Universal Freezing Transitions of Dipole-Conserving Chains

TL;DR

This paper identifies a universal, continuous freezing transition between strongly and weakly Hilbert-space-fragmented phases in dipole-conserving chains with range-$k$ interactions, with a critical filling $\nu_c=(k-2)^{-1}$ independent of on-site dimension $d$. The authors develop the Fully Extended State (FES) picture to locate blockages, prove strong fragmentation for $\nu<\nu_c$, and demonstrate blockage-free extended states dominating for $\nu>\nu_c$, supporting a $d$-independent phase diagram. They derive critical exponents (e.g., $\xi\sim(\nu_c-\nu)^{-2}$, $\rho_t\sim(\nu_c-\nu)$) and show area-law entanglement in typical states in the strongly fragmented phase, alongside signatures of inverse quantum many-body scars in the presence of type-2 blockages. Numerically, they present scalable algorithms to map states to FES/PC extended states, confirm weak fragmentation for $\nu>\nu_c$ across $d$, and discuss transport properties at the critical point. The results establish a minimal, universal framework for ergodicity breaking via Hilbert-space fragmentation in dipole-conserving 1D systems, with potential implications for tilted lattice realizations and beyond.

Abstract

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 $ν$ is varied. We show that these continuous phase transitions occur at a critical charge filling of $ν_c=(k-2)^{-1}$ independently of the on-site Hilbert space dimension $d$. To this end, we analytically prove that, for any $d$, any state with $ν<ν_c$ 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 characterise 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 $ν>ν_c$ 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 characterising the transition and numerically determine the density of blockages at $ν=ν_c$, with clear implications for transport at the critical point. Finally, we investigate the properties of special-case models for which no phase transitions occur.

Figures (16)

• Figure 1: Conjectured universal phase diagram of charge- and dipole-conserving quantum chains with range-$k$ interactions, which feature Hilbert space fragmentation (HSF). The relevant variables are the continuous particle filling $\nu\ge0$ and the discrete range of interactions $k\ge 3$. A first strong-to-weak fragmentation transition occurs at $\nu_c=1/(k-2)$, a value independent of the on-site Hilbert space dimension $d$. Note that for each value of $d$ we depict the corresponding phase diagram only below the half filling $(d-1)/2$ (vertical dashed lines). Beyond half filling the phase diagram is mirrored by particle-hole symmetry, leading to a second opposite transition (not depicted here) at $d-1-\nu_c$. Also note that for $d=2$ it is only meaningful to consider $k\ge4$, as there can be no dynamical evolution at $k=3$ which respects the global conservation laws.
• Figure 2: Examples of inward hop and outward hop range-$k$ gates ($k=4$) that conserve $N$ and $X$, acting on a local region of a chain with $d=4$.
• Figure 3: Schematic representation of type-2 blockages, as identified from the sub-FES that occupies a region $\mathcal{A}$ of the chain. (a) Example of region $\mathcal{A}$ enclosed by two type-1 blockages. Subregions separated by a type-2 edge (i.e., a sequence of $k-2$ holes in an FES) are associated with different colours. Any region consisting of two type-2 edges and the sites enclosed by them represents a type-2 blockage. (b) Close-up of the portion of region $\mathcal{A}$ highlighted by the dashed gray rectangle in panel (a), where two sequences of $k-2$ holes (type-2 edges) appear in the FES. In between the two type-2 edges, particles are separated only by sequences of $k-3$ holes. According to the colour scheme, regions separated by one type-2 edge have particles of different colours. (c) Example of a possible partition of the portion of $\mathcal{A}$ from panel (b) into disjoint subregions that can at most host two different colours. Note that it is irrelevant where exactly in-between two chosen particles of the FES we place a given partition cut.
• Figure 4: Example of a bipartition of the chain with $L$ sites and $N=\nu L$ particles. $W_{d,\nu}$ indicates the total number of configurations compatible with it.
• Figure 5: Partition of the chain into contiguous subregions ($\mathcal{A}_i$) separated by type-2 blockages of $L$-independent size ($\mathcal{B}_i$). Each vertical line represents a type-2 edge.