Symmetry-protected and symmetry-enriched topological (SPT/SET) phases in three dimensions are quantum systems that support nontrivial two-dimensional (2D) surface states. These surface states develop finite excitation energy gaps when the relevant symmetries are broken. On the other hand, one-dimensional (1D) gapless modes can populate along interfaces that separate adjacent gapped surface domains with distinct symmetry-breaking orders. A surface strip pattern in general reduces the low-energy SPT/SET surface degrees of freedom onto a 2D array of gapless 1D channels. These channels can be coupled to one another by quasiparticle tunneling, and these interwire interactions collectively provide an effective description of the surface state. In this paper, we study a general class of symmetry-preserving or -breaking SPT/SET surface states that admit finite excitation energy gaps and Abelian topological orders via the coupled-wire construction. In particular, we focus on the prototype Abelian surface topological orders that fall under the $ADE$ classification of simply laced Lie algebras. We also elaborate on the emergent symmetry and duality properties of the coupled-wire models.

• Received 30 July 2018
• Revised 12 April 2019

DOI:https://doi.org/10.1103/PhysRevB.99.235102