Weak integrability breaking in quantum and classical models on the lattice

Generic interacting many-body systems are expected to thermalize—classically through ergodic exploration of phase space, and quantum-mechanically through frameworks such as the eigenstate thermalization hypothesis (ETH). For perturbations of integrable models, Fermi’s golden rule typically predicts thermalization on timescales τ ∼ λ-2. However, this picture is modified in presence of quasi-conserved quantities which lead to parametrically slower dynamics. In this talk, I revisit the systematic framework for constructing weak integrability breaking (WIB) perturbations, which preserve quasi-conserved quantities to high order in the perturbation strength λ. I discuss the extension of this framework to classical integrable lattice models: the Ishimori chain, the Toda lattice, and the harmonic oscillator chain (HOC). A central result: the cubic nonlinearity of the Fermi–Pasta–Ulam–Tsingou (α-FPUT) model—a longstanding problem—emerges naturally as a WIB perturbation of the harmonic chain. Specifically, we identify the corresponding adiabatic gauge potential (AGP)—which serves both as a diagnostic for WIBs and as a proxy for its generators—as a nontrivial trilocal generator in real space, and showed that, more generally, any cubic, translationally invariant, momentum-conserving perturbation of the HOC admits such a generator and is therefore a WIB perturbation. This provides a concrete example of a WIB generator that does not fall into the previously known extensive local, boosted, or bilocal classes, and suggests the existence of a richer hierarchy of locality structures underlying weak integrability breaking in many-body systems.