Geometry of time-dependent $PT$ -symmetric quantum mechanics

A new type of quantum theory known as time-dependent $PT$-symmetric quantum mechanics has received much attention recently. It has a conceptually intriguing feature of equipping the Hilbert space of a $PT$-symmetric system with a time-varying inner product. In this work, we explore the geometry of time-dependent $PT$-symmetric quantum mechanics. We find that a geometric phase can emerge naturally from the cyclic evolution of a $PT$-symmetric system, and further formulate a series of related differential-geometry concepts, including connection, curvature, parallel transport, metric tensor, and quantum geometric tensor. These findings constitute a useful, perhaps indispensible, tool to investigate geometric properties of $PT$-symmetric systems with time-varying system's parameters. To exemplify the application of our findings, we show that the unconventional geometric phase [Phys. Rev. Lett. 91 187902 (2003)], which is the sum of a geometric phase and a dynamical phase proportional to the geometric phase, can be expressed as a single geometric phase unveiled in this work. (special topic)