We discuss topological invariants for Fermi systems that have time-reversal invariance. The TK${\mathrm{N}}^2$ integers (first Chern numbers) are replaced by second Chern numbers, and Berry's phase becomes a unit quaternion, or equivalently an element of SU(2). The canonical example playing much the same role as spin ½ in a magnetic field is spin ½ in a quadrupole electric field. In particular, the associated bundles are nontrivial and have ± 1 second Chern number. The connection that governs the adiabatic evolution coincides with the symmetric SU(2) Yang-Mills instanton.

• Received 6 July 1988

DOI:https://doi.org/10.1103/PhysRevLett.61.1329