Space-local memory in generalized master equations: Reaching the thermodynamic limit for the cost of a small lattice simulation

The exact quantum dynamics of lattice models can be computationally intensive, especially when aiming for large system sizes and extended simulation times necessary to converge transport coefficients. By leveraging finite memory times to access long-time dynamics using only short-time data, generalized master equations (GMEs) can offer a route to simulating the dynamics of lattice problems efficiently. However, such simulations are limited to small lattices whose dynamics exhibit finite-size artifacts that contaminate transport coefficient predictions. To address this problem, we introduce a novel approach that exploits finite memory in time \textit{and} space to efficiently predict the many-body dynamics of dissipative lattice problems involving short-range interactions. This advance enables one to leverage the short-time dynamics of small lattices to simulate arbitrarily large systems over long times. We demonstrate the strengths of this method by focusing on nonequilibrium polaron relaxation and transport in the dispersive Holstein model, successfully simulating lattice dynamics in one and two dimensions free from finite-size effects, reducing the computational expense of such simulations by multiple orders of magnitude. Our method is broadly applicable and provides an accurate and efficient means to investigate nonequilibrium relaxation with microscopic resolution over mesoscopic length and time scales that are relevant to experiment.