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.
Highlights
AND MOTIVATIONA topologically ordered quantum phase of matter in arbitrary spatial dimensions is defined as one which exhibits a finite gap to all excitations in the thermodynamic limit, has a finite but nontrivial ground-state degeneracy on a topologically nontrivial manifold—such that no local operators can distinguish between the degenerate ground states—and supports fractionalized quasiparticles which cannot be locally created
Our results establish the existence of non-Abelian foliated fracton phases. Investigating these models allows us to establish the structure of the ground-state wave function in the fracton phase, which we propose can be understood as a condensate of fluctuating “cage nets.”
We introduce a class of gapped d 1⁄4 3 nonAbelian fracton models, dubbed cage-net fracton models, 021010-20
Summary
A topologically ordered quantum phase of matter in arbitrary spatial dimensions is defined as one which exhibits a finite gap to all excitations in the thermodynamic limit, has a finite but nontrivial ground-state degeneracy on a topologically nontrivial manifold—such that no local operators can distinguish between the degenerate ground states—and supports fractionalized quasiparticles which cannot be locally created. The presence of such excitations, which we dub as being intrinsically subdimensional and inextricably non-Abelian, demonstrates that this model displays a novel non-Abelian fracton order Along these lines, recent work has introduced the notion of a foliated fracton phase [74]. III, we review the current understanding of fracton models, focusing, in particular, on the X-Cube model introduced by Vijay, Haah, and Fu [50] Using this model as an example, we introduce the layer-construction approach for studying certain fracton phases and elucidate the nature of the ground-state wave function. Borrowing ideas from anyon condensation in d 1⁄4 2 topological orders, we establish the excitation spectrum of the fracton phase, obtained from condensing extended one-dimensional stringlike excitations, and explicitly demonstrate the existence of non-Abelian dim-1 particles. We end by discussing implications of our work for the field of fractons and by exploring open questions and future directions
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