Percolation in three-dimensional random field Ising magnets

The structure of the three-dimensional (3D) random field Ising magnet is studied by ground-state calculations. We investigate the percolation of the minority-spin orientation in the paramagnetic phase above the bulk phase transition, located at $[\Delta /J{]}_c\simeq 2.27,$ where $\Delta$ is the standard deviation of the Gaussian random fields $\mathrm{}(J=1\mathrm{}).$ With an external field H there is a disorder-strength-dependent critical field $\pm H_c\left(\Delta \right)$ for the down (or up) spin spanning. The percolation transition is in the standard percolation universality class. $H_c\sim (\Delta -{\Delta }_p{)}^{\delta },$ where ${\Delta }_p=2.43\mathrm{}\pm 0.01$ and $\delta =1.31\mathrm{}\pm 0.03,$ implying a critical line for ${\Delta }_c<\Delta <~{\Delta }_p.$ When, with zero external field, $\Delta$ is decreased from a large value there is a transition from the simultaneous up- and down-spin spanning, with probability ${\Pi }_{\uparrow \downarrow }=1.00\mathrm{}$ to ${\Pi }_{\uparrow \downarrow }=0.\mathrm{}$ This is located at $\Delta =2.32\mathrm{}\pm 0.01,$ i.e., above ${\Delta }_c.$ The spanning cluster has the fractal dimension of standard percolation, $D_f=2.53\mathrm{}$ at ${H=H}_c\left(\Delta \right).\mathrm{}$ We provide evidence that this is asymptotically true even at $H=0\mathrm{}$ for ${\Delta }_c<\Delta <~{\Delta }_p$ beyond a crossover scale that diverges as ${\Delta }_c$ is approached from above. Percolation implies extra finite-size effects in the ground states of the 3D random field Ising model.