Critical behavior of systems with long-range interaction in restricted geometry

TL;DR

This review discusses finite-size scaling and critical behavior in systems with long-range interactions, focusing on renormalization group results, Monte Carlo simulations, and mathematical techniques for analyzing such systems.

Contribution

It provides a comprehensive overview of renormalization group analysis and mathematical methods for studying long-range interactions in finite systems, highlighting their effects on critical phenomena.

Findings

Finite-size scaling is significantly affected by long-range interactions.

Renormalization group results help understand critical properties in these systems.

Mathematical techniques enable equal treatment of long-range and short-range interactions.

Abstract

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 $1/r^{d+\sigma}$, $\sigma>0$. The attention is focused mainly on the renormalization group results in the framework of ${\cal O}(n)$ $\phi^{4}$ - theory for systems with fully finite (block) geometry under periodic boundary conditions. Some bulk critical properties and Monte Carlo results also are reviewed. The role of the cutoff effects as well their relation with those originating from the long-range interaction is also discussed. Special attention is paid to the description of the adequate mathematical technique that allows to treat the long-range and short-range interactions on equal ground. The review closes with short discussion of some open problems.