# Criticality of Spin Systems with Weak Long-Range Interactions

## TL;DR

This paper reviews the critical behavior of weak long-range interacting spin systems, analyzing their unique universality classes using renormalization group techniques across classical and quantum models.

## Contribution

It provides a comprehensive review of the critical properties of weak long-range spin systems, highlighting their distinct universality classes and analytical approaches.

## Key findings

- Identification of unique universality classes for weak long-range interactions
- Application of renormalization group methods to classical and quantum spin models
- Clarification of the critical behavior differences from short-range systems

## Abstract

The study of critical properties of systems with long-range interactions has attracted in the last decades a continuing interest and motivated the development of several analytical and numerical techniques, in particular in connection with spin models. From the point of view of the investigation of their criticality, a special role is played by systems in which the interactions are long-range enough that their universality class is different from the short-range case and, nevertheless, they maintain the extensivity of thermodynamical quantities. Such interactions are often called weak long-range. In this paper we focus on the study of the critical behaviour of spin systems with weak-long range couplings using renormalization group, and we review their remarkable properties. For the sake of clarity and self-consistency, we start from the classical $O(N)$ spin models and we then move to quantum spin systems.