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

TL;DR

This paper demonstrates that exact $U(1)$ higher-form symmetries in lattice systems inherently prevent ergodicity by causing Hilbert space fragmentation, leading to exponentially many Krylov sectors independent of Hamiltonian specifics.

Contribution

It establishes a general symmetry-based framework showing how $U(1)$ higher-form symmetries obstruct ergodicity and cause Hilbert space fragmentation in lattice models.

Findings

Presence of $U(1)$ higher-form symmetry leads to Hilbert space fragmentation.

Number of Krylov sectors scales exponentially with system size.

Emergent integrals of motion characterize fragmented sectors.

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.