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 broad applicability 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. Furthermore, by incorporating disorder-averaging into the formalism, we analyze discrete time-crystalline response using the IF of a bond-disordered kicked Ising chain. 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 to quantum statistical physics.

费曼-弗农影响泛函 (IF) 最初是为了描述量子环境对开放量子系统动力学的影响而引入的。我们应用 IF 方法来描述孤立自旋系统中的量子多体动力学，将系统视为其局部子系统的环境。虽然 IF 只能在某些多体模型中精确计算，但它通常满足自洽方程，前提是系统或系统集合是平移不变的。我们将 IF 视为时域中的虚构波函数，并使用矩阵乘积状态 (MPS) 对其进行近似。如果时间纠缠，这种方法是有效的IF 足够低。我们通过分析表现出一系列动力学行为（从热化到多体局部化）的几个模型来说明 IF 方法的广泛适用性。特别是，我们研究了 Floquet 和哈密顿设置下的量子 Ising 模型中的非平衡动力学。我们发现与空间纠缠相比，时间纠缠熵可能显着降低，并在连续时间限制内分析 IF。我们模拟了不同状态下局部可观测物的热力学极限演化，包括嵌入无限温度链中的杂质的弛豫，以及与激发限制相关的磁化的长寿命振荡动力学。此外，通过将无序平均纳入形式主义，我们使用键无序踢伊辛链的 IF 分析离散时间晶体响应。在这种情况下，我们发现时间纠缠熵随演化时间呈对数缩放。IF 方法为遍历和非遍历机制中的多体非平衡现象提供了一个新视角，将开放量子系统理论与量子统计物理学联系起来。