Topological Origin of the Phase Transition in a Mean-Field Model

We argue that the phase transition in the mean-field $\mathrm{XY}$ model is related to a particular change in the topology of its configuration space. The nature of this topological change can be discussed on the basis of elementary Morse theory using the potential energy per particle $V$ as a Morse function. The value of $V$ where such a topological change occurs equals the thermodynamic value of $V$ at the phase transition and the number of (Morse) critical points grows very fast with the number of particles $N$. Furthermore, as in statistical mechanics, the way the thermodynamic limit is taken is crucial in topology.