The second Chern class in Spinning System

TL;DR

This paper establishes a rigorous connection between the second Chern class and the wavefunction of a spinning system, revealing its dependence on topological invariants like the Hopf index and Brouwer degree.

Contribution

It provides a complete decomposition formula of SU(2) gauge potential in terms of the spinning wavefunction and proves the second Chern class's topological nature in such systems.

Findings

Second Chern class is inherent in spinning systems.

Topological invariant depends on Hopf index and Brouwer degree.

Decomposition formula of SU(2) gauge potential established.

Abstract

Topological property in a spinning system should be directly associated with its wavefunction. A complete decomposition formula of SU(2) gauge potential in terms of spinning wavefunction is established rigorously. Based on the $\phi $-mapping theory and this formula, one proves that the second Chern class is inherent in the spinning system. It is showed that this topological invariant is only determined by the Hopf index and Brouwer degree of the spinning wavefunction.