Scaling of temporal entanglement in proximity to integrability

Describing dynamics of quantum many-body systems is a formidable challenge due to rapid generation of quantum entanglement between remote degrees of freedom. A promising approach to tackle this challenge, which has been proposed recently, is to characterize the quantum dynamics of a many-body system and its properties as a bath via the Feynman-Vernon influence matrix (IM), which is an operator in the space of time trajectories of local degrees of freedom. Physical understanding of the general scaling of the IM's temporal entanglement and its relation to basic dynamical properties is highly incomplete to the present day. In this paper, we analytically compute the exact IM for a family of integrable Floquet models—the transverse-field kicked Ising chain—finding a Bardeen-Cooper-Schrieffer-like “wave function” on the Schwinger-Keldysh contour with algebraically decaying correlations. We demonstrate that the IM exhibits area-law temporal entanglement scaling for all parameter values. Furthermore, the entanglement pattern of the IM reveals the system's phase diagram, exhibiting jumps across transitions between distinct Floquet phases. Near criticality, a nontrivial scaling behavior of temporal entanglement is found. The area-law temporal entanglement allows us to efficiently describe the effects of sizable integrability-breaking perturbations for long evolution times by using matrix-product-state methods. This work shows that tensor-network methods are efficient in describing the effect of noninteracting baths on open quantum systems and provides an approach to studying quantum many-body systems with weakly broken integrability.