a short-range spin glass above the upper critical dimension (i.e. in the mean-field regime). The system is quenched to a temperature well below the transition temperature and the growth of correlations is observed. The spatial decay of the correlations at distances less than the dynamic correlation length agrees quantitatively with the predictions of a static theory, the ‘metastate’, evaluated according to the replica symmetry breaking (RSB) theory. We also compute the dynamic exponent z ( T ) defined by and find that it is compatible with the mean-field value of the critical dynamical exponent for short range spin glasses.
A model of the quantum (transverse) Ising spin glass with a long-range random interaction is proposed and studied here. In this model the random interaction between two spins at a distance rij{r}_{\mathrm{ij}}rij​ apart falls algebraically as ${1/r}_{\mathrm{ij}}^{(d+\ensuremath{\sigma})/2}.$ Here, apart from the strength of the quantum fluctuations, the interaction range $\ensuremath{\sigma}$ is also tunable. We have studied the ground-state properties of the model, extending the ``Droplet model'' of the short-range quantum spin glass. The model is also studied using a modified form of the effective Landau action describing the transition of the short-range quantum spin glass. The important features which are due to the long-range interaction are clearly mentioned. Field theoretical renormalization group calculations fail to locate any stable weak-coupling fixed point in the non-mean-field region. The simplest conjecture is that beyond the mean-field region, the critical behavior is governed by the infinite randomness fixed point. We extend the phenomenological scaling relation for the short-range quantum spin glass (based on the assumption of the existence of a dangerously irrelevant operator) to the present long-range interacting case. Most of the possible interesting aspects associated with the quantum transition in the present model are elaborately discussed.
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.
The one-dimensional Ising model in a random field and with power-law decaying ferromagnetic bonds is studied at zero temperature. Comparing the scaling of the energy contributions of the ferromagnetic domain wall flip and of the random field \`a la Imry-Ma, a threshold value for the power $\ensuremath{\rho}$ of the long-range interaction can be determined, beyond which no critical behavior occurs. The critical threshold value is ${\ensuremath{\rho}}_{c}=3/2$, at a difference with the zero field model in which ${\ensuremath{\rho}}_{c}=2$. This prediction is analyzed by numerical computation of the ground states below, at, and above this threshold value. The analogy between the critical behavior of long-range one-dimensional systems and in short-range DDD-dimensional systems is investigated. At the critical threshold value of $\ensuremath{\rho}$, corresponding to the lower critical dimension, numerical evidence is found for a zero temperature transition at a finite critical field. Possible finite size crossover effects are discussed in this case. Some implications for the critical behavior of spin glasses in a field are conjectured.
We have studied numerically the states reached in a quench from various temperatures in the one-dimensional fully-connected Kotliar, Anderson and Stein Ising spin glass model. This is a model where there are long-range interactions between the spins which falls off as a power σ\sigmaσ of their separation. We have made a detailed study in particular of th e energies of the states reached in a quench from infinite temperature and their overlaps, including the spin glass susceptibility. In the regime where σ≤1/2\sigma \le 1/2σ≤1/2, where the model is similar to the Sherrington-Kirkpatrick model, we find that the spin glass susceptibility diverges logarithmically with increasing NNN, the number of spins in the syste m, whereas for σ>1/2\sigma> 1/2σ>1/2 it remains finite. We attribute the behavior for σ≤1/2\sigma \le 1/2σ≤1/2 to \emph {self-organized critical behavior}, where the system after the quench is cl ose to the transition between states which have trivial overlaps and those with the non-trivial overlaps associated with replica symmetry breaking.
We study the critical behavior of the short-range ppp-state Potts spin glass in three and four dimensions using Monte Carlo simulations. In three dimensions, for p=3p=3p=3, a finite-size scaling analysis of the correlation length shows clear evidence of a transition to a spin-glass phase at ${T}_{\mathrm{c}}\ensuremath{\simeq}0.273$ for a Gaussian distribution of interactions and ${T}_{\mathrm{c}}\ensuremath{\simeq}0.377$ for a $\ifmmode\pm\else\textpm\fi{}J$ (bimodal) distribution. These results indicate that the lower critical dimension of the three-state Potts glass is below 3. By contrast, the correlation length of the ten-state (p=10)(p=10)(p=10) Potts glass in three dimensions remains small even at very low temperatures and thus shows no sign of a transition. In four dimensions we find that the p=3p=3p=3 Potts glass with Gaussian interactions has a spin-glass transition at ${T}_{\mathrm{c}}\ensuremath{\simeq}0.536$.