Strong Hilbert space fragmentation and fractons from subsystem and higher-form symmetries

TL;DR

This work develops a unified framework to realize Hilbert space fragmentation (HSF) in higher dimensions by lifting one-dimensional fragmentation through two complementary routes: subsystem symmetries and higher-form (gauge-like) symmetries. Subsystem-symmetric liftings yield strongly fragmented, fracton-like dynamics with lineon excitations, while higher-form liftings produce topologically robust weak fragmentation; combining both in 3D yields models such as the $Z_N$ X-cube with enhanced Krylov-sector structure. The construction uses a group-word formalism on words built from generator alphabets and relators, enabling explicit 1D, 2D, and 3D models (including a 2D subsystem pair-flip example and a 3D X-cube–type Hamiltonian via flatness constraints). The results unify previously known examples and open pathways to novel fracton phenomenology, including potential non-Abelian fractons, by providing a flexible, scalable lifting strategy across dimensions. Together, these methods offer a concrete route to engineer and study strong fragmentation and fracton physics in realistic lattice settings with controlled symmetry-encoded constraints.

Abstract

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.

Figures (2)

• Figure 1: (Left) The orientation conventions used to define 2D subsystem group models. The dynamics preserve the group elements $\varphi(w^\alpha_{i})$ associated with the words $w^{\alpha}_{i}$ read along each of the rows ($\alpha=x$, read from left to right) and columns ($\alpha=y$, read from top to bottom) of a 2D square lattice. Row and column indices satisfy $1 \leq i \leq L$ and open boundary conditions are imposed throughout the paper. (Right) A configuration belonging to the sector in which every row and column satisfies $\varphi(w_i^\alpha) = \mathtt{e}$. A bar above a character denotes the inverse.
• Figure 2: (a) In the 3D subsystem symmetry model, impose flatness on the paths $\gamma_{v, x}$, $\gamma_{v, y}$, and $\gamma_{v, z}$ around every vertex $v$. Proceeding in the direction indicated by the gray arrows, read off the character on edges with a $+$ and the inverse character on edges with a $-$. In flat configurations the resulting word maps to the identity group element. (b) When combined, the constraints require that the words read along two parallel rows map to the same group element.