Thermodynamics of the Ising model with long- and short-range couplings

Abstract Long-range interacting systems constitute a paradigmatic framework in which fundamental concepts of thermodynamics and statistical mechanics must be revisited. Characterized by interactions decaying at large distances as a power law r − α , these systems are intrinsically non-additive for α ≤ d , where d is the spatial dimension of the system. This leads to the breakdown of standard assumptions such as ensemble equivalence and the concavity of thermodynamic potentials. In this work, we discuss the main properties of long-range systems, using as a concrete case study the Nagle–Kardar model, describing an Ising system with competing nearest-neighbour and possibly next-nearest-neighbour short-range couplings and a long-range all-to-all coupling. This model provides a minimal yet rich framework to investigate the interplay between additivity-breaking effects and short-range correlations, allowing for a detailed analysis of phase diagrams, tricritical behavior, and ensemble inequivalence. The technical tools used to determine its partition function, such as the Hubbard–Stratonovich transformation and the transfer matrix method, are introduced and discussed.