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This paper shows that a wide class of “generalized” symmetries can split a many‑body quantum system into exponentially many dynamically disconnected pieces. That splitting is known as Hilbert space fragmentation. Each piece, called a Krylov sector, is the small part of the full state space that a given initial product state can explore under the system’s dynamics. Fragmentation has often been taken as evidence that a system fails to thermalize, or is nonergodic. The authors argue that this conclusion can be too quick: generalized symmetries alone can produce many sectors without necessarily breaking thermalization in every sector.
The authors give a simple, general mechanism. If a conserved operator U can be translated many times so that the translated copies act on disjoint regions, and if U is diagonal in the product basis used to build the many‑body states with at least two distinct eigenvalues, then the different local eigenvalues combine like independent choices across space. That combinatorics produces exponentially many symmetry sectors. Since Krylov sectors sit inside symmetry sectors whenever the symmetry is diagonal in the same product basis, the number of Krylov sectors grows exponentially as well. They formalize this with a proposition and a counting argument that links the geometry of the symmetry operators to the number of sectors.
The paper shows this mechanism applies to several types of generalized symmetry. Gauge symmetries, higher‑form or subsystem symmetries, and certain non‑invertible symmetries can meet the conditions needed for fragmentation. As a concrete example, the authors discuss a three‑dimensional U(1) quantum link model on a cubic lattice. A local gauge generator in that model can take seven values and can be translated about L3/2 times, giving at least 7·L3/2 symmetry sectors in one product basis. A subsystem “1‑form” operator that measures planar magnetization has L2+1 eigenvalues on each plane and can be translated L times, yielding roughly (L2+1)^L symmetry sectors when counted across planes. The paper also shows how non‑invertible operators called partial isometries can further fragment individual symmetry sectors.
Why this matters: many recent studies have used the presence of exponentially many Krylov or symmetry sectors as a diagnostic of ergodicity breaking. This work changes that interpretation. Exponential fragmentation can be a direct consequence of symmetry geometry rather than a signature of unavoidable nonthermal behavior. The authors also point out a related consequence for so‑called disorder‑free localization: when symmetry sectors are not translationally uniform, thermalization restricted to a Krylov sector can look like localized behaviour without disorder and without invoking full ergodicity breaking or gauge constraints.
Important caveats and limits are stated in the paper. The counting argument requires symmetry operators (or the projectors that define sectors) to be diagonal in a tensor‑product basis and to admit many disjoint translations; not every generalized symmetry has those properties. Open boundary conditions change the counting only by boundary corrections, but the bulk argument assumes many independent translated copies. Also, fragmentation of the product basis does not by itself guarantee nonthermal dynamics inside each sector. Individual sectors can still thermalize or show other nonergodic phenomena such as quantum many‑body scars. The excerpted text presents proofs and explicit examples, but a full assessment of dynamical consequences requires examination of particular Hamiltonians and sectors beyond the counting argument.