Non-Abelian Holonomy of BCS and SDW Quasi-particles

In this work we investigate properties of fermions in the SO(5) theory of high Tc superconductivity. We show that the adiabatic time evolution of a SO(5) superspin vector leads to a non-Abelian SU(2) holonomy of the SO(5) spinor states. Physically, this non-trivial holonomy arises from the non-zero overlap between the SDW and BCS quasi-particle states. While the usual Berry's phase of a SO(3) spinor is described by a Dirac magnetic monopole at the degeneracy point, the non-Abelian holonomy of a SO(5) spinor is described by a Yang monopole at the degeneracy point, and is deeply related to the existence of the second Hopf map from $S^7$ to $S^4$. We conclude this work by extending the bosonic SO(5) nonlinear sigma model to include the fermionic states around the gap nodes as 4 component Dirac fermions coupled to SU(2) gauge fields in 2+1 dimensions.