A Geometrical Phase Transition

Many theoretical studies on non-Hermitian quantum systems have reported to violated some well-accepted quantum phenomena or theorems. Rather than sweeping these facts under the rug, we decided to decode the enigma by using some simple geometrical tools. To be more specic, using the fact that Shroedinger's equation is formally the same as parallel transport in a fiber bundle, we surmise that the Hilbert spaces of the quantum states should possess some geometrical structures. Further investigation shows that the paradoxes were indeed removed from non-Hermitian quantum systems by allowing such geometrical structures. Further inspection shows that the geometries between similar systems could behave very dierently which is very similar to a phase transition. In this talk, we'll first introduce the idea of the geometry of the quantum space and then move on to the phase transition-like properties of the structure.