  In this work, we introduce an approach to study quantum many-body dynamics,
inspired by the Feynman-Vernon influence functional. Focusing on a family of
interacting, Floquet spin chains, we consider a Keldysh path-integral
description of the dynamics. The central object in our approach is the
influence matrix (IM), which describes the effect of the system on the dynamics
of a local subsystem. For translationally invariant models, we formulate a
self-consistency equation for the influence matrix. For certain special values
of the model parameters, we obtain an exact solution which represents a perfect
dephaser (PD). Physically, a PD corresponds to a many-body system that acts as
a perfectly Markovian bath on itself: at each period, it measures every spin.
For the models considered here, we establish that PD points include
dual-unitary circuits investigated in recent works. In the vicinity of PD
points, the system is not perfectly Markovian, but rather acts as a bath with a
short memory time. In this case, we demonstrate that the self-consistency
equation can be solved using matrix-product states (MPS) methods, as the IM
temporal entanglement is low. A combination of analytical insights and MPS
computations allows us to characterize the structure of the influence matrix in
terms of an effective "statistical-mechanics" description. We finally
illustrate the predictive power of this description by analytically computing
how quickly an embedded impurity spin thermalizes. The influence matrix
approach formulated here provides an intuitive view of the quantum many-body
dynamics problem, opening a path to constructing models of thermalizing
dynamics that are solvable or can be efficiently treated by MPS-based methods,
and to further characterizing quantum ergodicity or lack thereof.
