[Paper Review] Designer non-Abelian fractons from topological layers

Dominic J. Williamson, Meng Cheng|arXiv (Cornell University)|Apr 15, 2020

Theoretical and Computational Physics44 references16 citations

TL;DR

This paper proposes a novel construction of type-I fracton models by gauging planar subsystem symmetries in stacked two-dimensional topological phases. By promoting defects of Abelian symmetries in 2D symmetry-enriched topological orders to fractons, the authors realize non-Abelian fractons with non-integer quantum dimension and chiral boundaries, including a lineon model hosting non-Abelian surface fractons.

ABSTRACT

We formulate a construction of type-I fracton models based on gauging planar subsystem symmetries of topologically ordered two dimensional layers that have been stacked in three ambient spatial dimensions. Via our construction, any defect of an Abelian symmetry group in a two dimensional symmetry-enriched topological order can be promoted into a fracton. This allows us to construct fracton models supporting chiral boundaries and fractons of noninteger quantum dimension. We also find a lineon model supporting non-Abelian surface fractons on its boundary.

Motivation & Objective

• To develop a systematic construction of type-I fracton models using gauged planar subsystem symmetries in 3D stacks of 2D topological phases.
• To realize non-Abelian fractons by promoting symmetry defects in 2D symmetry-enriched topological orders to fractonic excitations.
• To demonstrate the existence of fracton models with chiral boundaries and non-integer quantum dimension.
• To extend the gauging construction to anomalous 1-form symmetries, such as those in fermionic topological orders.
• To construct a 2D non-Abelian fracton model on the surface of a 3D lineon model via the gauged layer construction.

Proposed method

• The construction begins with 2D topological phases possessing 1-form symmetries generated by Wilson lines of Abelian anyons.
• Planar subsystem symmetries are identified as subgroups of these 1-form symmetries, supported on rigid planes in 3D space.
• The gauging procedure is applied to these subsystem symmetries, promoting symmetry defects (fluxes) to fractons in the 3D bulk.
• The resulting models exhibit fractonic excitations with restricted mobility due to loop-like domain-wall condensation on symmetry planes.
• The method is applied to various 2D models, including Ising string-net, non-Abelian gauge theories (e.g., $\mathbb{S}_3$, twisted $\mathbb{Z}_2^3$), SU(2)$_{4k}$, and Kitaev honeycomb models.
• For anomalous 1-form symmetries (e.g., fermionic anyons), the gauging construction is shown to still yield consistent fracton models.

Experimental results

Research questions

• RQ1 Can non-Abelian fractons be systematically constructed from 2D topological orders via gauging of planar subsystem symmetries?
• RQ2 What types of topological order and anyonic statistics emerge in the resulting 3D fracton models?
• RQ3 Can chiral boundaries and fractons with non-integer quantum dimension be realized in this framework?
• RQ4 How does the gauging construction behave when applied to anomalous 1-form symmetries, such as those with fermionic anyons?
• RQ5 Can a 2D non-Abelian fracton model be realized on the surface of a 3D lineon model using this construction?

Key findings

• The construction successfully generates type-I fracton models with non-Abelian fractons from 2D non-Abelian topological phases, including $\mathbb{S}_3$ and twisted $\mathbb{Z}_2^3$ gauge theories.
• Fractons with non-integer quantum dimension are realized through the gauging of symmetry defects in 2D layers, such as in the Ising string-net model.
• A lineon model is constructed that supports non-Abelian surface fractons, demonstrating the potential for exotic surface topological order.
• The gauging procedure remains consistent even when applied to anomalous 1-form symmetries, such as in the Kitaev honeycomb model where the anyon is a fermion.
• A 2D non-Abelian fracton model is realized on the surface of a 3D abelian lineon model via the gauged layer construction, extending the framework to surface fractons.
• The construction provides a unified mechanism to engineer fractons with exotic topological properties, including chiral boundaries and non-Abelian statistics.