The topological structure of the integral quantum Hall effect in magnetic semiconductor-superconductor hybrids

An unconventional integer quantum Hall regime was found in magnetic semiconductor-superconductor hybrids. By making use of the decomposition of the gauge potential on a $U(1)$ principal fibre bundle over $k$-space, we study the topological structure of the integral Hall conductance. It is labeled by the Hopf index $\beta$ and the Brouwer degree $\eta$. The Hall conductance topological current and its evolution is discussed.