Dynamical phase diagram of the quantum Ising model with cluster interaction under noiseless and noisy driven field

In most lattice models, gap closing typically occurs at high-symmetry points in the Brillouin zone. In the transverse field Ising model with cluster interaction, the gap closes at high-symmetry points, as well as at the phase transition between paramagnetic and cluster phases, where the gap-closing mode can be moved by tuning the strength of the cluster interaction. We take advantage of this property to examine the nonequilibrium dynamics of the model in the framework of dynamical quantum phase transitions (DQPTs) after a noiseless and noisy ramp of the transverse magnetic field. The numerical results show that DQPTs always happen if the starting or ending point of the quench field is restricted between two critical points. In other ways, there is always critical sweep velocity above which DQPTs disappear. Our finding reveals that noise modifies drastically the dynamical phase diagram of the model. We find that the critical sweep velocity decreases by enhancing the noise intensity and scales linearly with the square of noise intensity for weak and strong noise. Moreover, the region with multi-critical modes (MCMs) induced in the dynamical phase diagram by noise. The sweep velocity under which the system enters the MCMs region increases by enhancing the noise and scales linearly with the square of noise intensity.