Topological Entanglement Entropy of Fracton Stabilizer Codes

TL;DR

This work probes 3D fracton topological orders by computing entanglement entropy in stabilizer-based models, revealing a universal topological contribution that scales linearly with subsystem size $R$ in both the X-cube model and Haah’s code. By employing 3D generalizations of ABC and PQWT prescriptions and a stabilizer formalism, the authors quantify the nonlocal constraints underlying entanglement and demonstrate robustness under arbitrary local perturbations via Schrieffer–Wolff-type transformations. They further argue that, in disordered systems, localization-protected fracton order can persist in excited states, with topological entanglement entropy serving as a diagnostic despite many-body localization. Overall, the paper provides a concrete, entanglement-based characterization of fracton phases, linking stabilizer structure to nonlocal correlations and robustness against perturbations, and points to extensions to excited-state fracton order.

Abstract

Entanglement entropy provides a powerful characterization of two-dimensional gapped topological phases of quantum matter, intimately tied to their description by topological quantum field theories (TQFTs). Fracton topological orders are three-dimensional gapped topologically ordered states of matter, but the existence of a TQFT description for these phases remains an open question. We show that three-dimensional fracton phases are nevertheless characterized, at least partially, by universal structure in the entanglement entropy of their ground state wave functions. We explicitly compute the entanglement entropy for two archetypal fracton models --- the `X-cube model' and `Haah's code' --- and demonstrate the existence of a topological contribution that scales linearly in subsystem size. We show via Schrieffer-Wolff transformations that the topological entanglement of fracton models is robust against arbitrary local perturbations of the Hamiltonian. Finally, we argue that these results may be extended to characterize localization-protected fracton topological order in excited states of disordered fracton models.

Figures (12)

• Figure 1: Illustration of the two types of prescriptions used to obtain the topological entanglement entropy. An ABC prescription is illustrated in (a), while (b) illustrates a PQWT prescription.
• Figure 2: Two-dimensional toric code on the square lattice. The region enclosed by red the line is the subregion $A$, which has size $2 \times 2$ measured in edges of the lattice. Spins on links cut by the red line lie outside $A$. The vertex and plaquette terms $A_p$ and $B_v$ are also shown.
• Figure 3: Regions for a $d=3$ ABC prescription to compute topological entanglement entropy. This choice of regions picks out a preferred axis (arrow). $A$ is a $2R \times R \times R$ rectangular prism, and regions $B$ and $C$ both have dimensions $R \times R \times R$. The union $ABC$ of the three regions is a $2R \times 2R \times R$ rectangular prism, with $R$ the linear size along the preferred axis.
• Figure 4: Regions for a PQWT prescription with a preferred axis denoted by arrows in $d=3$ to extract topological entanglement entropy. Lengths are measured in terms of the lattice distance (i.e. the number of links).
• Figure 5: Nonlocal stabilizer (red square) for $d=3$ toric code, which is a product of plaquette terms in the $B = \bar{P}$ subregion and those on the boundary but only acts nontrivially on the $P$ subsystem.