Non-Abelian Berry Gauge Field and Topological Invariant in Parity-Time Symmetric Non-Hermitian Spin-1/2 Quantum Systems

Recently developed parity (P) and time-reversal (T) symmetric non-Hermitian quantum theory is envisioned to have far-reaching implications and applications. It is known that the PT-inner product is defined with respect to a non-canonical, system generated symmetry, namely the C symmetry. We show that the PT symmetric equation of motion is defined by the simultaneous time evolution of the state $\psi(t)$ and the operator C(t) to manifests unitarity - a situation analogous to the Dirac/interaction picture. The time-dependent C operator lends itself into a new term in the Berry phase, setting a platform for novel and exotic topological phases. We also point out that the gauge invariance is achieved by a more generic CPT gauge transformation, not by the usual unitary gauge transformation. The PT symmetric theory is not generally applicable for spin-1/2 fermions, since here PT inner product becomes undefined due to Kramer's theory. We propose a realizable non-Hermitian setup for spin-1/2 fermions which acquires the combined PT^2=+1 symmetry, despite T^2=-1 and P^2=+1. The Hamiltonian inherits non-Abelian Berry gauge fields for non-interacting fermions without magnetic field. The corresponding edge states are found to have unique supersymmetric oscillator solutions but with complex energy levels.