Criticality of spin systems with weak long-range interactions

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. I. INTRODUCTIONIn this Special Issue several aspects of the equilibrium and dynamical properties of systems with long-range (LR) interactions are discussed. In the paper we apply concepts and tools developed for LR systems, including the modern renormalization group approach, to the study of spin models with weak LR interactions. Such interactions are able to modify the universal properties of the systems in which they act, but anyway preserve the extensivity of the thermodynamic quantities. Our main goal is to present and apply the formalism of the functional renormalization group (FRG) to weak LR systems and show the capability of the FRG to clarify their critical properties, which are in many cases still unknown and sometimes diffucult to obtain with other approaches.Given the wide framework already presented in the other contributions to this Special Issue, it is not necessary to extensively emphasize that LR interactions play an important and paradigmatic role in the study of many body interacting systems, such as O(N ) symmetric models. Indeed, the importance of LR interactions is motivated by their presence in several systems ranging from plasma physics to astrophysics and cosmology [1,2].In order to understand the typical phenomena occurring in spin models with power-law LR couplings and define the weak LR regime, let us first consider the classical O(N ) symmetric models, whose Hamiltonian reads(1)The spin variables S i are unit vectors with N components, placed at the sites, labeled by the index i, of a d dimensional lattice. The coupling constant J is constant for a decay exponent σ > 0, conversely for σ ≤ 0 the coupling constant J needs to be rescaled by an appropriate power of the system size to absorb the divergence of the interaction energy density the thermodynamic limit [1,2]. When σ > 0 the model may have a second order phase transition. The main result is that three different regimes can occur as a function of the parameter σ [3, 4]:• for σ ≤ d/2 the mean-field approximation correctly describes the universal behavior;• for σ greater than a threshold value, σ * , the model has the same critical...

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