We introduce a new route to Hilbert space fragmentation in high dimensions leveraging the group-word formalism. We show that taking strongly fragmented models in one dimension and "lifting" to higher dimensions using subsystem symmetries can yield strongly fragmented dynamics in higher dimensions, with subdimensional (e.g., lineonic) excitations. This provides a new route to higher-dimensional strong fragmentation, and also a new route to fractonic behavior. Meanwhile, lifting one-dimensional strongly fragmented models to higher dimensions using higher-form symmetries yields models with topologically robust weak fragmentation. In three or more spatial dimensions, one can also "mix and match" subsystem and higher-form symmetries, leading to canonical fracton models such as X-cube. We speculate that this approach could also yield a new route to non-Abelian fractons. These constructions unify a number of phenomena that have been discussed in the literature, as well as furnishing models with novel properties.
In this paper, we propose strongly correlated gapless states (or critical states) of spin-1/2 electrons in
1+1 dimensions, such as the doped ferromagnetic and antiferromagnetic spin-1/2 Ising chains. We find that the metallic phases in the doped ferromagnetic and antiferromagnetic Ising chain are different strongly correlated gapless phases, despite the fact that the two phases have the same symmetry. The doped antiferromagnetic Ising chain has a finite energy gap for all charge-1 fermionic excitations even without pairing caused by the attractive interaction, resembling the pseudogap phase of underdoped high
Tc superconductors. Applying a transverse field to the ferromagnetic and antiferromagnetic metallic phase can restore the
Z2 symmetry, which gives rise to two distinct critical points despite the fact that the two transitions have exactly the same symmetry-breaking pattern. We also propose chiral metallic states. Some of these gapless states are strongly correlated in the sense that they do not belong to the usual Tomonaga-Luttinger phase of fermions, i.e., they cannot be smoothly deformed into noninteracting fermion systems with the same symmetry. Our nonperturbative results are obtained by noting that gapless quantum systems have emergent categorical symmetries (i.e., noninvertible gravitational anomalies), which are described by multicomponent partition functions that are modular covariant. This allows us to calculate the scaling dimensions and quantum numbers of all the low-energy operators for those strongly correlated gapless states. This demonstrates an application of emergent categorical symmetries in determining low-energy properties of strongly correlated gapless states, which are hard to obtain otherwise.