Percolation in three-dimensional random field Ising magnets

Abstract

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