Topological Invariants in Fermi Systems with Time-Reversal Invariance

Abstract

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 \textonehalf{} in a magnetic field is spin \textonehalf{} in a quadrupole electric field. In particular, the associated bundles are nontrivial and have \ifmmode\pm\else\textpm\fi{} 1 second Chern number. The connection that governs the adiabatic evolution coincides with the symmetric SU(2) Yang-Mills instanton.