Analysis of the exactness of mean-field theory in long-range interacting systems

Relationships between general long-range interacting classical systems on a lattice and the corresponding mean-field models (infinitely long-range interacting models) are investigated. We study systems in arbitrary dimension d for periodic boundary conditions and focus on the free energy for fixed value of the total magnetization. As a result, it is shown that the equilibrium free energy of the long-range interacting systems are exactly the same as that of the corresponding mean-field models (exactness of the mean-field theory). Moreover, the mean-field metastable states can be also preserved in general long-range interacting systems. It is found that in the case that the magnetization is conserved, the mean-field theory does not give correct property in some parameter region.