We study the phase transition of the
±J Heisenberg model in three dimensions. Using a dynamical simulation method that removes a drift of the system, the existence of the spin-glass (SG) phase at low temperatures is suggested. The transition temperature is estimated to be
TSG∼0.18J from both equilibrium and off-equilibrium Monte-Carlo simulations. Our result contradicts the chirality mechanism of the phase transition reported recently by Kawamura which claims that it is not the spins but the chiralities of the spins that are ordered in Heisenberg SG systems.
We studied the phase transition of the $\ifmmode\pm\else\textpm\fi{}J$ Heisenberg model with and without a random anisotropy on four-dimensional lattice $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}(L+1)(L<~9).$ We showed that the Binder parameters
g(L,T)'s for different sizes do not cross even when the anisotropy is present. On the contrary, when a strong anisotropy exists,
g(L,T) exhibits a steep negative dip near the spin-glass phase-transition temperature
TSG similarly to the p-state infinite-range Potts glass model with
p> 3, in which the one-step replica-symmetry breaking (RSB) occurs. We speculated that a one-step RSB-like state occurs below
TSG, which breaks the usual crossing behavior of
g(L,T).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.
We re-examine the spin-glass (SG) phase transition of the ± J Heisenberg models with and without random anisotropy D in three dimensions ( d = 3) using two complementary methods, i.e., (i) the defect energy method and (ii) the Monte Carlo method. We reveal that the conventional defect energy method is not convincing and propose a new method which considers the stiffness of the lattice itself. Using the method, we show that the stiffness exponent θ has a positive value (θ > 0) even when D = 0. Considering the stiffness at finite temperatures, we obtain the SG phase transition temperature of T SG ∼ 0.19 J for D = 0. On the other hand, a large scale MC simulation shows that, in contrast to the previous results, a scaling plot of the SG susceptibility χ SG for D = 0 is obtained using almost the same transition temperature of T SG ∼ 0.18 J . Hence we believe that the SG phase transition occurs in the Heisenberg SG model in d = 3.
A new method to numerically calculate the n th moment of the spin overlap of the two-dimensional ± J Ising model is developed using the identity derived by one of the authors (HK) several years ago. By using the method, the n th moment of the spin overlap can be calculated as a simple average of the n th moment of the total spins with a modified bond probability distribution. The values of the Binder parameter etc have been extensively calculated with the linear size, L , up to L = 23. The accuracy of the calculations in the present method is similar to that in the conventional transfer matrix method with about 10 5 bond samples. The simple scaling plots of the Binder parameter and the spin-glass susceptibility indicate the existence of a finite-temperature spin-glass phase transition. We find, however, that the estimation of T c is strongly affected by the corrections to scaling within the present data ( L ⩽ 23). Thus, there still remains the possibility that T c = 0, contrary to the recent results which suggest the existence of a finite-temperature spin-glass phase transition.