Magnetic-glassy multicritical behavior of the three-dimensional ±J Ising model

We consider the three-dimensional ±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-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^*=0.768 20(4), 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, y₁=1.02(5), y₂=0.61(2), and the susceptibility exponent η=−0.114(3). The temperature and crossover exponents are ν=1∕y₂=1.64(5) and ϕ=y₁∕y₂=1.67(10), respectively. We also investigate the model-A dynamics, obtaining the dynamic critical exponent z=5.0(5).