Fig. 5: Truncated long-range Ising model.

Demonstration of initialization and possible evolution of the Truncated long-range Ising model. Each particle can be in two possible states: spin up (yellow) or spin down (blue). Here, the particles are on a one-dimensional chain, where the nearest-neighbor interactions are present (blue). Furthermore, for the subchains of the same spin, the long-range interactions (red) of the strength are proportional to \(\frac{1}{{r}^{\alpha }}\), where r is the distance between the particles. Thus, the interaction network is effectively evolving due to the spin flips. In the right-hand figure, we see simulations of the phase transition of the finite system with n = 400 spins for C = 1, J = 1, and α = 1.5. Again, we observe an abrupt phase transition, but since the interactions are long-range, the critical temperature is size-dependent. We show the estimated critical temperature for the finite size of the system, and for comparison, we also depict the theoretical critical temperature that can be obtained from the theory in the thermodynamic limit.