Numerical studies in random systems are plagued with strong finite-size effects and boundary effects. We introduce a window-measurement method as a practical solution to these difficulties. We observe physical quantities only within a subsystem located in the midst of a whole system and scale them with the correlation length estimated in the subsystem. Both equilibrium data and nonequilibrium data with different system sizes and different window sizes fall onto a single scaling function. It suggests that the correction-to-scaling terms become very small. We confirm the validity in the
±J Heisenberg spin glass model in three dimensions. The spin-glass and chiral-glass transition temperatures are estimated to be very close to each other.
An extensive equilibrium Monte Carlo simulation is performed on the three-dimensional isotropic Heisenberg spin-glass (SG) model with the random nearest-neighbor Gaussian coupling, with particular interest in its chiral-glass (CG) and SG orderings. For this model, the possibility of the spin-chirality decoupling, i.e., the CG order setting in at a higher temperature than that of the SG order, was suggested earlier but still remains controversial. We simulate the model up to the maximum size (linear dimension)
L=48 under both periodic and open boundary conditions (BCs). In locating the CG and SG transition temperatures
TCG and
TSG with the $L\ensuremath{\rightarrow}\ensuremath{\infty}$ extrapolation, a variety of independent physical quantities under both BCs are computed and utilized to get larger number of degrees of freedom (NDF). Because of the large NDF up to
NDF=43, we succeed in obtaining stable and accurate estimates of the CG and SG transition temperatures, ${T}_{\mathrm{CG}}=0.142\ifmmode\pm\else\textpm\fi{}0.001$ and ${T}_{\mathrm{SG}}=0.{131}_{\ensuremath{-}0.006}^{+0.001}$. No sign of the size crossover is observed. For larger
L, the CG correlation length progressively outgrows the SG correlation length at low temperatures. These results provide strong numerical support for the spin-chirality decoupling. The critical exponents associated with the CG and SG transitions are evaluated using the finite-size scaling with the scaling correction. For the CG transition, we get the CG exponents, ${\ensuremath{\nu}}_{\mathrm{CG}}=1.36\ifmmode\pm\else\textpm\fi{}0.10$ and ${\ensuremath{\eta}}_{\mathrm{CG}}=0.49\ifmmode\pm\else\textpm\fi{}0.10$, consistent with the corresponding experimental exponents of canonical SG. Implications for the chirality scenario of experimental SG ordering are discussed.
A spin-glass transition has been investigated for a long time but we have not reached a conclusion yet due to difficulties in the simulation studies. They are slow dynamics, strong finite-size effects, and sample-to-sample dependencies. We found that a size of the spin-glass order reaches a lattice boundary within a very short Monte Carlo step. A competition between the spin-glass order and a boundary condition causes these difficulties. Once the boundary effect was removed, physical quantities exhibited quite normal behaviors. They became self-averaging in a limit of large replica numbers. These findings suggest that the nonequilibrium relaxation method is a good choice for solving the difficulties if a lattice size and a replica number are set sufficiently large. A dynamic scaling analysis on nonequilibrium relaxation functions gave a result that the spin-glass transition and the chiral-glass transition occurs at the same temperature in the Heisenberg model in three dimensions. The estimated critical exponent ν agrees with the experimental result.
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.
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}}