  Feynman-Vernon influence functional (IF) was originally introduced to
describe the effect of a quantum environment on the dynamics of an open quantum
system. We apply the IF approach to describe quantum many-body dynamics in
isolated spin systems, viewing the system as an environment for its local
subsystems. While the IF can be computed exactly only in certain many-body
models, it generally satisfies a self-consistency equation, provided the
system, or an ensemble of systems, are translationally invariant. We view the
IF as a fictitious wavefunction in the temporal domain, and approximate it
using matrix-product states (MPS). This approach is efficient provided the
temporal entanglement of the IF is sufficiently low. We illustrate the
versatility of the IF approach by analyzing several models that exhibit a range
of dynamical behaviors, from thermalizing to many-body localized. In
particular, we study the non-equilibrium dynamics in the quantum Ising model in
both Floquet and Hamiltonian settings. We find that temporal entanglement
entropy may be significantly lower compared to the spatial entanglement and
analyze the IF in the continuous-time limit. We simulate the
thermodynamic-limit evolution of local observables in various regimes,
including the relaxation of impurities embedded in an infinite-temperature
chain, and the long-lived oscillatory dynamics of the magnetization associated
with the confinement of excitations. By incorporating disorder-averaging into
the formalism, we analyze discrete time-crystalline response using the IF
method. In this case, we find that the temporal entanglement entropy scales
logarithmically with evolution time. The IF approach offers a new lens on
many-body non-equilibrium phenomena, both in ergodic and non-ergodic regimes,
connecting the theory of open quantum systems theory to quantum statistical
physics.
