Three-dimensional random-field ising magnet: Interfaces, scaling, and the nature of states

A. Alan Middleton, Daniel S. Fisher

Research output: Contribution to journalArticlepeer-review

163 Scopus citations


The nature of the zero-temperature ordering transition in the three-dimensional Gaussian random-field Ising magnet is studied numerically, aided by scaling analyses. Various numerical calculations are used to consistently infer the location of the transition to a high precision. A variety of boundary conditions are imposed on large samples to study the order of the transition and the number of states in the large volume limit. In the ferromagnetic phase, where the domain walls have fractal dimension ds = 2, the scaling of the roughness of the domain walls, w∼Lξ, is consistent with the theoretical prediction ξ=2/3. As the randomness is increased through the transition, the probability distribution of the interfacial tension of domain walls scales in a manner that is clearly consistent with a single second-order transition. At the critical point, the fractal dimensions of domain walls and the fractal dimension of the outer surface of spin clusters are investigated: there are at least two distinct physically important fractal dimensions that describe domain walls. These dimensions are argued to be related by scaling to combinations of the energy scaling exponent θ, which determines the violation of hyperscaling, the correlation length exponent v, and the magnetization exponent β. The value β=0.017 ±0.005 computed from finite-size scaling of the magnetization is very nearly zero: this estimate is supported by the study of the spin cluster size distribution at criticality. The variation of configurations in the interior of a sample with boundary conditions is consistent with the hypothesis that there is a single transition separating the disordered phase with one ground state from the ordered phase with two ground states. The array of results, including values for several exponents, are shown to be consistent with a scaling picture and a geometric description of the influence of boundary conditions on the spins. The details of the algorithm used and its implementation are also described.

Original languageEnglish (US)
Article number134411
Pages (from-to)1344111-13441131
Number of pages12097021
JournalPhysical Review B - Condensed Matter and Materials Physics
Issue number13
StatePublished - Apr 1 2002

ASJC Scopus subject areas

  • Electronic, Optical and Magnetic Materials
  • Condensed Matter Physics


Dive into the research topics of 'Three-dimensional random-field ising magnet: Interfaces, scaling, and the nature of states'. Together they form a unique fingerprint.

Cite this