Finite-size scaling in anisotropic systems

We present analytical results for the finite-size scaling in d-dimensional O(N) systems with strong anisotropy where the critical exponents (e.g., nu(parallel to) and nu(perpendicular to)) depend on the direction. Prominent examples are systems with long-range interactions, decaying with the interparticle distance r as r(-d-sigma) with different exponents sigma in corresponding spatial directions, systems with space-"time" anisotropy near a quantum critical point, and systems with Lifshitz points. The anisotropic properties involve also the geometry of the systems. We consider O(N) systems in the N ->infinity limit, confined to a d-dimensional layer with geometry L(m)x infinity(n);m+n=d and periodic boundary conditions across the finite m dimensions. The arising difficulties are avoided using a technique of calculations based on the analytical properties of the generalized Mittag-Leffler functions.