PIF2025 -Tensor networks, Temporal entanglement, and the Real-time dynamics of out-of-Equilibrium quantum Systems - (PID2024-160172NB-I00)

Project summary. How can simple microscopic constituents give rise to the rich diversity of phases and phenomena we see in nature? Much of this beauty stems from collective emergence in many-body systems. Equilibrium physics is by now well charted and dictated by a balance of energy and entropy. Genuinely quantum effects (e.g., superconductivity, superfluidity, topologically ordered phases) typically show up at very low temperatures where the entropy is sufficiently suppressed. Out of equilibrium, new questions arise: What strongly correlated states can we produce that are not present at equilibrium? In general, systems locally thermalize and thus localized quantum effects are washed out. Can specific protocols evade relaxation and protect genuine quantum features? What is the ultimate computational complexity we need to face to describe the evolution of such local quantum properties? We aim to tackle these questions by better understanding the role of temporal and generalized temporal entropies in out-of-equilibrium dynamics and by building tensor-network tools that leverage this understanding. Approach. We combine analytical methods (field theory, process tensors, influence functionals) with state-of-the-art tensor-network simulations in 1D and 2D. Temporal entropies serve as key quantifiers of quantum complexity and guide the design of scalable algorithms for long-time evolution, thermalization, and transport.