Aging of two-dimensional (2D) Ising spin glasses with short-range interactions
Rb2${\mathrm{Cu}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$
CoxF4 (
Tc=0) is studied in the critical regime via the low-frequency ac susceptibility and via the dc magnetization for x=0.22 and 0.33. After quenching to low temperatures, slow logarithmic relaxation is observed for both the in-phase and out-of-phase ac susceptibilities. Using the droplet scaling model, the spin-glass correlation length R is concluded to increase after a temperature quench as R\ensuremath{\propto}(lnt${)}^{1/\mathrm{\ensuremath{\psi}}}$, with t in units of some microscopic time. The exponent \ensuremath{\psi}=1.0\ifmmode\pm\else\textpm\fi{}0.1 compares with recent theoretical results and numerical simulations for 2D short-range Ising spin glasses. The waiting-time dependence found for the field-cooled and zero-field-cooled dc magnetizations points to a markedly slower crossover from the quasiequilibrium to the nonequilibrium regime than in 3D. From experiments with different cooling histories, finally, the equilibrium spin-glass state is inferred to be strongly dependent on the temperature.
We study the nonequilibrium aging behavior of the $\ifmmode\pm\else\textpm\fi{}J$ Edwards-Anderson model in three dimensions for samples of size up to
N=1283 and for up to
108 Monte Carlo sweeps. In particular we are interested in the change of the aging when crossing from the spin-glass phase to the ferromagnetic phase. The necessary long simulation times are reached by employing a CUDA-based GPU implementation, which allows for single-spin flip times as small as 8 ps. We measure typical spin-glass correlation functions in space and time to determine the growing length scale and extract the constituting exponents. We observe a clear signature of the disorder-driven equilibrium transition in the nonequilibrium behavior.
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\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.
Relaxation functions of the spin-glass correlation length are investigated in the three-dimensional ±J XY model. The algebraic divergence is observed at the spin-glass transition temperature, Tsg≃0.45. The finite-time scaling analysis provides results consistent with our previous scaling analysis of the spin-glass susceptibility. The relaxation functions in the low-temperature phase exhibit the same behavior with the Ising spin glass: the exponent is linearly dependent on the temperature as 1/z(T)≃0.21T/Tsg. Relaxation functions of the chiral-glass correlation length do not exhibit such behavior as the spin-glass ones do. All the findings suggest that the transition is driven by the spin degrees of freedom.
As an analysis of equilibrium phase transitions, the nonequilibrium relaxation method is extended to the spin glass (SG) transition. The $\ifmmode\pm\else\textpm\fi{}J$ Ising SG model is analyzed for three-dimensional (cubic) lattices up to the linear size of
L=127 and for four-dimensional (hypercubic) lattice up to
L=41. These sizes of systems are quite large as compared with those calculated, so far, by equilibrium simulations. As a dynamical order parameter, we calculate the clone correlation function (CCF) ${Q(t,t}_{\mathrm{w}})\ensuremath{\equiv}[〈{S}_{i}^{(1)}{(t+t}_{\mathrm{w}}){S}_{i}^{(2)}{(t+t}_{\mathrm{w}}){〉}^{\mathrm{F}}],$ which is a spin correlation of two replicas produced after the waiting time
tw from a simple starting state. It is found that the CCF shows an exponential decay in the paramagnetic phase, and a power-law decay after aginglike development $(t\ensuremath{\gg}{t}_{\mathrm{w}})$ in the SG phase. This provides a reliable upper bound of the transition temperature
Tg. It is also found that a scaling relation, ${Q(t,t}_{\mathrm{w}}{)=t}_{\mathrm{w}}^{\ensuremath{-}{\ensuremath{\lambda}}_{q}}\overline{q}{(t/t}_{\mathrm{w}}),$ holds just around the transition point providing the lower bound of
Tg. Together with these two bounds, we propose a new dynamical way for the estimation of
Tg from much larger systems. In the SG phase, the power-law behavior of the CCF for $t\ensuremath{\gg}{t}_{\mathrm{w}}$ suggests that the SG phase in short-range Ising models has a rugged phase space.
The low temperature dynamics of the three-dimensional Ising spin-glass in zero field with a discrete bond distribution is investigated via MC simulations. The thermoremanent magnetization is found to decay algebraically and the temperature dependent exponents agree very well with the experimentally determined values. The nonequilibrium autocorrelation function C(t, t w ) shows a crossover at the waiting (or aging) time t w from algebraic quasiequilibrium decay for times t<<t w to another, faster algebraic decay for t>>t w with an exponent similar to one for the remanent magnetization.