A three-dimensional $\ifmmode\pm\else\textpm\fi{}J$
XY 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.
Nonequilibrium dynamics of the
±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 (
Tsg), and becomes temperature-dependent below
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.
Dynamical scaling study for the estimation of dynamical exponent
z of three-dimensional XY spin glass model
To analyze the
±J XY spin-glass in three dimensions, we verified a method aimed at obtaining a high-precision dynamical exponent
z from the correlation length in the nonequilibrium relaxation process. The obtained
z 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 Ising model. Building on these previous studies, we used this method and the dynamical scaling method to analyze the 3D
±J XY model and obtained highly precise critical temperatures and exponents. The estimated transition temperatures satisfy $T_{\mathrm{SG}}
The critical behavior of the gauge-glass and the
XY spin-glass models in three dimensions is studied by analyzing their nonequilibrium aging dynamics. A new numerical method, which relies on the calculation of the two-time correlation and integrated response functions, is used to determine both the critical temperature and the nonequilibrium scaling exponents, both for spin and chiral degrees of freedom. First, the ferromagnetic
XY model is studied to validate this nonequilibirum aging method (NAM) since for this nondisordered system we can compare with known results obtained with standard equilibrium and nonequilibrium techniques. When applied to the case of the gauge-glass model, we show that the NAM allows us to obtain precise and reliable values of its critical quantities, improving previous estimates. The
XY spin-glass model with both Gaussian and bimodal bond distributions is analyzed in more detail. The spin and the chiral two-time correlation and integrated response functions are calculated in our simulations. The results obtained mainly for Gaussian and, to a lesser extent, for bimodal interactions support the existence of a spin-chiral decoupling scenario, where the chiral order occurs at a finite temperature while the spin degrees of freedom order at very low or zero temperature.
The ordering of the three-dimensional isotropic
XY spin glass with nearest-neighbor random Gaussian coupling is studied by extensive Monte Carlo simulations. To investigate the ordering of the spin and the chirality, we compute several independent physical quantities including the glass order parameter, the Binder parameter, the correlation-length ratio, the overlap distribution, and the non-self-averageness parameter, etc., for both the spin-glass (SG) and the chiral-glass (CG) degrees of freedom. Evidence of spin-chirality decoupling, i.e., that the CG and SG order occur at two separate temperatures,
0<TSG<TCG, is obtained from the glass order parameter, and is fully corroborated by the Binder parameter. By contrast, the CG correlation-length ratio yields a rather pathological and inconsistent result in the range of sizes we studied, which may originate from the finite-size effect associated with a significant short-length dropoff of the spatial CG correlations. Finite-size-scaling analysis yields the CG exponents ${\ensuremath{\nu}}_{\mathrm{CG}}=1.{36}_{\ensuremath{-}0.37}^{+0.15}$ and ${\ensuremath{\eta}}_{\mathrm{CG}}=0.{26}_{\ensuremath{-}0.26}^{+0.29}$, and the SG exponents ${\ensuremath{\nu}}_{\mathrm{SG}}=1.{22}_{\ensuremath{-}0.06}^{+0.26}$ and ${\ensuremath{\eta}}_{\mathrm{SG}}=\ensuremath{-}0.{54}_{\ensuremath{-}0.52}^{+0.24}$. The exponents obtained are close to those of the Heisenberg SG, but are very different from those of the Ising SG. The chiral overlap distribution and the chiral Binder parameter exhibit the feature of a continuous one-step replica-symmetry breaking (1RSB), consistently with previous reports. Such a 1RSB feature is again like that of the Heisenberg SG, but is different from the Ising SG, which may be the cause of the difference in the CG critical properties from those of the Ising SG despite the common
Z2 symmetry.