Chern numbers for fermionic quadrupole systems

The authors analyse families of quantum quadrupole Hamiltonians H= Sigma alpha beta Qalpha beta Jalpha Jbeta for half-odd-integer spin, and calculate the second Chern numbers of the energy levels. Each non-zero integer occurs only a finite number of times. The adiabatic time evolution, the non-Abelian generalisation of Berry's phase, is different for each system, in contrast to Berry's example. The j=3/2 and j=1/2 cases previously analysed are the only ones with self-dual curvatures and SO(5) symmetry.