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_{||} and \nu_{\perp}) 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"a anisotropy near a quantum critical point and systems with Lifshitz points. The anisotropic properties involve also the geometry of the systems. We consider systems confined to a d-dimensional layer with geometry L^{m}\times\infty^{n}; m+n=d and periodic boundary conditions across the finite m dimensions. The arising difficulties are avoided using a technics of calculations based on the analytical properties of the generalized Mittag-Leffler functions.