Fractional excitations in foliated fracton phases

TL;DR

Fractional excitations in foliated fracton phases are characterized by quotient superselection sectors (QSS) that mod out planon content, yielding a finite, universal data set to classify 3D fracton orders up to layering with 2D topological states. The authors define interferometric statistics for these sectors, organizing them into an S-matrix analogue that captures universal phase data while remaining insensitive to attached 2D quasiparticles. They demonstrate this framework on multiple exactly solvable models, notably showing the X-cube and semionic X-cube share the same QSS and interferometric data, and they map between these models via the inclusion of 2D layers and local unitaries. The work extends to various lattice geometries (kagome, hyperkagome), rotor generalizations ($\mathbb{Z}_N$ X-cube), and the checkerboard model, establishing a robust, extensible approach to compare and relate foliated fracton orders and paving the way for broader non-Abelian and cage-net generalizations.

Abstract

Fractional excitations in fracton models exhibit novel features not present in conventional topological phases: their mobility is constrained, there are an infinitude of types, and they bear an exotic sense of 'braiding'. Hence, they require a new framework for proper characterization. Based on our definition of foliated fracton phases in which equivalence between models includes the possibility of adding layers of gapped 2D states, we propose to characterize fractional excitations in these phases up to the addition of quasiparticles with 2D mobility. That is, two quasiparticles differing by a set of quasiparticles that move along 2D planes are considered to be equivalent; likewise, 'braiding' statistics are measured in a way that is insensitive to the attachment of 2D quasiparticles. The fractional excitation types and statistics defined in this way provide a universal characterization of the underlying foliated fracton order which can subsequently be used to establish phase relations. We demonstrate as an example the equivalence between the X-cube model and the semionic X-cube model both in terms of fractional excitations and through an exact mapping.

Figures (24)

• Figure 1: (a) Cube and (b) cross terms of the X-cube Hamiltonian.
• Figure 2: (a) A rigid string operators in the X-cube model. Lineons, represented as red dots, are created at the endpoints and corner. (b) A flexible string operator. Lineon dipoles, which are free to move in a 2D plane, are created at the endpoints.
• Figure 3:
• Figure 4: A graphical representation of the unitary operators (a) $S_1$ and (b) $S_2$. In this figure only a single unit cell is depicted, although $S_1$ and $S_2$ act uniformly along an $xy$ plane. The finite depth quantum circuit $S=S_1S_2$ disentangles the blue $xy$ layer from the bulk X-cube system. The qubits represented by dashed edges in (b) are decoupled ancilla qubits stabilized by $H_0$ of Eq. (\ref{['eqn:conjugation']}).
• Figure 5: (a) A flexible string operator $W_\varepsilon(\gamma)$ and a ribbon operator $W_\mu(\lambda)$ of the X-cube model, which are mapped under conjugation by the finite-depth circuit $S$ to (b) electric and magnetic string operators $W_e(\gamma)$ and $W_m(\lambda)$ acting on a decoupled toric code layer lying along the $z=z_0$ plane, which is the back plane pictured in (a). These operator are respectively defined as tensor products of Pauli $X$ operators over the yellow edges and Pauli $Z$ operators over the blue edges. $\gamma$ and $\lambda$ are paths on the direct and dual lattices respectively of the $z=z_0$ plane. The red dots represent X-cube lineons in (a) and $\mathbb{Z}_2$ charges in (b). Conversely, the shaded green cubes in (a) represent fractons, whereas the green squares in (b) represent $\mathbb{Z}_2$ fluxes.