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arXiv:cond-mat/0203316 · Journal of Physics A Mathematical and General

Low T dynamical properties of spin glasses smoothly extrapolate to T $equal$ 0

Enzo Marinari, Giorgio Parisi A, Juan Jesus Ruiz-Lorenzo

Published Jul 30, 2002• 4 citations
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Abstract

We compare ground state properties of 3D Ising spin glasses with Gaussian couplings using results from off-equilibrium numerical simulations at non-zero (but low) temperatures. We find that the non-zero temperature properties of the system smoothly connect to the T = 0 behaviour, confirming the point of view that results established at T = 0 typically also give relevant information about the T ≠ 0 physics of the system.

DOI
10.1088/0305-4470/35/32/303

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Low T dynamical properties of spin glasses smoothly extrapolate to T equalequalequal 0
We compare ground state properties of 3D Ising spin glasses with Gaussian couplings using results from off-equilibrium numerical simulations at non-zero (but low) temperatures. We find that the non-zero temperature properties of the system smoothly connect to the T = 0 behaviour, confirming the point of view that results established at T = 0 typically also give relevant information about the T ≠ 0 physics of the system.
Experimentally derived physical properties of 3d Ising spin glasses are discussed and compared to what is anticipated from theory and simulations of model systems. It is found that the spin glass experiences a zero-field phase transition to a low-temperature spin-glass phase, that there is no phase transition in a magnetic field and that nonequilibrium and chaos effects often dominate the dynamics of the low-temperature glassy phase. These results are in agreement with findings from Monte Carlo simulations and with properties derived from the droplet model.
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.
We study the low-energy excitations of the three dimensional spin glass through a large-scale Monte Carlo simulation on lattices up to L=18L=18L=18. We find smooth extrapolations down to zero temperature, which, in the case of the energy and of the link overlap, can be directly -- and favourably -- compared with previous investigations featuring ground states (i.e., at zero temperature). The best fit for the fractal dimension of the excitations is provided by Replica-Symmetry Breaking theory, but we also consider the alternative TNT description. The P(q)P(q)P(q) is found to verify the Parisi-Toulouse temperature scaling. Our data provides a spectacular confirmation of the overlap-equivalence hypothesis.
We present a high-statistic systematic study of the overlap correlation function well below the critical temperature in the three-dimensional Gaussian spin glass. The off-equilibrium correlation function has been studied confirming the power law behaviour for the dynamical correlation length. In particular, we have computed the dynamical critical exponent z in a wide range of temperatures, 0.35 T 0.9, obtaining a dependence z ( T ) = 6.2/ T in very good agreement with recent experiments. Moreover, we report a study of the violation of the fluctuation-dissipation theorem for very low temperatures T = 0.5 and 0.35. All our numerical results avoid a droplet model interpretation even when T is as low as T = 0.35.
An analysis is given of numerical simulation data to size L=128 on the archetype square lattice Ising spin glasses (ISGs) with bimodal (±J) and Gaussian interaction distributions. It is well established that the ordering temperature of both models is zero. The Gaussian model has a nondegenerate ground state and thus a critical exponent η≡0, and a continuous distribution of energy levels. For the bimodal model, above a size-dependent crossover temperature T(*)(L) there is a regime of effectively continuous energy levels; below T(*)(L) there is a distinct regime dominated by the highly degenerate ground state plus an energy gap to the excited states. T(*)(L) tends to zero at very large L, leaving only the effectively continuous regime in the thermodynamic limit. The simulation data on both models are analyzed with the conventional scaling variable t=T and with a scaling variable τ(b)=T(2)/(1+T(2)) suitable for zero-temperature transition ISGs, together with appropriate scaling expressions. The data for the temperature dependence of the reduced susceptibility χ(τ(b),L) and second moment correlation length ξ(τ(b),L) in the thermodynamic limit regime are extrapolated to the τ(b)=0 critical limit. The Gaussian critical exponent estimates from the simulations, η=0 and ν=3.55(5), are in full agreement with the well-established values in the literature. The bimodal critical exponents, estimated from the thermodynamic limit regime analyses using the same extrapolation protocols as for the Gaussian model, are η=0.20(2) and ν=4.8(3), distinctly different from the Gaussian critical exponents.
We use a nonequilibrium Monte Carlo simulation method and dynamical scaling to study the phase transition in three-dimensional Ising spin glasses. The transition point is repeatedly approached at finite velocity v (temperature change versus time) in Monte Carlo simulations starting at a high temperature. This approach has the advantage that the equilibrium limit does not have to be strictly reached for a scaling analysis to yield critical exponents. For the dynamic exponent we obtain z=5.85(9) for bimodal couplings distribution and z=6.00(10) for the Gaussian case. Assuming universal dynamic scaling, we combine the two results and obtain z=5.93±0.07 for generic 3D Ising spin glasses.