FINITE-SIZE SCALING FOR SYSTEMS WITH LONG-RANGE INTERACTIONS

The present review is devoted to the fundamental problems of finite-size scaling due to the presence of long-range interactions. The attention is focused on the precise formulation of critical finite-size scaling in the case that the bulk correlation length is identically infinite. The hypotheses, based on the notion of a finite-size scaling length, are formulated in a way which treats the cases of short- and long-range interactions on equal grounds, as well as the standard and modified (above the upper critical dimensionality) finite-size scaling. The general conjectures are tested on the exactly solvable mean spherical model with power-law interaction. Special attention is paid to the adequate mathematical techniques. The formulation of finite-size scaling at first-order phase transitions is also discussed, since the finite-size scaling length enters into the definition of the relevant scaled external field variable provided the system has d' > 0 infinite dimensions. The review closes with a summary and short discussion of some open problems.