Symmetry-Protected Topological relationship between $SU(3)$ and $SU(2)\times U(1)$ in Two Dimension

Abstract

Symmetry-protected topological $\left(SPT\right)$ phases are gapped short-range entangled states with symmetry $G$, which can be systematically described by group cohomology theory. $SU(3)$ and $SU(2)\times U(1)$ are considered as the basic groups of Quantum Chromodynamics and Weak-Electromagnetic unification, respectively. In two dimension $(2D)$, nonlinear-sigma models with a quantized topological Theta term can be used to describe nontrivial SPT phases. By coupling the system to a probe field and integrating out the group variables, the Theta term becomes the effective action of Chern-Simons theory which can derive the response current density. As a result, the current shows a spin Hall effect, and the quantized number of the spin Hall conductance of SPT phases $SU(3)$ and $SU(2)\times U(1)$ are same. In addition, relationships between $SU(3)$ and $SU(2)\times U(1)$ which maps $SU(3)$ to $SU(2)$ with a rotation $U(1)$ will be given.

References in corpus (2)

• Non-Abelian Anyons and Topological Quantum Computation
• Symmetry protected Spin Quantum Hall phases in 2-Dimensions