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Physical review. B, Condensed matter

Dynamical properties of a cluster model of spin-glasses

Wolfgang Kinzel

Published Dec 1, 1982• 26 citations
Publisher

Abstract

The dynamics of small decoupled Ising clusters with Gaussian-distributed interactions is examined. The exactly solved case of 4-spin clusters shows many dynamic properties of spinglasses. With a smoothly increasing cluster size the nonequilibrium susceptibility $\ensuremath{\chi}$ is in quantitative agreement with computer simulations of the Edwards-Anderson model of spin-glasses. A smooth change of the microscopic time constant gives a sharp peak in $\ensuremath{\chi}(T)$. Our results support the dynamical nature of the spin-glass transition.

DOI
10.1103/physrevb.26.6303

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The dynamics of small decoupled Ising clusters with Gaussian-distributed interactions is examined. The exactly solved case of 4-spin clusters shows many dynamic properties of spinglasses. With a smoothly increasing cluster size the nonequilibrium susceptibility \ensuremathχ\ensuremath{\chi}\ensuremathχ is in quantitative agreement with computer simulations of the Edwards-Anderson model of spin-glasses. A smooth change of the microscopic time constant gives a sharp peak in \ensuremathχ(T)\ensuremath{\chi}(T)\ensuremathχ(T). Our results support the dynamical nature of the spin-glass transition.
A simple approximation procedure is introduced which quantitatively accounts for the low-temperature logarithmic decay in time of the autocorrelation function for the Edwards-Anderson model of an Ising spin glass. The relationship between the autocorrelation function and the susceptibility is used to derive the dependence on observation time of the susceptibility. The order parameter q(t) is calculated and is in good agreement with the Monte Carlo simulations on two- and three-dimensional systems with Gaussian bond distributions, which show that q(t) tends to zero for long times t.
We examine spin dynamics just above a spin-glass transition for two classical soft-spin models with random exchange interactions. The statics of the two models are identical; in mean-field theory we find an Edwards-Anderson transition of the usual sort. The dynamics are described by Langevin equations: in one case we use a purely dissipative equation of motion for a nonconserved scalar spin, while in the other we include a "mode-coupling" term to describe the precession of a vector spin in the field of its neighbors and impose conservation of the total spin. In the first case we find critical slowing down as the transition at Tg{T}_{g}Tg​ is approached from above (and ${t}^{\ensuremath{-}\frac{1}{2}}$ tails below Tg{T}_{g}Tg​), in agreement with the result of Ma and Rudnick for a different model. In the second case, we find a divergent spin-diffusion constant: $D\ensuremath{\propto}{(T\ensuremath{-}{T}_{g})}^{\ensuremath{-}\frac{1}{2}}$ (d>4d>4d>4) and $D\ensuremath{\propto}{(T\ensuremath{-}{T}_{g})}^{\frac{2}{d}\ensuremath{-}1}$ (2<d<42<d<42<d<4), in agreement with our previous calculation for a mode-coupling version of the Ma-Rudnick model.
We investigate numerically disorder chaos in spin glasses, i.e. the sensitivity of the ground state to small changes of the random couplings. Our study focuses on the Edwards-Anderson model in d=1,2,3 and in mean-field. We find that in all cases, simple scaling laws, involving the size of the system and the strength of the perturbation, are obeyed. We characterize in detail the distribution of overlap between ground states and the geometrical properties of flipped spin clusters in both the weak and strong chaos regime. The possible relevance of these results to temperature chaos is discussed.
We numerically study the nonequilibrium dynamics of the three-dimensional Heisenberg Edwards-Anderson spin glass following a sudden quench to its low-temperature phase. The subsequent aging behavior of the system is analyzed in detail, and the scaling behavior of various space-time correlation functions is investigated for both spin and chiral degrees of freedom. We carefully compare our results with those obtained from simulations of the more studied Ising version of the model, and with experiments on real spin glasses in which the spins have vectorial character. Finally, the present dynamical study offers perspectives into the possibility of spin-chirality decoupling at low temperature in vectorial spin glasses.
We characterize numerically the properties of the phase transition of the three-dimensional Ising spin glass with Gaussian couplings and of the low-temperature phase. We compute critical exponents on large lattices. We study in detail the overlap probability distribution and the equilibrium overlap-overlap correlation functions. We find a clear agreement with off-equilibrium results from previous work. These results strongly support the existence of a continuous spontaneous replica symmetry breaking in three-dimensional spin glasses.