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Recently developed parity ($\mathcal{P}$) and time-reversal ($\mathcal{T}$)symmetric non-Hermitian quantum theory is envisioned to have far-reachingimplications in basic science and applications. It is known that the $PT$-innerproduct is defined with respect to a non-canonical, system-generated dynamicalsymmetry, namely the $C$ symmetry. Here, we show that the $PT$ invariantequation of motion is defined by the simultaneous time evolution of the state$\psi(t)$ and the operator $C(t)$ to manifest unitarity. The dynamical $C$operator lends itself to a new term in the Berry phase. The $PT$ symmetrictheory is not generally applicable for spin-1/2 fermions, since here $PT$ innerproduct vanishes due to Kramer's degeneracy. We consider a spin-1/2non-Hermitian setup which acquires the combined $(PT)^2=+1$ symmetry, despite$T^2=-1$ and $P^2=+1$. The Hamiltonian inherits a non-Abelian adiabatictransporter and the topological degeneracy via the combined evolution of the$\psi(t)$ state and the $C(t)$ operator. The putative dynamical $C$ symmetrycan be a novel springboard for many other exotic quantum and topologicalphases.

A new Berry phase term in parity-time symmetric non-Hermitian spin-1/2 quantum systems