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Physical review. B, Condensed matter

Dynamic scaling and the field-dependent critical line in a fractal cluster model of spin glasses

M. A. Contínentino, A. P. Malozemoff

Published Mar 1, 1986• 44 citations
Publisher

Abstract

We extend an earlier fractal cluster model of spin glasses to study (1) the magnetization relaxation M(t) induced by a magnetic field, (2) magnetic noise, and (3) dynamics near a critical line ${T}_{g}$(H). Above the zero-field transition temperature ${T}_{g}$, M(t) follows a stretched exponential form exp(-${t}^{1\mathrm{\ensuremath{-}}n}$) with n=\ensuremath{\nu}z/(\ensuremath{\varphi}+\ensuremath{\nu}z), where \ensuremath{\varphi},\ensuremath{\nu}, and z are standard static and dynamical critical exponents. A dynamic scaling relation is derived for the entire region ${T}_{g}$(H)${T}_{g}$ in agreement with experiments. The equations associated with lines of constant relaxation time are obtained and it is shown how nonuniversality of exponents along the critical line can be tested.

DOI
10.1103/physrevb.33.3591

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We extend an earlier fractal cluster model of spin glasses to study (1) the magnetization relaxation M(t) induced by a magnetic field, (2) magnetic noise, and (3) dynamics near a critical line Tg{T}_{g}Tg​(H). Above the zero-field transition temperature Tg{T}_{g}Tg​, M(t) follows a stretched exponential form exp(-t1\ensuremath−n{t}^{1\mathrm{\ensuremath{-}}n}t1\ensuremath−n) with n=\ensuremath{\nu}z/(\ensuremath{\varphi}+\ensuremath{\nu}z), where \ensuremath{\varphi},\ensuremath{\nu}, and z are standard static and dynamical critical exponents. A dynamic scaling relation is derived for the entire region Tg{T}_{g}Tg​(H)Tg{T}_{g}Tg​ in agreement with experiments. The equations associated with lines of constant relaxation time are obtained and it is shown how nonuniversality of exponents along the critical line can be tested.
The relaxation behavior of spin glasses has been analyzed in the framework of the critical fractal-cluster model. It is found that the relaxation rate of the magnetization can be expressed as $\frac{\ensuremath{\partial}m}{\ensuremath{\partial}\mathrm{ln}t}\ensuremath{\propto}{t}^{\ensuremath{-}\frac{\ensuremath{\beta}}{z\ensuremath{\nu}}}\mathrm{exp}[\ensuremath{-}{(\frac{t}{{\ensuremath{\tau}}_{\ensuremath{\xi}}})}^{\frac{\ensuremath{\beta}\ensuremath{\delta}}{z\ensuremath{\nu}}}]$, where $\ensuremath{\beta}$, $\ensuremath{\delta}$, zzz, $\ensuremath{\nu}$ are standard static and dynamic critical exponents and ${\ensuremath{\tau}}_{\ensuremath{\xi}}$ is the relaxation time of a characteristic cluster which diverges at the spin-glass freezing temperature Tg{T}_{g}Tg​. The expression is valid through Tg{T}_{g}Tg​ and is found to describe excellently the fundamental features of the relaxation in real spin glasses.
A semiphenomenological cluster theory is developed for dynamic critical properties, which is not limited to small deviations from equilibrium. Explicit numerical expressions are derived for linear and nonlinear response functions of the kinetic Ising model, which are compatible with dynamic scaling and recent calculations of dynamic critical exponents. An uniaxial magnet (or Ising system) and its critical fluctuations are approximated by a "droplet" or "cluster" model, the critical part of the free energy being the sum of the contributions of clusters each containing lll spins. These clusters are assumed to grow and shrink with a phenomenological rate $\ensuremath{\propto}{l}^{r}$, $r\ensuremath{\approx}1$. These rates are compatible with direct Monte Carlo simulations in two and three dimensions. The resulting cluster reaction and diffusion equation is an approximation to the master equation of the Glauber kinetic Ising model (in which the magnetization is not conserved), and has the same structure as used in nucleation theories. In linear response to space- and time-dependent fields the relaxation times of energy and magnetization are expressed as triple integrals. All relaxation times diverge as ${|1\ensuremath{-}\frac{T}{{T}_{c}}|}^{\ensuremath{-}(2\ensuremath{-}r)\ensuremath{\beta}\ensuremath{\delta}}$. This treatment is consistent with dynamic scaling and universality, and is more general than the conventional (Van Hove) theory of critical slowing down. Also the (smaller) exponents found for the response to localized variations ("autocorrelation functions") agree with dynamic scaling. The wave-vector dependence of the relaxation times arises from the static correlation function only, and not from the cluster diffusion term. If the equilibrium cluster distribution is assumed to be that of the Fisher droplet model, and for special values of rrr, the (discrete) eigenvalue spectrum and eigenfunctions (generalized Laguerre polynomials) of the cluster reaction equation are found explicitly. Relaxation functions and frequency-dependent susceptibilities are expressed by hypergeometric series, even in the case of nonlinear response. At the critical point, the spectrum becomes continuous and the exponential decay for large times is replaced by power-law behavior. Similarly, it is found that dynamic scaling applies also to the nucleation rate for this model, and the scaling exponent of the nucleation rate is $2\ensuremath{-}\ensuremath{\alpha}+(2\ensuremath{-}r)\ensuremath{\beta}\ensuremath{\delta}$.
Non-exponential slow dynamics in the `Griffiths phase' of the three-dimensional (3D) ± J Ising spin-glass model is studied by means of Monte Carlo simulation. The distribution of relaxation times averaged over samples, \bar P ( x ) with x = lnτ, is calculated from spin auto-correlation functions simulated. It is found that \bar P ( x ) consists of two branches. The cross-over value between the branches, x c , diverges as T → T c , T c being the spin-glass transition temperature. Therefore τ c (= e x c ) is regarded as the critical relaxation time associated with the onset of the spin-glass order at T c , and the branch of \bar P ( x ) at x < x c is interpreted due to the ordinary critical slowing down process ( critical dynamics ). The other branch at x > x c causes the unusual slow dynamics peculiar to spin glasses. Various aspects of \bar P ( x ) of this branch examined are consistent with the following picture; compact regions with less frustrations than the system as a whole are thermally fluctuating with very small frequencies, and independently of each others ( cluster dynamics ).
The ±J Ising model in three dimensions is investigated using dynamical scaling analysis for a nonequilibrium relaxation process. The spin-glass (SG) transition is analyzed under an assumed dynamical exponent z proposed for the scaling analysis of the SG transition. To estimate dynamical exponent z precisely, we introduce an extrapolation scheme for the asymptotic value of the logarithmic derivative of the dynamical Binder parameter. We obtain the phase diagram in the temperature–concentration (p–T) space, as well as the p-dependence of critical exponents γ, ν, and z. For the static exponents of the SG transition, the universality is strongly indicated numerically along the whole region of the paramagnetic (PM)–SG phase boundary. A remarkable crossover behavior is observed for estimations of critical exponents at the boundary close to the multicritical point (MCP) of PM–ferromagnetic (FM)–SG phases. In addition, the dynamical assumption for z resulting from the slow relaxation in the SG region disappears in the FM region, and the crossover between them occurs around the Nishimori Line.