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arXiv:cond-mat/0411305 · Physical Review B

Correlation length of the two-dimensional Ising spin glass with bimodal interactions

Helmut G. Katzgraber, L. W. Lee

Published Apr 8, 2005• 44 citations
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Abstract

We study the correlation length of the two-dimensional Edwards-Anderson Ising spin glass with bimodal interactions using a combination of parallel tempering Monte Carlo updates and a rejection-free cluster algorithm in order to speed up equilibration. Our results show that the correlation length grows $\ensuremath{\sim}\mathrm{exp}(2J∕T)$ suggesting through hyperscaling that the degenerate ground state is separated from the first excited state by an energy gap $\ensuremath{\sim}4J$, as would naively be expected.

DOI
10.1103/physrevb.71.134404

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Physical review. B, Condensed matter•Jan 1, 1994•13 citations
We study the correlation length of the two-dimensional Edwards-Anderson Ising spin glass with bimodal interactions using a combination of parallel tempering Monte Carlo updates and a rejection-free cluster algorithm in order to speed up equilibration. Our results show that the correlation length grows \ensuremath∼exp(2J∕T)\ensuremath{\sim}\mathrm{exp}(2J∕T)\ensuremath∼exp(2J∕T) suggesting through hyperscaling that the degenerate ground state is separated from the first excited state by an energy gap \ensuremath∼4J\ensuremath{\sim}4J\ensuremath∼4J, as would naively be expected.
A detailed analysis of Monte Carlo data on the two-dimensional Ising spin glass with bimodal interactions shows that the free energy of the model has a nontrivial scaling. In particular, we show by studying the correlation length that much larger system sizes and lower temperatures are required to see the true critical behavior of the model in the thermodynamic limit. Our results agree with data by Lukic et al. in that the degenerate ground state is separated from the first excited state by an energy gap of 2J.
An exact algorithm is used to compute the degeneracies of the excited states of the bimodal Ising spin glass in two dimensions. It is found that the specific heat at arbitrary low temperature is not a self-averaging quantity and has a distribution that is neither normal nor lognormal. Nevertheless, it is possible to estimate the most likely value and this is found to scale as L 3 T -2 exp (−4 J / kT ), for a L × L lattice. Our analysis also explains, for the first time, why a correlation length ξ∼exp (2 J / kT ) is consistent with an energy gap of 2 J . Our method allows us to obtain results for up to 10 5 disorder realizations with L ⩽96. Distributions of second and third excitations are also shown.
We carry out simulated annealing and employ a generalized Kibble-Zurek scaling hypothesis to study the two-dimensional Ising spin glass with normal-distributed couplings. The system has an equilibrium glass transition at temperature T=0. From a scaling analysis when T→0 at different annealing velocities v, we find power-law scaling in the system size for the velocity required in order to relax toward the ground state, v∼L^{-(z+1/ν)}, the Kibble-Zurek ansatz where z is the dynamic critical exponent and ν the previously known correlation-length exponent, ν≈3.6. We find z≈13.6 for both the Edwards-Anderson spin-glass order parameter and the excess energy. This is different from a previous study of the system with bimodal couplings [Rubin et al., Phys. Rev. E 95, 052133 (2017)2470-004510.1103/PhysRevE.95.052133] where the dynamics is faster (z is smaller) and the above two quantities relax with different dynamic exponents (with that of the energy being larger). We argue that the different behaviors arise as a consequence of the different low-energy landscapes: for normal-distributed couplings the ground state is unique (up to a spin reflection), while the system with bimodal couplings is massively degenerate. Our results reinforce the conclusion of anomalous entropy-driven relaxation behavior in the bimodal Ising glass. In the case of a continuous coupling distribution, our results presented here also indicate that, although Kibble-Zurek scaling holds, the perturbative behavior normally applying in the slow limit breaks down, likely due to quasidegenerate states, and the scaling function takes a different form.
We perform Monte Carlo simulations of large two-dimensional Gaussian Ising spin glasses down to very low temperatures $\ensuremath{\beta}=1∕T=50$. Equilibration is ensured by using a cluster algorithm including Monte Carlo moves consisting of flipping fundamental excitations. We study the thermodynamic behavior using the Binder cumulant, the spin-glass susceptibility, the distribution of overlaps, the overlap with the ground state, and the specific heat. We confirm that Tc=0{T}_{c}=0Tc​=0. All results are compatible with an algebraic divergence of the correlation length with an exponent $\ensuremath{\nu}$. We find $\ensuremath{-}1∕\ensuremath{\nu}=\ensuremath{-}0.295(40)$, which is compatible with the value for the domain-wall and droplet exponent $\ensuremath{\theta}\ensuremath{\approx}\ensuremath{-}0.29$ found previously in ground-state studies. Hence the thermodynamic behavior of this model seems to be governed by one single exponent.
In order to determine if the effects of an overlap length are visible in an Ising Edwards-Anderson spin-glass model, we have performed Monte Carlo simulations that mimic experimental procedures. The spin system was quenched from a random initial condition to a temperature T1{\mathit{T}}_{1}T1​, equilibrated in zero field for a time tw1{\mathit{t}}_{\mathit{w}1}tw1​, whereupon the system was subjected to a perturbation in field \ensuremath{\Delta}H, temperature \ensuremath{\Delta}T, or bond \ensuremath{\Delta}J with a duration tw2{\mathit{t}}_{\mathit{w}2}tw2​. After the perturbation of the original field, the temperature or bonds were restored. After waiting for a time tw3{\mathit{t}}_{\mathit{w}3}tw3​, a weak magnetic field H was applied. The susceptibility \ensuremath{\chi}(t)=[M+{\mathit{M}}_{+}M+​(t)-${\mathit{M}}_{\mathrm{\ensuremath{-}}}$(t)]/2H, where ${\mathit{M}}_{\ifmmode\pm\else\textpm\fi{}}$(t) is the magnetization due to positive or negative field, and the spin autocorrelation function q(t) were studied as functions of time. For small perturbations, \ensuremath{\Delta}HH0{\mathit{H}}_{0}H0​, \ensuremath{\Delta}TT0{\mathit{T}}_{0}T0​, or \ensuremath{\Delta}JJ0{\mathit{J}}_{0}J0​, respectively, the system is undisturbed. For larger perturbations, the correlations built up during tw1{\mathit{t}}_{\mathit{w}1}tw1​ are gradually destroyed. Similar results are found in all three types of simulations and the qualitative features are in agreement with experimental results. These features are found for both two-dimensional and three-dimensional spin-glass systems. The results can be interpreted in the domain models for spin glasses. \ensuremath{\Delta}H0{\mathit{H}}_{0}H0​ is found to be strongly dependent on the duration tw2{\mathit{t}}_{\mathit{w}2}tw2​ of the perturbation. If tw1{\mathit{t}}_{\mathit{w}1}tw1​ is constant and tw2{\mathit{t}}_{\mathit{w}2}tw2​ increases, \ensuremath{\Delta}H0{\mathit{H}}_{0}H0​ decreases. If tw1{\mathit{t}}_{\mathit{w}1}tw1​ is set equal to tw2{\mathit{t}}_{\mathit{w}2}tw2​ and tw1{\mathit{t}}_{\mathit{w}1}tw1​=tw2{\mathit{t}}_{\mathit{w}2}tw2​ is increased, \ensuremath{\Delta}H0{\mathit{H}}_{0}H0​ decreases, in agreement with the expected behavior. \ensuremath{\Delta}H0{\mathit{H}}_{0}H0​ was found to increase with increasing temperature T, which is in agreement with experimental results, but contrary to the theoretical expectations.