A Rigorous Theory of Many-Body Prethermalization for Periodically Driven and Closed Quantum Systems

Prethermalization refers to the transient phenomenon where a system thermalizes according to a Hamiltonian that is not the generator of its evolution. We provide here a rigorous framework for quantum spin systems where prethermalization is exhibited for very long times. First, we consider quantum spin systems under periodic driving at high frequency ν. We prove that up to a quasi-exponential time τ∗∼ecνlog3ν, the system barely absorbs energy. Instead, there is an effective local Hamiltonian D^ that governs the time evolution up to τ_∗, and hence this effective Hamiltonian is a conserved quantity up to τ_∗. Next, we consider systems without driving, but with a separation of energy scales in the Hamiltonian. A prime example is the Fermi–Hubbard model where the interaction U is much larger than the hopping J. Also here we prove the emergence of an effective conserved quantity, different from the Hamiltonian, up to a time τ_∗ that is (almost) exponential in U/ J.