For certain fine-tuned Floquet systems, remarkably simple exact solutions appear, which represent perfect dephasers (PD), i.e., many-body systems acting as perfectly Markovian baths on their parts. Such PDs include dual-unitary quantum circuits investigated in recent works. Systems detuned away from PD points are not perfectly Markovian, but rather act as a quantum bath with a short memory time. This opens the door to efficient representations of the IM via matrix-product-states (MPS), as the underlying “principle of efficiency” only relies on the weakness of temporal correlations and entanglement.
I will then survey our current understanding of the classification of dynamical phases through the structure and entanglement properties of the IM.
For ergodic dynamics, we characterize the structure of the IM in terms of an effective "statistical-mechanics" description for local quantum trajectories and illustrate its predictive power by analytically computing the relaxation rate of an impurity embedded in the system.
For integrable dynamics, we outline a quasiparticle picture of temporal entanglement scaling. In the absence of interactions, we prove long-time saturation (area-law scaling). We analyze the effect of elastic scattering, demonstrating logarithmic scaling in interacting integrable systems. We finally clarify the role of protected strong zero modes in phase transitions, and discuss integrability breaking.
If time permits, in the last part of the talk I will briefly describe how to use these ideas to study the many-body localization (MBL) in strongly disordered interacting spin chains. The IM approach allows to study exact disorder-averaged time evolution in the thermodynamic limit. MBL systems fail to act as efficient baths, and their IM encodes the onset of characteristic temporal long-range order.

 
<sub>[1] Influence matrix approach to many-body Floquet dynamics, Phys. Rev. X 11 (2), 021040 (2021)
[2] Characterizing many-body localization via exact disorder-averaged quantum noise, arXiv:2012.00777 (2020)
[3] Influence functional of many-body systems: temporal entanglement and matrix-product state representation, Ann. Phys. 431, 168552  (2021)
[4] Scaling of temporal entanglement in proximity to integrability, Phys. Rev. B 104, 035137 (2021)
+ works in preparation</sub>
 

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