Twisted fracton models in three dimensions

Abstract

We study novel three-dimensional gapped quantum phases of matter which support quasiparticles with restricted mobility, including immobile "fracton" excitations. So far, most existing fracton models may be instructively viewed as generalized Abelian lattice gauge theories. Here, by analogy with Dijkgraaf-Witten topological gauge theories, we discover a natural generalization of fracton models, obtained by twisting the gauge symmetries. Introducing generalized gauge transformation operators carrying an extra phase factor depending on local configurations, we construct a plethora of exactly solvable three-dimensional models, which we dub "twisted fracton models." A key result of our approach is to demonstrate the existence of rich non-Abelian fracton phases of distinct varieties in a three-dimensional system with finite-range interactions. For an accurate characterization of these novel phases, the notion of being inextricably non-Abelian is introduced for fractons and quasiparticles with one-dimensional mobility, referring to their new behavior of displaying braiding statistics that is, and remains, non-Abelian regardless of which quasiparticles with higher mobility are added to or removed from them. We also analyze these models by embedding them on a three-torus and computing their ground state degeneracies, which exhibit a surprising and novel dependence on the system size in the non-Abelian fracton phases. Moreover, as an important advance in the study of fracton order, we develop a general mathematical framework which systematically captures the fusion and braiding properties of fractons and other quasiparticles with restricted mobility. CONTENTS

Gauging permutation symmetries as a route to non-Abelian fractons

“…There has been recent progress on this front [62,75,76]: in Ref. [62], layers of 2D stringnets were coupled to produce fracton phases with non-Abelian excitations mobile only along lines, while non-Abelian fractons were found by twisting the layers of Abelian fracton models in Ref.…”

“…Moreover, all of the aforementioned examples fall into the category of foliated type-I fracton phases, where the presence of excitations mobile along lines or planes allows generalized notions of braiding and statistics to be defined [19,21]. Unlike these examples, where notions of "non-Abelianness" 2 have been extended to subdimensional excitations [62,76], whether such notions extend to type-II models, lacking any mobile excitations, remains unclear. An obvious obstruction to studying this question is the lack of any non-Abelian type-II model in the literature.…”

Quantum error correction with the semion code

“…A new way of constructing quantum codes opens up with this work. The tools introduced here for models like the DS based on Abelian lattice gauge theories can be generalized to other Levin-Wen models [16,67,68], like doubled Fibonacci models, or twisted versions of fracton models [69]. This is the subject of further study.…”

On duality between Cosserat elasticity and fractons

“…The phenomenon of restricted mobility of excitations has recently attracted attention in the study of topological phases of matter, spin liquids and self-correcting quantum memory. Namely, a new type of topological order, fracton order was found and established [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35]. Fracton systems exhibit certain similarities to traditional topologically ordered phases, namely topological 1 groundstate degeneracy on a torus and robustness to local perturbations.…”