Cluster mean-field dynamics of the long-range interacting Ising chain

In this thesis we study the dynamics of the long-range interacting Ising chain using a cluster mean-field approach, a mean-field theory that accounts for short range correlations. The principal result is that, if the system is isolated from an external environment, the model exhibits two different behaviors depending on the range of the interactions. Whenever the system is truly long-range, it exhibits a sharp mean-field dynamical phase transition from a dynamical ferromagnetic to a dynamical paramagnetic phase. Reducing the range of the interactions a critical region, showing hypersensitivity to initial conditions, appears. Interestingly, this chaotic region shares the same physics of a classical tossed coin that is allowed to bounce on the floor. In a second part of the work we derive the cluster mean-field equations of motion describing the dynamics of a fully connected Ising chain connected to an external bath. In particular we focused on the dynamics in presence of a dissipation generated by string of Glauber operators acting on one or more sites of the chain. This model, in presence of global dissipative processes, exhibits persistent oscillations in time revealing the existence of a boundary time-crystal and we studied the stability of the boundary time-crystal showing that global dissipative processes are a key ingredient for their existence.