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Prethermalization refers to the transient phenomenon where a systemthermalizes according to a Hamiltonian that is not the generator of itsevolution. We provide here a rigorous framework for quantum spin systems whereprethermalization is exhibited for very long times. First, we consider quantumspin systems under periodic driving at high frequency $\nu$. We prove that upto a quasi-exponential time $\tau_* \sim e^{c \frac{\nu}{\log^3 \nu}}$, thesystem barely absorbs energy. Instead, there is an effective local Hamiltonian$\hat D$ that governs the time evolution up to $\tau_*$, and hence thiseffective Hamiltonian is a conserved quantity up to $\tau_*$. Next, we considersystems without driving, but with a separation of energy scales in theHamiltonian. 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 aneffective conserved quantity, different from the Hamiltonian, up to a time$\tau_*$ that is (almost) exponential in $U/J$.

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