Fig. 7: Microscopic origin of the abrupt, mixed-order phase transition.

Left: a typical trajectory of all accepted changes starting from the state ∣m∣ = 1 just above critical temperature T_c. Here, M denotes the total magnetization (M = m ⋅ n) and τ^ac is the number of accepted MC steps (most of the MC suggested MC steps are rejected, as discussed in the text). After the initial period of small microscopic fluctuations, the trajectory finally converges to its equilibrium state with m ≈ 0. We denote the different types of transitions by different colors: adding a molecule in red, removing a molecule in green, flipping a spin up in yellow, and flipping a spin down in purple. In this case, due to the large average degree (k = 50), most changes include adding a molecule or removing a molecule, which affects the underlying interaction network. These microscopic steps of adding and removing molecules change the effective network topology (green links) from a very dense (a) over less dense networks (b, c) to the final state (d) where most particles form molecules (in red) and the effective network becomes very sparse showing zero macroscopic magnetization.