Entanglement dynamics and phase transitions of the Floquet cluster spin chain

Cluster states were introduced in the context of measurement-based quantum computing. In one dimension, the cluster Hamiltonian possesses topologically protected states. We investigate the Floquet dynamics of the cluster spin chain in an external field, interacting with a particle. We explore the entanglement properties of the topological and magnetic phases, first in the integrable spin lattice case and then in the interacting quantum walk case. We find, in addition to thermalization, dynamical phase transitions separating low-and high-entanglement nonthermal states, reminiscent of the ones present in the integrable case, but differing in their magnetic properties. The nonergodic phases are characterized by the emergence of magnetic order, persistent at long times.