Criticality of spin systems with weak long-range interactions

The study of critical properties of systems with long-range interactions has attracted, in recent 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 functional renormalization group, and we review their remarkable properties. For the sake of clarity and self-consistency, we start from classical spin models and we then move to quantum spin systems.