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arXiv:cond-mat/0603062

Nonequilibrium dynamic exponent and spin-glass transitions

Tota Nakamura

Published Mar 3, 2006• 0 citations
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Abstract

Nonequilibrium dynamics of the $\pm J$ Ising, the {\it XY}, and the Heisenberg spin-glass models are investigated in three dimensions. A nonequilibrium dynamic exponent is calculated from the dynamic correlation length. The spin-glass dynamic exponent continuously depends on the temperature. There is no anomaly at the critical temperature as is recently reported by Katzgraber and Campbell. On the other hand, the chiral-glass dynamic exponent takes a constant value above the spin-glass transition temperature ($T_\mathrm{sg}$), and becomes temperature-dependent below $T_\mathrm{sg}$.The finite-time scaling analyses on the spin- and the chiral-glass susceptibility are performed using the temperature dependence of the dynamic exponent. A difference of the spin- and the chiral-glass transition temperatures is resolved in the Heisenberg model. The dynamic critical exponent takes almost the same value for all transitions. It suggests that the spin-glass and the chiral-glass transitions in three dimensions are dynamically universal.

DOI
10.48550/arxiv.cond-mat/0603062

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Nonequilibrium dynamics of the ±J\pm J±J Ising, the {\it XY}, and the Heisenberg spin-glass models are investigated in three dimensions. A nonequilibrium dynamic exponent is calculated from the dynamic correlation length. The spin-glass dynamic exponent continuously depends on the temperature. There is no anomaly at the critical temperature as is recently reported by Katzgraber and Campbell. On the other hand, the chiral-glass dynamic exponent takes a constant value above the spin-glass transition temperature (TsgT_\mathrm{sg}Tsg​), and becomes temperature-dependent below TsgT_\mathrm{sg}Tsg​.The finite-time scaling analyses on the spin- and the chiral-glass susceptibility are performed using the temperature dependence of the dynamic exponent. A difference of the spin- and the chiral-glass transition temperatures is resolved in the Heisenberg model. The dynamic critical exponent takes almost the same value for all transitions. It suggests that the spin-glass and the chiral-glass transitions in three dimensions are dynamically universal.
Dynamic scaling analyses are performed in the spin-glass phase of the ±J\pm J±J Ising, the {\it XY}, and the Heisenberg models in three dimensions. We found a crossover from the critical dynamics to the ground-state dynamics in the Ising model and the Heisenberg model. The ground-state dynamics of the Ising model is characterized by an activation law with a finite energy gap: the typical time diverges exponentially. On the other hand, the typical time in the Heisenberg model diverges algebraically with the inverse temperature. Algebraic relaxation with a finite dynamic exponent is observed after the typical time in both models. The ground-state dynamic exponent is estimated to be z0≃13z_0\simeq 13z0​≃13, which is common to both models. There is no crossover in the {\it XY} model. The critical dynamics is considered to continue to the ground-state.
Dynamical scaling study for the estimation of dynamical exponent zzz of three-dimensional XY spin glass model
To analyze the ±J\pm J±J XY spin-glass in three dimensions, we verified a method aimed at obtaining a high-precision dynamical exponent zzz from the correlation length in the nonequilibrium relaxation process. The obtained zzz yielded consistent and highly accurate results in previous studies for relatively well-studied models---specifically, the three-dimensional (3D) ferromagnetic Ising model and the 3D ±J\pm J±J Ising model. Building on these previous studies, we used this method and the dynamical scaling method to analyze the 3D ±J\pm J±J XY model and obtained highly precise critical temperatures and exponents. The estimated transition temperatures satisfy $T_{\mathrm{SG}}
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.
A three-dimensional $\ifmmode\pm\else\textpm\fi{}J$ XYXYXY spin-glass model is investigated using the nonequilibrium relaxation method. We have introduced a criterion for the finite-time-scaling analysis. The transition temperature is estimated by a crossing point of the obtained data. The scaling analysis of the relaxation functions of the spin-glass susceptibility and the chiral-glass susceptibility shows that both transitions occur simultaneously. The result is checked by the relaxation functions of the Binder parameters and the glass correlation lengths of the spin and the chirality. All results are consistent if we consider that the transition is driven by the spin degrees of freedom.
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.