Finite-Size Scaling and Long-Range Interactions

The present review is devoted to the problems of finite-size scaling due to the presence of long-range interaction decaying at large distance as 1/rd+σ, where d is the spatial dimension and the long-range parameter σ > 0. Classical and quantum systems are considered. 1. Finite-size systems and critical phenomena A common wisdom is that the singularities in the thermodynamic functions at a critical point may occur only in the thermodynamic limit. If the system is fully finite (as any real system is), no such singularities exist and, strictly speaking, no phase transitions occur. These facts naturally give rise to the following questions: a) Why the bulk theory turns out to be adequate to the experimental evidence for finite objects? b) How the singularities appear when no phase transitions occur in any fully finite system? The answer to the first question lies in the fact that a macroscopic body is close, under some circumstances, to the idealized thermodynamic limit. The answer to the second question becomes evident if one recalls that the limit function of a sequence of analytic functions needs not to be analytic. Therefore, what happens is that in the thermodynamic limit some thermodynamic functions of the system develop singularities which are attributes of phase transitions. So, in a finite system one expects an appreciable rounding of the critical point singularities. The bulk correlation length ξ∞(T ) measures the distance over which particles in a system are significantly correlated. Then, as the temperature T approaches the critical temperature Tc, ξ∞(T ) diverges. When ξ∞(T ) attains a magnitude of the order of the characteristic size L of the finite system, then the boundary particles at the opposite sides of the system become well correlated, and ordering cannot continue to build up further in the restricted dimensions. It is certainly reasonable to expect that the rounding of the phase transition is controlled by the criterion: ξ∞(T ) ∼ L. The description of this rounding and crossover was first formulated by M. Fisher (1972) and is called Finite-Size Scaling (FSS)( see e.g.[1]). An important application of FSS is to analyze numerical data obtained from simulations on relatively small finite systems, and, thereby to obtain knowledge of the bulk systems. Finite-size effects are much stronger in systems of particles with long-range (LR) interactions because every particle directly feels the influence of the boundaries.