Spin models on random lattices

Abstract

Under certain conditions phase transitions in systems with quenched disorder are expected to exhibit a di erent behaviour than in the corresponding pure system. Here we discuss a series of Monte Carlo studies of a special type of such disordered systems, namely spin models deÿned on quenched, random lattices exhibiting geometrical disorder in the connectivity of the lattice sites. In two dimensions we present results for the q-state Potts model on random tri-valent ( 3 ) planar graphs, which appear quite naturally in the dynamically triangulated random surface (DTRS) approach to quantum gravity, as well as on Poissonian random lattices of Voronoi=Delaunay type. Both cases, q64 and ¿ 4, are discussed which, in the pure model without disorder, give rise to second-and ÿrst-order phase transitions, respectively. In three dimensions results for the Ising model on Poissonian random lattices are brie y described. We conclude with a comparison of the two types of connectivity disorder with the more standard case of bond disorder, and a discussion of the distinguishing di erences.

Dimensionality crossover in critical behaviour of ultrathin ferromagnetic films

“…So [34]. This is because, the system demonstrates the 3D critical behaviour when the correlation length is increased with approaching to the critical temperature in all directions before it reaches a film boundary, but after this, the correlation length continues to increase only inplane direction of the film and system shows the critical behaviour of the 2D system.…”

Unexpected spin-off from quantum gravity

“…The original approach of (Euclidean) Dynamical Triangulations (EDT -see [4,5] for reviews) turns out to be unsuitable for our purposes, because the contributing triangulated lattices are highly curved, with an effectively fractal structure, for any dimension d ≥ 2. Their geometry is so radically different from the usual flat lattices that it alters the universal behaviour of matter systems defined on them, as has been well documented for two-dimensional spin systems (see [6] and references therein).…”