The ±J Ising model in three dimensions is investigated using dynamical scaling analysis for a nonequilibrium relaxation process. The spin-glass (SG) transition is analyzed under an assumed dynamical exponent z proposed for the scaling analysis of the SG transition. To estimate dynamical exponent z precisely, we introduce an extrapolation scheme for the asymptotic value of the logarithmic derivative of the dynamical Binder parameter. We obtain the phase diagram in the temperature–concentration (p–T) space, as well as the p-dependence of critical exponents γ, ν, and z. For the static exponents of the SG transition, the universality is strongly indicated numerically along the whole region of the paramagnetic (PM)–SG phase boundary. A remarkable crossover behavior is observed for estimations of critical exponents at the boundary close to the multicritical point (MCP) of PM–ferromagnetic (FM)–SG phases. In addition, the dynamical assumption for z resulting from the slow relaxation in the SG region disappears in the FM region, and the crossover between them occurs around the Nishimori Line.
We consider the three-dimensional $\ifmmode\pm\else\textpm\fi{}J$ model defined on a simple cubic lattice and study its behavior close to the multicritical Nishimori point, where the paramagnetic-ferromagnetic, the paramagnetic-glassy, and the ferromagnetic-glassy transition lines meet in the $T\text{\ensuremath{-}}p$ phase diagram (
p characterizes the disorder distribution and gives the fraction of ferromagnetic bonds). For this purpose, we perform Monte Carlo simulations on cubic lattices of size $L\ensuremath{\leqslant}32$ and a finite-size-scaling analysis of the numerical results. The magnetic-glassy multicritical point is found at
p∗=0.76820(4), along the Nishimori line given by $2p\ensuremath{-}1=\mathrm{tanh}(J∕T)$. We determine the renormalization-group dimensions of the operators that control the renormalization-group flow close to the multicritical point,
y1=1.02(5),
y2=0.61(2), and the susceptibility exponent $\ensuremath{\eta}=\ensuremath{-}0.114(3)$. The temperature and crossover exponents are $\ensuremath{\nu}=1∕{y}_{2}=1.64(5)$ and $\ensuremath{\phi}={y}_{1}∕{y}_{2}=1.67(10)$, respectively. We also investigate the model-
A dynamics, obtaining the dynamic critical exponent
z=5.0(5).
The ferromagnetic transition of ± J -Ising spin-glass model on square lattice is studied by using Monte Carlo simulation. At ferromagnetic-bond density p =0.96 where the system shows ferromagnetic transition, the spontaneous magnetization, the ratio < m 4 >/ < m 2 > 2 and the non-equilibrium relaxation (NER) are studied. The transition is estimated to be at K c =0.5100 (1). The analysis by NER gives the most reliable estimation with the least CPU time for simulations among the three methods. The value of magnetization exponent β is estimated to be 0.130(10) which coincides with that of the purely ferromagnetic Ising model ( p =1). The dynamical critical exponent z is estimated to be 2.418(23) from the NER analysis which is different from that of the pure model. The size of simulated lattices is up to 2001×2000, and the number of totally updated Ising spins is 5.5×10 15 .
The non-equilibrium critical dynamics of the Baxter model is studied through Monte Carlo simulations. The single-spin-flip stochastic dynamics is introduced on the basis of the formulation of the model as an Ising-type model on the square lattice with next-nearest-neighbor and four-spin interactions. At several points on the critical line, the dynamic critical exponent z and the critical exponent \bar λ c , which characterizes the non-equilibrium critical relaxation, are estimated for the two order parameters of the model, the magnetization and the polarization. For both order parameters, the estimated \bar λ c varies systematically along the critical line. On the other hand, the estimated z is almost constant. This suggests that while the exponent z is universal, the exponent \bar λ c is not.
We use transfer-matrix and finite-size scaling methods to investigate the location and properties of the multicritical point of two-dimensional Ising spin glasses on square, triangular, and honeycomb lattices with both binary and Gaussian disorder distributions. For square and triangular lattices with binary disorder, the estimated position of the multicritical point is in numerical agreement with recent conjectures regarding its exact location. For the remaining four cases, our results indicate disagreement with the respective versions of the conjecture, though by very small amounts, never exceeding 0.2%. Our results for (i) the correlation-length exponent $\ensuremath{\nu}$ governing the ferroparamagnetic transition, (ii) the critical domain-wall energy amplitude $\ensuremath{\eta}$, (iii) the conformal anomaly
c, (iv) the finite-size susceptibility exponent $\ensuremath{\gamma}/\ensuremath{\nu}$, and (v) the set of multifractal exponents ${{\ensuremath{\eta}}_{k}}$ associated to the moments of the probability distribution of spin-spin correlation functions at the multicritical point are consistent with universality as regards lattice structure and disorder distribution and in good agreement with existing estimates.
Ferromagnetic (FM) transition of the ± J Ising model is studied for square and simple-cubic lattices by Monte Carlo simulations. Non-equilibrium relaxation (NER) process of magnetization from the all-aligned spin state is analyzed. The transition temperature is estimated at several values of FM bond concentrations to provide the boundary of the FM phase. The estimated NER exponent of magnetization shows non-universal behavior and it becomes larger smoothly when the transition temperature becomes higher.