On Exponentially Long Prethermalization Timescales in Isolated Quantum Systems

We study prethermalization in time-independent quantum many-body systems on a $d$-dimensional lattice with an extensive local Hamiltonian $H=N+\epsilon P$, in the regime where $\epsilon \ll 1$. We show that the prethermalization time is exponentially large in ${\epsilon }_0\mathrm{/}\epsilon$, where ${\epsilon }_0$ is the ratio between an effective spectral gap width and the local norm of $P$. We prove also that for exponentially long times, there exist two quasi-conserved quantities up to exponentially small errors.