Finite-size scaling in anisotropic systems

TL;DR

This paper derives analytical finite-size scaling results for anisotropic d-dimensional O(N) systems with direction-dependent critical exponents, including long-range interactions, quantum critical points, and Lifshitz points, using Mittag-Leffler functions.

Contribution

It introduces a novel analytical approach to finite-size scaling in anisotropic systems with complex interactions and geometries, expanding understanding of critical phenomena.

Findings

Derived explicit finite-size scaling forms for anisotropic systems.

Applied Mittag-Leffler functions to handle anisotropic boundary conditions.

Provided insights into systems with long-range interactions and quantum criticality.

Abstract

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.