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Journal of Physics A Mathematical and General

On a scaling law for the nonequilibrium relaxation of spin glasses

H. Eissfeller, Wolfgang Kinzel

Published Mar 21, 1992• 4 citations
Publisher

Abstract

A scaling relation between remanent magnetization and excess energy of spin-glasses is tested numerically. Based on a simple domain picture of non-equilibrium states of spin-glasses a power law can be derived, which relates the excess energy to the remanent magnetization Delta E varies as M x . The exponent x is related to the interface exponent y, which determines whether a spin-glass phase exists for T=0 or not. Therefore equilibrium properties can be determined by measurement of non-equilibrium quantities, x is calculated numerically for different temperatures T and the spatial dimensions d=2 and d=3. The numerical results are consistent with the suggested scaling relation.

DOI
10.1088/0305-4470/25/6/009

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A scaling relation between remanent magnetization and excess energy of spin-glasses is tested numerically. Based on a simple domain picture of non-equilibrium states of spin-glasses a power law can be derived, which relates the excess energy to the remanent magnetization Delta E varies as M x . The exponent x is related to the interface exponent y, which determines whether a spin-glass phase exists for T=0 or not. Therefore equilibrium properties can be determined by measurement of non-equilibrium quantities, x is calculated numerically for different temperatures T and the spatial dimensions d=2 and d=3. The numerical results are consistent with the suggested scaling relation.
The remanent magnetization Mr{M}_{r}Mr​ and the excess energy \ensuremath{\Delta}E are studied for the Sherrington-Kirpatrick model of spin glasses using Monte Carlo simulations. At zero temperature T the system relaxes to a stationary nonequilibrium state with nonzero Mr{M}_{r}Mr​ and \ensuremath{\Delta}E; both quantities are self-averaging. For 0Tc{T}_{c}Tc​ the simulations indicate that Mr{M}_{r}Mr​ and \ensuremath{\Delta}E decay to zero as a power law in time.
The relaxation of the Sherrington-Kirkpatrick model for spin glasses is studied at low temperatures. A recently developed numerical method is used by which an infinitely large system is simulated, hence finite size effects are avoided. For parallel dynamics the decay of the energy as well as of the magnetization is investigated. For low temperatures we find evidence for a relaxation into a state which is characterized by non-equilibrium values of the energy and a non-vanishing remanent magnetization.
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.
The statics-dynamics correspondence in spin glasses relate non-equilibrium results on large samples (the experimental realm) with equilibrium quantities computed on small systems (the typical arena for theoretical computations). Here we employ statics-dynamics equivalence to study the Ising spin-glass critical behavior in three dimensions. By means of Monte Carlo simulation, we follow the growth of the coherence length (the size of the glassy domains), on lattices too large to be thermalized. Thanks to the large coherence lengths we reach, we are able to obtain accurate results in excellent agreement with the best available equilibrium computations. To do so, we need to clarify the several physical meanings of the dynamic exponent close to the critical temperature.
We determine accurate values of ordering temperatures and critical exponents for Ising-spin-glass transitions in four dimensions, using a combination of finite-size scaling and nonequilibrium scaling techniques. We find that the exponents $\ensuremath{\eta}$ and zzz vary with the form of the interaction distribution, indicating nonuniversality at Ising-spin-glass transitions. These results confirm conclusions drawn from numerical data for three dimensions.
We have studied numerically the dynamics of spin glasses with Ising and XYXYXY symmetry (gauge glass) in space dimensions 2, 3, and 4. The nonequilibrium spin-glass susceptibility ${\ensuremath{\chi}}_{\mathrm{ne}}({t}_{\mathrm{w}},T)$ and the nonequilibrium energy per spin, ene(tw,T){e}_{\mathrm{ne}}({t}_{\mathrm{w}},T)ene​(tw​,T), of samples of large size Lb{L}_{\mathrm{b}}Lb​ are measured as a function of anneal time tw{t}_{\mathrm{w}}tw​ after a quench to temperatures TTT. The two observables are compared to the equilibrium spin-glass susceptibility ${\ensuremath{\chi}}_{\mathrm{eq}}(L,T)$ and the equilibrium energy eeq(L,T){e}_{\mathrm{eq}}(L,T)eeq​(L,T), respectively, measured as functions of temperature TTT and system size LLL for a range of system sizes. For any time and temperature a nonequilibrium time-dependent length scale L∗(tw,T){L}^{*}({t}_{\mathrm{w}},T)L∗(tw​,T) can be defined by writing ${\ensuremath{\chi}}_{\mathrm{ne}}({t}_{\mathrm{w}},T)={\ensuremath{\chi}}_{\mathrm{eq}}({L}^{*},T)$ (or the equivalent expression for the energy). Our analysis shows that for all systems studied, an ``effective dynamical critical exponent'' parametrization L∗(tw,T)=A(T)t1∕z(T){L}^{*}({t}_{\mathrm{w}},T)=A(T){t}^{1∕z(T)}L∗(tw​,T)=A(T)t1∕z(T) fits the data well at each temperature within the whole temperature range studied, which extends from well above the critical temperature Tc{T}_{\mathrm{c}}Tc​ to near T=0T=0T=0 for dimension 2 or to well below Tc{T}_{\mathrm{c}}Tc​ for the other space dimensions studied. In addition, the data suggest that the dynamical exponent zzz varies smoothly when crossing the transition temperature.