  A quantum integrable system slightly perturbed away from integrability is
typically expected to thermalize on timescales of order $\tau\sim
\lambda^{-2}$, where $\lambda$ is the perturbation strength. We here study
classes of perturbations that violate this scaling, and exhibit much longer
thermalization times $\tau\sim \lambda^{-2\ell}$ where $\ell>1$ is an integer.
Systems with these "weak integrability breaking" perturbations have an
extensive number of quasi-conserved quantities that commute with the perturbed
Hamiltonian up to corrections of order $\lambda^\ell$. We demonstrate a
systematic construction to obtain families of such weak perturbations of a
generic integrable model for arbitrary $\ell$. We then apply the construction
to various models, including the Heisenberg, XXZ, and XYZ chains, the Hubbard
model, models of spinless free fermions, and the quantum Ising chain. Our
analytical framework explains the previously observed evidence of weak
integrability breaking in the Heisenberg and XXZ chains under certain
perturbations.
