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We present a generalization of the geometric phase to pure and thermal states in $$\mathcal{PT}$$-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. \textcolor{red}{At certain critical points, the generalized IGP may exhibit discrete jumps at finite temperatures, signaling a geometric phase transition. We illustrate the IGP of PTQM through two examples and compare their differences}. 

The structures of multiply quantized vortices (MQVs) of an equal-population atomic Fermi superfluid in a rotating spherical bubble trap approximated as a thin shell are analyzed by solving the Bogoliubov-de Gennes (BdG) equation throughout the BCS-Bose Einstein condensation (BEC) crossover. Consistent with the Poincare-Hopf theorem, a pair of vortices emerge at the poles of the rotation axis in the presence of azimuthal symmetry, and the compact geometry provides confinement for the MQVs. While the single-vorticity vortex structure is similar to that in a planar geometry, higher-vorticity vortices exhibit interesting phenomena at the vortex center, such as a density peak due to accumulation of a normal Fermi gas and reversed circulation of current due to in-gap states carrying angular momentum, in the BCS regime but not the BEC regime because of the subtle relations between the order parameter and density. The energy spectrum shows the number of the in-gap state branches corresponds to the vorticity of a vortex, and an explanation based on a topological correspondence is provided. 

We first compare the geometric frameworks behind the Uhlmann andBerry phases in a fiber-bundle language and then evaluate the Uhlmannphases of bosonic and fermionic coherent states. The Uhlmann phases ofboth coherent states are shown to carry geometric information anddecrease smoothly with temperature. Importantly, the Uhlmann phasesapproach the corresponding Berry phases as temperature decreases.Together with previous examples in the literature, we propose acorrespondence between the Uhlmann and Berry phases in thezero-temperature limit as a general property except some special casesand present a conditional proof of the correspondence.