Interferometric geometric phases of $\mathrm{PT}$-symmetric quantum mechanics

We present a generalization of the geometric phase to pure and thermal states in $\mathrm{PT}$-symmetric quantum mechanics (PTQM) based on the approach of the interferometric geometric phase (IGP). The formalism first introduces the parallel-transport conditions of quantum states and reveals two geometric phases, ${\theta }^1$ and ${\theta }^2$, for pure states in PTQM according to the states under parallel-transport. Due to the non-Hermitian Hamiltonian in PTQM, ${\theta }^1$ is complex and ${\theta }^2$ is its real part. The imaginary part of ${\theta }^1$ plays an important role when we generalize the IGP to thermal states in PTQM. The generalized IGP modifies the thermal distribution of a thermal state, thereby introducing effective temperatures. At certain critical points, the generalized IGP may exhibit discrete jumps at finite temperatures, signaling a geometric phase transition. We illustrate this phenomenon in PTQM through two examples and compare their differences.