Magnetic-glassy multicritical behavior of the three-dimensional $\pm J$ Ising model

We consider the three-dimensional $\pm J$ 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 $T\text{−}p$ 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 $L⩽32$ and a finite-size-scaling analysis of the numerical results. The magnetic-glassy multicritical point is found at $p^{*}=0.76820\left(4\right)$, along the Nishimori line given by $2p-1=\tanh (J∕T)$. We determine the renormalization-group dimensions of the operators that control the renormalization-group flow close to the multicritical point, $y_1=1.02\left(5\right)$, $y_2=0.61\left(2\right)$, and the susceptibility exponent $\eta =-0.114\left(3\right)$. The temperature and crossover exponents are $\nu =1∕y_2=1.64\left(5\right)$ and $\varphi =y_1∕y_2=1.67\left(10\right)$, respectively. We also investigate the model-$A$ dynamics, obtaining the dynamic critical exponent $z=5.0\left(5\right)$.

Figure 1

(Color online) Phase diagram of the three-dimensional $\pm J$ Ising model in the $T\text{−}p$ plane. The phase diagram is symmetric for $p\to 1-p$.

Figure 2

(Color online) MC data of $R_{\xi }\equiv \xi ∕L$, $U_4$, and $U_d$ vs $p$.

Figure 3

(Color online) Estimates of the exponential autocorrelation time $\tau$ at the MGP vs $L$.