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

TL;DR

This paper explores the relationship between $SU(3)$ and $SU(2) imes U(1)$ symmetry-protected topological phases in two dimensions, revealing they share the same quantized spin Hall conductance and establishing a mapping between these symmetry groups.

Contribution

It demonstrates a topological relationship between $SU(3)$ and $SU(2) imes U(1)$ SPT phases in 2D, including a mapping of the groups and shared physical responses.

Findings

$SU(3)$ and $SU(2) imes U(1)$ SPT phases have identical quantized spin Hall conductance.

A mapping between $SU(3)$ and $SU(2)$ with a $U(1)$ rotation is established.

The effective response is described by Chern-Simons theory derived from nonlinear sigma models.

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.