Scaling of the Random--Field Ising Model in Two Dimensions

Being one of the simplest models of magnetic systems with quenched disorder, the random-field Ising model shows surprisingly rich critical behavior. Only recently has it been possible with the help of large-scale numerical simulations to shed some light on a range of fundamental questions in three and higher dimensions, such as universality, critical scaling and dimensional reduction. The two-dimensional model
has received less attention, but is no less fascinating. We solve a long-standing puzzle by presenting compelling numerical evidence for the scaling behavior of the correlation length ξ. Results for two lattice geometries, square and triangular, consistently support the form ξ ~ exp[A/h²], where h denotes the random-field strength, in line with early theoretical work [1], but at variance with some more recent numerical and analytical results [2,3]. We also investigate the more widely used break-up length scale of the system, which we however find to be afflicted by much stronger scaling corrections and hence a rather less useful quantity.

[1] K. Binder, Z. Phys. B50, 343 (1983).
[2] G. P. Shrivastav, M. Kumar, V. Banerjee, and S. Puri,Phys. Rev. E90, 032140 (2014).
[3] L. Hayden, A. Raju, and J. P. Sethna, Phys. Rev. Res.1, 033060 (2019).