Topological Quantum Liquids with Quaternion Non-Abelian Statistics

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.

5 pages, 3 figures

References in corpus (6)

• Superconducting proximity effect and Majorana fermions at the surface of a topological insulator
• The quadrupolar phases of the S=1 bilinear-biquadratic Heisenberg model on the triangular lattice
• Quadrupolar correlations and spin freezing in S = 1 triangular lattice antiferromagnets
• SU(2) gauge theory of the Hubbard model and application to the honeycomb lattice
• Octupolar order in the multiple spin exchange model on a triangular lattice
• Quantum Spin Hall, triplet Superconductor, and topological liquid on the honeycomb lattice

Cited by in corpus (12)

• Hardware efficient quantum simulation of non-abelian gauge theories with qudits on Rydberg platforms
• Fault-Tolerant Quantum Error Correction for non-Abelian Anyons
• Generalized liquid crystals: giant fluctuations and the vestigial chiral order of $I$, $O$ and $T$ matter
• Digital quantum simulation of a (1+1)D SU(2) lattice gauge theory with ion qudits
• Quantum solitons with emergent interactions in a model of cold atoms on the triangular lattice
• Emergent generalized symmetries in ordered phases and applications to quantum disordering
• Classification of point-group-symmetric orientational ordering tensors
• $\mathbb{C}{P}^2$ Skyrmion Crystals in an SU(3) Magnet with a Generalized Dzyaloshinskii-Moriya Interaction
• Hierarchy of orientational phases and axial anisotropies in the gauge theoretical description of generalized nematics
• Generalized symmetries in singularity-free nonlinear $\sigma$ models and their disordered phases
• Constructing Non-Abelian Quantum Spin Liquids Using Combinatorial Gauge Symmetry
• Topological phase diagram of $\mathcal{D}(S_3)$ induced by forbidding charges and fluxes