Symmetric Fracton Matter: Twisted and Enriched

TL;DR

This work tackles how planar subsystem symmetry in three dimensions can protect symmetry-protected topological phases and, upon gauging, give rise to twisted fracton orders with novel lineon statistics. It builds exactly solvable lattice models—including a topological plaquette paramagnet and its twisted gauged version—and connects them to a higher-rank U(1) Chern-Simons field theory that encodes gapless boundary modes and lineon braiding. A key advance is the demonstration that partial gauging yields symmetry-enriched fracton phases where lineons carry fractional symmetry charges, realized via decorations such as the Valence Tube Solid that endow endpoints with Kramers degeneracy under subsystem time reversal. The results establish a unified framework linking SSPT, twisted fracton order, and higher-rank topological field theories, with implications for 3D quantum phases, boundary phenomena, and potential fault-tolerant architectures.

Abstract

In this paper, we explore the interplay between symmetry and fracton order, motivated by the analogous close relationship for topologically ordered systems. Specifically, we consider models with 3D planar subsystem symmetry, and show that these can realize subsystem symmetry protected topological phases with gapless boundary modes. Gauging the planar subsystem symmetry leads to a fracton order in which particles restricted to move along lines exhibit a new type of statistical interaction that is specific to the lattice geometry. We show that both the gapless boundary modes of the ungauged theory, and the statistical interactions after gauging, are naturally captured by a higher-rank version of Chern-Simons theory. We also show that gauging only part of the subsystem symmetry can lead to symmetry-enriched fracton orders, with quasiparticles carrying fractional symmetry charge.

Figures (20)

• Figure 1: Comparison of the relationship between subsystem SPT phase, twsited fracton theory and higher rank Chern-Simons term with their respective counterparts in topologically ordered systems. Here the Higgs phases are condensates of objects with charge $2$, leading to the nontrivial orders.
• Figure 2: Domain frame condensate for the plaquette Ising model in the paramagnetic phase.
• Figure 3: Couplings in X-cube model. The 12 spin interaction on the cube indicates the Gauss law constraint. The four spin in the vertex in the $\alpha-\beta$ plane describes the gauge fluctuation.
• Figure 4: L: The charge excitation generated by the 2d membrane(red links) operator. At each corner of the membrane, there is a cube where $\prod_{i\in c} \sigma^x_i =-1$, which contains a charge (fracton) excitation; R: The lineon(flux) excitation generated by a straight string(green). The lineon excitation lives at the end of string.
• Figure 5: The spin model on the BCC lattice. Each corner of the cube contains an Ising spin $S_0$. The cube center contains 3 spin dipoles $(S_a,S_b,S_c)$. The spin interaction appears between four spin $S_0$ on the same cube face, as well as two $S_0$ together with the spin dipole $(S_\alpha)$ on the triangle.