Phase transitions in systems with non-additive long-range interactions

We consider spin systems with long-range interactions in a non-additive regime. When the non-additive scaling limit is employed, the energy and the entropy compete and the system exhibits some phase transitions. Such systems do not satisfy the additivity, which results in some unfamiliar properties related to phase transitions. In this paper, the concept of additivity and its consequences are explained, and recent progress on the statistical mechanics of long-range interacting systems is reviewed. It is shown that parameter space is clearly decomposed into three regions according to the stability of the uniform state predicted by the mean-field (MF) theory. Based on this parameter space decomposition, recent results on the exactness of the MF theory are explained. When the interaction is non-negative (ferromagnetic), the analysis of the MF theory is exact and a typical spin configuration is always uniform in the canonical ensemble. However, in the restricted canonical ensemble, i.e., the canonical ensemble with a restriction of the value of the magnetization, it is shown that the MF theory does not necessarily give the exact description of the system and phase transitions between the MF uniform states (MF phase) and the inhomogeneous states (non-MF phase) occur. A new finding is that when the interaction potential changes its sign depending on the distance, the non-MF phase appears even in the canonical ensemble.