Weak integrability breaking perturbations of integrable models

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.