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Journal of Applied Physics

Physical model for static and dynamic scaling in spin glasses

A. P. Malozemoff, B. Barbara

Published Apr 15, 1985• 48 citations
Publisher

Abstract

We extend earlier critical fractal cluster models of a spin glass phase transition. At the transition temperature in finite field, a negative excluded-volume term dominates a positive saturated-cluster term, giving the negative H2/δ nonlinear susceptibility observed experimentally. The two competing effects in general have different relaxation times and therefore novel implications for dynamic scaling.

DOI
10.1063/1.335057

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We extend earlier critical fractal cluster models of a spin glass phase transition. At the transition temperature in finite field, a negative excluded-volume term dominates a positive saturated-cluster term, giving the negative H2/δ nonlinear susceptibility observed experimentally. The two competing effects in general have different relaxation times and therefore novel implications for dynamic scaling.
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(-${t}^{1\mathrm{\ensuremath{-}}n}$) 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 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.
Following a brief introduction to spin-glass systems, the current status of experimental evidence on the occurrence of a phase transition is reviewed. Recent analyses of dynamical properties of spin glasses are discussed in terms of several model predictions, including the conclusion of a divergent relaxation time τ at a finite temperature, although with large, non-mean-field indices z and v. The role of regular terms (leading to corrections to scaling) in the nonlinear response are reviewed, stressing their influence on the evaluation of the susceptibility and order-parameter exponents γ′ and β′.
We examine the behavior of the dynamic nonlinear response of a quantum spin glass in an exactly solvable fully connected quantum spherical model with random Gaussian bond distribution. In the quantum critical regime where the microscopic energy scale is set entirely by temperature the nonlinear response was found frequency independent and nonsingular. On the contrary, the genuine static nonlinear susceptibility diverges everywhere on the critical boundary with unusual violation of the universal scaling by the double logarithms at the zero-temperature critical point. Implications for experiments on quantum dipolar spin glasses are also noted.
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.