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We propose a topological approach suitable to establish a connection betweenthermodynamics and topology in the microcanonical ensemble. Indeed, we reporton results that point to the possibility of describing {\it interactingclassical spin systems} in the thermodynamic limit, including the occurrence ofa phase transition, using topology arguments only. Our approach relies on Morsetheory, through the determination of the critical points of the potentialenergy, which is the proper Morse function. Our main finding is to show that,in the context of the studied classical models, the Euler characteristic$\chi(E)$ embeds the necessary features for a correct description of severalmagnetic thermodynamic quantities of the systems, such as the magnetization,correlation function, susceptibility, and critical temperature. Despite theclassical nature of the studied models, such quantities are those that do notviolate the laws of thermodynamics [with the proviso that Van der Waals loopstates are mean field (MF) artifacts]. We also discuss the subtle connectionbetween our approach using the Euler entropy, defined by the logarithm of themodulus of $\chi(E)$ per site, and that using the {\it Boltzmann}microcanonical entropy. Moreover, the results suggest that the loss ofregularity in the Morse function is associated with the occurrence of unstableand metastable thermodynamic solutions in the MF case. The reliability of ourapproach is tested in two exactly soluble systems: the infinite-range and theshort-range $XY$ models in the presence of a magnetic field. In particular, weconfirm that the topological hypothesis holds for both the infinite-range ($T_c\neq 0$) and the short-range ($T_c = 0$) $XY$ models. Further studies are verydesirable in order to clarify the extension of the validity of our proposal.

Topological Approach to Microcanonical Thermodynamics and Phase Transition of Interacting Classical Spins