For spin coherent and one-axis-squeezed states, the Uhlmann phase and interferometric geometric phase show finite-temperature jumps; for a two-axis-squeezed state, the paper claims smooth variation.
Of course, such a path also respects the condition (19)
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Mixed-state geometric phases of coherent and squeezed spin states
For spin coherent and one-axis-squeezed states, the Uhlmann phase and interferometric geometric phase show finite-temperature jumps; for a two-axis-squeezed state, the paper claims smooth variation.