Obstruction to Ergodicity from Locality and $U(1)$ Higher Symmetries on the Lattice

Abstract

We argue that the presence of \emph{any} exact $U(1)$ higher-form symmetry, under mild assumptions, presents a fundamental obstruction to ergodicity under unitary dynamics in lattice systems with local interactions and finite on-site Hilbert space dimension. Focusing on the two-dimensional case, we show that such systems necessarily exhibit Hilbert space fragmentation and explicitly construct Krylov sectors whose number scales exponentially with system size. While these sectors cannot be distinguished by symmetry quantum numbers, we identify the emergent integrals of motion which characterize them. Our symmetry-based approach is insensitive to details of the Hamiltonian and the lattice, providing a systematic explanation for ergodicity-breaking in a range of systems, including quantum link models.