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 $${\nu}$$ν. We prove that up to a quasi-exponential time $${\tau_* \sim {\rm e}^{c \frac{\nu}{\log^3 \nu}}}$$τ∗∼ecνlog3ν, the system barely absorbs energy. Instead, there is an effective local Hamiltonian $${\widehat D}$$D^ that governs the time evolution up to $${\tau_*}$$τ∗, and hence this effective Hamiltonian is a conserved quantity up to $${\tau_*}$$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 $${\tau_*}$$τ∗ that is (almost) exponential in $${U/J}$$U/J.