We demonstrate the existence of a universal phase diagram of quantum chains with range-$k$ interactions subject to the conservation of a total charge and its dipole moment. These systems exhibit “freezing” transitions between strongly and weakly Hilbert-space-fragmented phases as the charge filling $\nu$ is varied. We show that these continuous phase transitions occur at a critical charge filling of ${\nu }_c={(k-2)}^{-1}$ independently of the on-site Hilbert-space dimension $d$. To this end, we analytically prove that, for any $d$, any state with $\nu <{\nu }_c$ hosts a finite density of sites belonging to “blockages,” which we define as subregions of the chain across which transport of charge and dipole moment cannot occur. Some blockages arise from sequences of frozen sites, i.e., sites with an unchanging on-site charge, while others do not involve frozen sites at all. We prove that the presence of blockages implies strong fragmentation of typical symmetry sectors into Krylov subspaces, each of which forms an exponentially vanishing fraction of the total sector. By studying the distribution of blockages we analytically characterize how typical states are subdivided into dynamically disconnected local “active bubbles” and prove that typical eigenstates at these charge fillings exhibit area-law entanglement entropy, while there exist rare eigenstates featuring non-area-law scaling. We also numerically show that for $\nu >{\nu }_c$ and arbitrary $d$, typical symmetry sectors are weakly fragmented, with their dominant Krylov sectors constituted of states that are free of blockages. We analytically derive some critical exponents characterizing the transition and numerically determine the density of blockages at $\nu ={\nu }_c$, with clear implications for transport at the critical point. Finally, we investigate the properties of special-case models for which no phase transitions occur.

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• Anomalous diffusion
• Critical phenomena
• Eigenstate thermalization
• Entanglement entropy
• Fractons
• Nonequilibrium statistical mechanics
• Phase diagrams
• Quantum statistical mechanics
• Second order phase transitions
• 1-dimensional systems
• Quantum many-body systems
• Strongly correlated systems
• Kinetically constrained models