Ferromagnetic random-bond Ising model: Metastable states and complexity of the energy surface.

Abstract

Metastable states of the ferromagnetic random-bond Ising model are produced by simulated quenches from infinite to zero temperature. Two-dimensional systems (square lattice) show many domains: the many-valley picture of the energy surface of spin glasses applies to these unfrustrated systems as well. The number of the domains, which serves as a rough measure of the complexity of the energy surface in two dimensions (but not in three), is only 30% greater for a broad bond distribution (uniform from 0 to 1) than for a narrow one (uniform from 0.499 to 0.501). We conclude that the uniform-bond model also has a complex energy surface; for a square lattice the surface has many terraces (or steps) rather than many valleys, but for the honeycomb lattice it has a many-valley structure. The large domains in our two-dimensional systems have holes on many length scales and are highly ramified, with total perimeter proportional to the number of sites; their dimensions from the capacity, information, and radius-of-gyration definitions are 1.82\ifmmode\pm\else\textpm\fi{}0.06, 1.84\ifmmode\pm\else\textpm\fi{}0.04, and 1.83\ifmmode\pm\else\textpm\fi{}0.07, respectively. In three dimensions (simple cubic lattice), the initial concentrations of both up and down spins exceed the percolation threshold and the metastable states are dominated by two large spanning domains; only a few, small, embedded domains are found.