Quantum many-body attractors

Real-world complex systems often show robust, persistent oscillatory dynamics, e.g.~non-trivial attractors. On the quantum level this behaviour has only been found in semi-classical or weakly correlated systems under restrictive assumptions. However, strongly interacting systems without classical limits, e.g.~electrons on a lattice or spins, typically relax quickly to a stationary state (trivial attractors). This raises the puzzling question of how non-trivial attractors can arise from the quantum laws. Here, we introduce strictly local dynamical symmetries that lead to extremely robust and persistent oscillations in quantum many-body systems without a classical limit. Observables that do not have overlap with the symmetry operators can relax, losing memory of their initial conditions. The remaining observables enter complex dynamical cycles, signalling the emergence of a quantum many-body attractor. We provide a recipe for constructing Hamiltonians featuring local dynamical symmetries. As an example, we introduce the spin lace – a model of a quasi-1D quantum magnet.