In this work we investigate properties of fermions in the SO(5) theory of high T c 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 quasiparticle 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 _ 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. 1999 Academic Press 1. INTRODUCTION Recently, a unified theory based on SO(5) symmetry between antiferromagnetism (AF) and d wave superconductivity (dSC) has been proposed[1] for the high T c cuprates. Initially, this theory was formulated in terms of a nonlinear _ model which describes the effective bosonic degrees of freedom below the pseudogap temperature. This theory gives a unified description of the high T c phase diagram and offers a natural explanation of the ? resonance mode [2, 3] observed in the YBCO superconductors. With the exception of exact microscopic SO(5) models [4 7], both numerical investigations [8, 9] and the experimental proposals [10 12] have primarily focused on the bosonic sector of the SO(5) theory.
However, it is clear that a complete theory of high T c superconductivity has to properly account for the fermionic degrees of freedom as well. Some key experiments on the pseudogap physics, e.g., the ARPES experiments, primarily probe the single electron properties rather than the collective modes. Within the SO(5) theory, the pseudogap regime is identified with the fluctuations of the orientation of the SO(5) superspin vector. Therefore, it is essential to understand how the fluctuations of the superspin couple to single particle fermionic degrees of freedom. Because of the d wave nodes, there are fermionic excitations with low energy, and they make important contributions to thermodynamics and to the damping of the collective modes. It is interesting to note that similarities in the dispersion and the character of the fermionic quasiparticles at half-filling and in