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Describing dynamics of quantum many-body systems is a formidable challengedue to rapid generation of quantum entanglement between remote degrees offreedom. A promising approach to tackle this challenge, which has been proposedrecently, is to characterize the quantum dynamics of a many-body system and itsproperties as a bath via the Feynman-Vernon influence matrix (IM), which is anoperator in the space of time trajectories of local degrees of freedom.Physical understanding of the general scaling of the IM's temporal entanglementand its relation to basic dynamical properties is highly incomplete to presentday. In this Article, we analytically compute the exact IM for a family ofintegrable Floquet models - the transverse-field kicked Ising chain - finding aBardeen-Cooper-Schrieffer-like "wavefunction" on the Schwinger-Keldysh contourwith algebraically decaying correlations. We demonstrate that the IM exhibitsarea-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. Nearcriticality, a non-trivial scaling behavior of temporal entanglement is found.The area-law temporal entanglement allows us to efficiently describe theeffects of sizeable integrability-breaking perturbations for long evolutiontimes by using matrix product state methods. This work shows that tensornetwork methods are efficient in describing the effect of non-interacting bathson open quantum systems, and provides a new approach to studying quantummany-body systems with weakly broken integrability.

Scaling of temporal entanglement in proximity to integrability