Generalized Mittag-Leffler functions in the theory of finite-size scaling for systems with strong anisotropy and/or long-range interaction

TL;DR

This paper introduces generalized Mittag-Leffler functions to analyze finite-size scaling in anisotropic and long-range interacting systems, simplifying integral representations and asymptotic analysis.

Contribution

It develops a unified approach using generalized Mittag-Leffler functions to handle complex finite-size scaling in systems with anisotropy and long-range interactions.

Findings

Integral representations for lattice sums are simplified.

Generalized Mittag-Leffler functions facilitate asymptotic analysis.

Unified treatment of classical and quantum systems with anisotropy.

Abstract

The difficulties arising in the investigation of finite-size scaling in $d$--dimensional O(n) systems with strong anisotropy and/or long-range interaction, decaying with the interparticle distance $r$ as $r^{-d-\sigma}$ ($0<\sigma\leq2$), are discussed. Some integral representations aiming at the simplification of the investigations are presented for the classical and quantum lattice sums that take place in the theory. Special attention is paid to a more general form allowing to treat both cases on an equal footing and in addition cases with strong anisotropic interactions and different geometries. The analysis is simplified further by expressing this general form in terms of a generalization of the Mittag-Leffler special functions. This turned out to be very useful for the extraction of asymptotic finite-size behaviours of the thermodynamic functions.