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

Recently developed parity ($\mathcal{P}$) and time-reversal ($\mathcal{T}$) symmetric non-Hermitian quantum theory is envisioned to have far-reaching implications in basic science and applications. It is known that the $PT$-inner product is defined with respect to a non-canonical, system-generated dynamical symmetry, namely the $C$ symmetry. Here, we show that the $PT$ invariant equation 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 $PT$ symmetric theory is not generally applicable for spin-1/2 fermions, since here $PT$ inner product vanishes due to Kramer's degeneracy. We consider a spin-1/2 non-Hermitian setup which acquires the combined $(PT)^2=+1$ symmetry, despite $T^2=-1$ and $P^2=+1$. The Hamiltonian inherits a non-Abelian adiabatic transporter and the topological degeneracy via the combined evolution of the $\psi(t)$ state and the $C(t)$ operator. The putative dynamical $C$ symmetry can be a novel springboard for many other exotic quantum and topological phases.