Lower Critical Dimension of the Random-Field Ising Model: A Monte Carlo Study

This paper presents extensive Monte Carlo simulations of the random-field Ising model in various dimensions for long times in moderately large systems, and specifically addresses the question of whether the lower critical dimension is 2 or 3. The authors find long-range order for $d=3\mathrm{}$ and no long-range order for $d=2\mathrm{}$. The marginality of the $d=2\mathrm{}$ case is further checked by studying a system in $d=\frac{\ln 8}{\ln 3}\approx 1.89$ dimensions simulated by a fractal; the authors thus conclude that the lower critical dimension is 2.