Cage-Net Fracton Models

TL;DR

This work constructs cage-net fracton models by stacking 2D string-net layers and condensing 1D flux-strings, thereby generalizing anyon condensation to three dimensions. The approach yields gapped 3D phases with immobile Abelian fractons and, crucially, non-Abelian dim-1 excitations confined to lines, as demonstrated explicitly in the doubled Ising case and generalized to SU(2)$_k$ theories. The ground state is interpreted as a cage-net condensate, a fluctuating network of cages and strings that encodes fracton order and restricted mobility. These intrinsically 3D non-Abelian excitations open avenues for non-Abelian fracton physics, foliated fracton phase distinctions, and potential three-dimensional topological quantum computation.

Abstract

We introduce a class of gapped three-dimensional models, dubbed "cage-net fracton models," which host immobile fracton excitations in addition to non-Abelian particles with restricted mobility. Starting from layers of two-dimensional string-net models, whose spectrum includes non-Abelian anyons, we condense extended one-dimensional "flux-strings" built out of point-like excitations. Flux-string condensation generalizes the concept of anyon condensation familiar from conventional topological order and allows us to establish properties of the fracton ordered (equivalently, flux-string condensed) phase, such as its ground state wave function and spectrum of excitations. Through the examples of doubled Ising and SU(2)$_k$ cage-net models, we demonstrate the existence of strictly immobile Abelian fractons and of non-Abelian particles restricted to move only along one dimension. In the doubled Ising cage-net model, we show that these restricted-mobility non-Abelian excitations are a fundamentally three-dimensional phenomenon, as they cannot be understood as bound states amongst two-dimensional non-Abelian anyons and Abelian particles. We further show that the ground state wave function of such phases can be understood as a fluctuating network of extended objects -- cages -- and strings, which we dub a cage-net condensate. Besides having implications for topological quantum computation in three dimensions, our work may also point the way towards more general insights into quantum phases of matter with fracton order.

Figures (18)

• Figure 1: (a) The dual string $i^*$ has an orientation opposite to that of $i$. (b) The branching rules associated with null strings are defined such that $\delta_{ij0} = 1$ iff $i=j^*$ and vanishes otherwise.
• Figure 2: Two-dimensional truncated square lattice on which we define the string-net models. The vertex projector $A_v$ acts on the three spins adjacent to the vertex $v$ and enforces the branching rules. The plaquette term $B_p$ acts on the 16 spins adjacent to the plaquette $p$ and provides dynamics to the string-net configurations.
• Figure 3: Action of the fundamental string $W_l^{\phi_a}$ on the link $l$ separating plaquettes $p_1$ and $p_2$, where $l$ carries the string-label $b$. Assuming that we start from a state in which there are no fluxes at $p_1$ and $p_2$, the operator $W_l^{\phi_a}$ creates a pair of fluxes $\phi_a$ and $\bar{\phi}_a$ on these plaquettes.
• Figure 4: Path-independent string operator which creates a pair of fluxes, with a flux isolated at each end of the string.
• Figure 5: The X-Cube model is represented by spins $\sigma$ placed on the links of a cubic lattice and is given by the sum of a twelve-spin Pauli-$x$ operator at each cube $c$ and planar four-spin Pauli-$z$ operators at each vertex $v$.