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arXiv:1309.2154 · Physical Review B

Scaling of the largest dynamical barrier in the one-dimensional long-range Ising spin glass

Cécile Monthus, Thomas Garel

Published Jan 9, 2014• 11 citations
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Abstract

The long-range one-dimensional Ising spin glass with random couplings decaying as $J(r)\ensuremath{\propto}{r}^{\ensuremath{-}\ensuremath{\sigma}}$ presents a spin-glass phase ${T}_{c}(\ensuremath{\sigma})>0$ for $0\ensuremath{\le}\ensuremath{\sigma}<1$ (the limit $\ensuremath{\sigma}=0$ corresponds to the mean-field Sherrington-Kirkpatrick model). We use the eigenvalue method introduced in our previous work (C. Monthus and T. Garel, J. Stat. Mech. 2009, P12017) to measure the equilibrium time ${t}_{eq}(N)$ at temperature $T={T}_{c}(\ensuremath{\sigma})/2$ as a function of the number $N$ of spins. We find the activated scaling $\overline{ln{t}_{eq}(N)}\ensuremath{\sim}{N}^{\ensuremath{\psi}}$ with the same barrier exponent $\ensuremath{\psi}\ensuremath{\simeq}0.33$ in the whole region $0\ensuremath{\le}\ensuremath{\sigma}<1$.

DOI
10.1103/physrevb.89.014408

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The long-range one-dimensional Ising spin glass with random couplings decaying as J(r)\ensuremath∝r\ensuremath−\ensuremathσJ(r)\ensuremath{\propto}{r}^{\ensuremath{-}\ensuremath{\sigma}}J(r)\ensuremath∝r\ensuremath−\ensuremathσ presents a spin-glass phase Tc(\ensuremathσ)>0{T}_{c}(\ensuremath{\sigma})>0Tc​(\ensuremathσ)>0 for 0\ensuremath≤\ensuremathσ<10\ensuremath{\le}\ensuremath{\sigma}<10\ensuremath≤\ensuremathσ<1 (the limit \ensuremathσ=0\ensuremath{\sigma}=0\ensuremathσ=0 corresponds to the mean-field Sherrington-Kirkpatrick model). We use the eigenvalue method introduced in our previous work (C. Monthus and T. Garel, J. Stat. Mech. 2009, P12017) to measure the equilibrium time teq(N){t}_{eq}(N)teq​(N) at temperature T=Tc(\ensuremathσ)/2T={T}_{c}(\ensuremath{\sigma})/2T=Tc​(\ensuremathσ)/2 as a function of the number NNN of spins. We find the activated scaling lnteq(N)‾\ensuremath∼N\ensuremathψ\overline{ln{t}_{eq}(N)}\ensuremath{\sim}{N}^{\ensuremath{\psi}}lnteq​(N)​\ensuremath∼N\ensuremathψ with the same barrier exponent \ensuremathψ\ensuremath≃0.33\ensuremath{\psi}\ensuremath{\simeq}0.33\ensuremathψ\ensuremath≃0.33 in the whole region 0\ensuremath≤\ensuremathσ<10\ensuremath{\le}\ensuremath{\sigma}<10\ensuremath≤\ensuremathσ<1.
We present results of a Monte Carlo study of the equilibrium dynamics of the one dimensional long-range Ising spin glass model. By tuning a parameter , this model interpolates between the mean field Sherrington–Kirkpatrick model and a proxy of the finite dimensional Edward-Anderson model. Activated scaling fits for the behavior of the relaxation time as a function of the number of spins N (Namely ) give values of that are not stable against inclusion of subleading corrections. Critical scaling ( ) gives more stable fits, at least in the non mean field region. We also present results on the scaling of the time decay of the critical remanent magnetization of the Sherrington–Kirkpatrick model, a case where the simulation can be done with quite large systems and that shows the difficulties in obtaining precise values for dynamical exponents in spin glass models.
A detailed numerical study is made of relaxation at equilibrium in the Sherrington-Kirkpatrick Ising spin glass (ISG) model, at and above the critical temperature Tg{T}_{g}Tg​. The data show a long time stretched exponential relaxation $q(t)\ensuremath{\sim}\mathrm{exp}{\ensuremath{-}{[t/\ensuremath{\tau}(T)]}^{\ensuremath{\beta}(T)}}$ with an exponent $\ensuremath{\beta}(T)$ tending to $\ensuremath{\approx}$1/3 at Tg{T}_{g}Tg​. The results are compared to those which were observed by A. T. Ogielski, [Phys. Rev. B 32, 7384 (1985)] in the three-dimensional ISG model and are discussed in terms of a phase space percolation transition scenario.
We consider the nonequilibrium behavior of the spin-glass ordered phase within the droplet scaling theory introduced previously. The fundamental long-time nonequilibrium process is assumed to be the thermally activated growth of spin-glass ordered domains. The remanent magnetization, m(t), in zero field is found to decay at long times as m(t)\ensuremath{\sim}${R}_{t}^{\mathrm{\ensuremath{-}}\ensuremath{\lambda}}$, where Rt{R}_{t}Rt​\ensuremath{\sim}(lnt${)}^{1/\ensuremath{\psi}}$ is the linear domain size, \ensuremath{\psi} is the previously introduced barrier exponent describing the growth of activation-barrier heights with length scale, and \ensuremath{\lambda} is a new nonequilibrium dynamic exponent, satisfying the relation \ensuremath{\lambda}\ensuremath{\ge}d/2 for d-dimensional systems. The effects of waiting for partial equilibration before making a measurement are studied in various regimes. The effects of quenching first to one temperature and then to another are also examined. Such experiments can, in principle, be used to obtain information about the relative rate of dynamic evolution as well as the overlap between the equilibrium states at different temperatures. In particular, the length scale ${L}_{\ensuremath{\Delta}T}$, below which equilibrium correlations at temperatures T and T+\ensuremath{\Delta}T are similar, plays an important role. The decay of m(t) and the growth of spin-glass order after a quench are examined in Monte Carlo simulations of the Sherrington-Kirkpatrick model.
I present an analysis of the dynamic behavior of short-range Ising spin glasses observed in stochastic simulations. The time dependence of the order parameter q(t)=〈Sx{S}_{x}Sx​(0)Sx{S}_{x}Sx​(t)〉\ifmmode\bar\else\textasciimacron\fi{}---which is the same as that of the structure factor---and the time dependence of the related dynamic correlation functions have been recorded with good statistics and very long observation times. The spin-glass model with a symmetric distribution of discrete nearest-neighbor \ifmmode\pm\else\textpm\fi{}J interactions on a simple-cubic lattice was used. Simulations were performed with a special fast computer, allowing for the first-time investigation of the equilibrium dynamics for a wide range of temperatures (0.7\ensuremath{\le}kT/J\ensuremath{\le}5.0) and lattice sizes (83{8}^{3}83, 163{16}^{3}163, 323{32}^{3}323, and 643{64}^{3}643). I have found that the empirical formula q(t)=${\mathrm{ct}}^{\mathrm{\ensuremath{-}}x}$exp(-\ensuremath{\omega}${t}^{\ensuremath{\beta}}$) with temperature-dependent exponents x(T) and \ensuremath{\beta}(T) describes the decay very well at all temperatures above the spin-glass transition. In the spin-glass phase, only the algebraic decay q(t)=${\mathrm{ct}}^{\mathrm{\ensuremath{-}}x}$ could be observed, with different temperature dependences of the exponent x(T). The dynamic scaling hypothesis and finite-size scaling explain well the observed temperature and size dependence of the data, and the functional form of the correlation functions is com- patible with the scaling form if corrections to scaling are taken into account. The scaling behavior and the dynamic and static critical exponents found in my simulations are in reasonable agreement with recent experiments performed on insulating spin glasses, showing that despite its simplicity the discrete model of spin glasses analyzed in this work displays behavior similar to that seen in nature.
We study in Ising spin glasses the finite-size effects near the spin-glass transition in zero field and at the de Almeida-Thouless transition in a field by Monte Carlo methods and by analytical approximations. In zero field, the finite-size scaling function associated with the spin-glass susceptibility of the Sherrington-Kirkpatrick mean-field spin-glass model is of the same form as that of one-dimensional spin-glass models with power-law long-range interactions in the regime where they can be a proxy for the Edwards-Anderson short-range spin-glass model above the upper critical dimension. We also calculate a simple analytical approximation for the spin-glass susceptibility crossover function. The behavior of the spin-glass susceptibility near the de Almeida-Thouless transition line has also been studied, but here we have only been able to obtain analytically its behavior in the asymptotic limit above and below the transition. We have also simulated the one-dimensional system in a field in the non-mean-field regime to illustrate that when the Imry-Ma droplet length scale exceeds the system size one can then be erroneously lead to conclude that there is a de Almeida-Thouless transition even though it is absent.
For the long-range one-dimensional Ising spin-glass with random couplings decaying as J ( r ) ∝ r − σ , the scaling of the effective coupling defined as the difference between the free energies corresponding to periodic and antiperiodic boundary conditions J R ( N ) ≡ F (P) ( N ) − F (AP) ( N ) ∼ N θ ( σ ) defines the droplet exponent θ ( σ ). Here we study numerically the instability of the renormalization flow of the effective coupling J R ( N ) with respect to magnetic, disorder and temperature perturbations respectively, in order to extract the corresponding chaos exponents ζ H ( σ ), ζ J ( σ ) and ζ T ( σ ) as a function of σ . Our results for ζ T ( σ ) are interpreted in terms of the entropy exponent θ S ( σ ) ≃ 1/3 which governs the scaling of the entropy difference S (P) ( N ) − S (AP) ( N ) ∼ N θ S ( σ ) . We also study the instability of the ground state configuration with respect to perturbations, as measured by the spin overlap between the unperturbed and the perturbed ground states, in order to extract the corresponding chaos exponents and .