Imaginary vector potential in non-Hermitian spin-orbit coupled systems

Quantum dynamics in Hermitian systems are described by the unitary time evolution where the total probability is conserved. When non-Hermitian terms are introduced to the Hamiltonian, it becomes a question whether or not the probability conservation still holds. In this talk, I will discuss conserved quantities in a non-Hermitian system with spin-orbit coupling. We show that the dynamics are governed by imaginary vector potentials and conserved quantities arise. These conserved quantities can be understood from a geometric perspective.