Susceptibility and percolation in two-dimensional random field Ising magnets

The ground-state structure of the two-dimensional random field Ising magnet is studied using exact numerical calculations. First we show that the ferromagnetism, which exists for small system sizes, vanishes with a large excitation at a random field strength-dependent length scale. This breakup length scale $L_b$ scales exponentially with the squared random field, $\exp (A/{\Delta }^2).\mathrm{}$ By adding an external field H, we then study the susceptibility in the ground state. If $L>\mathrm{}{L}_b,$ domains melt continuously and the magnetization has a smooth behavior, independent of system size, and the susceptibility decays as $L^{-2}.$ We define a random field strength-dependent critical external field value $\pm H_c\left(\Delta \right)$ for the up and down spins to form a percolation type of spanning cluster. The percolation transition is in the standard short-range correlated percolation universality class. The mass of the spanning cluster increases with decreasing Δ and the critical external field approaches zero for vanishing random field strength, implying the critical field scaling (for Gaussian disorder) $H_c\sim (\Delta -{\Delta }_c{)}^{\delta },$ where ${\Delta }_c=1.65\mathrm{}\pm 0.05$ and $\delta =2.05\mathrm{}\pm \mathrm{0.10.}$ Below ${\Delta }_c$ the systems should percolate even when $H=0.\mathrm{}$ This implies that even for $H=0\mathrm{}$ above $L_b$ the domains can be fractal at low random fields, such that the largest domain spans the system at low random field strength values and its mass has the fractal dimension of standard percolation $D_f=91/48.\mathrm{}$ The structure of the spanning clusters is studied by defining red clusters, in analogy to the “red sites” of ordinary site percolation. The sizes of red clusters define an extra length scale, independent of L.