Topological Quantum Liquids with Quaternion Non-Abelian Statistics

TL;DR

This paper explores how three types of two-dimensional tetrad magnetic orders, characterized by different ground state manifolds, lead to distinct gapped quantum liquids with Abelian and non-Abelian topological orders upon disordering.

Contribution

It introduces three new classes of tetrad orders with unique ground state manifolds and demonstrates their transition into topologically ordered liquids described by gauge theories, including a non-Abelian quaternion gauge theory.

Findings

Disorder of SO(3) tetrad order yields Z_2 topological order.

Disorder of S^3/Z_4 tetrad order yields Z_4 topological order.

Disorder of S^3/Q_8 tetrad order yields a non-Abelian quaternion gauge theory with 22-fold degeneracy.

Abstract

Noncollinear magnetic order is typically characterized by a "tetrad" ground state manifold (GSM) of three perpendicular vectors or nematic-directors. We study three types of tetrad orders in two spatial dimensions, whose GSMs are SO(3) = S^3/Z_2, S^3/Z_4, and S^3/Q_8, respectively. Q_8 denotes the non-Abelian quaternion group with eight elements. We demonstrate that after quantum disordering these three types of tetrad orders, the systems enter fully gapped liquid phases described by Z_2, Z_4, and non-Abelian quaternion gauge field theories, respectively. The latter case realizes Kitaev's non-Abelian toric code in terms of a rather simple spin-1 SU(2) quantum magnet. This non-Abelian topological phase possesses a 22-fold ground state degeneracy on the torus arising from the 22 representations of the Drinfeld double of Q_8.