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 phasetransition, located at [Δ/J] c ≃2.27, where Δ is the standard deviation of the Gaussian random fields (J= 1). With an external field H there is a disorder-strength-dependent critical field ′ H c (Δ) for the down (or up) spin spanning. The percolation transition is in the standard percolation universality class. H c ∼(Δ - Δ p ) δ , where Δ p = 2.43′0.01 and δ= 1.31′0.03, implying a critical line for Δ c <Δ≤Δ p . When, with zero external field, A is decreased from a large value there is a transition from the simultaneous up- and down-spin spanning, with probability ΠđĐ=1.00 to ΠđĐ=0. This is located at Δ=2.32′0.01, i.e., above Δ c . The spanning cluster has the fractal dimension of standard percolation, D f =2.53 at H=H c (Δ). We provide evidence that this is asymptotically true even at H=0 for Δ c <Δ≤Δ p beyond a crossover scale that diverges as Δ c is approached from above. Percolation implies extra finite-size effects in the ground states of the 3D random field Ising model.