Mixed-state geometric phases of coherent and squeezed spin states

Two mixed-state geometric phases, known as the Uhlmann phase and interferometric geometric phase (IGP), of spin coherent states (CSSs) and spin squeezed states (SSSs) are analyzed. Exact solutions and numerical results of selected examples are presented. For the $j = 3/2$ CSS, the Uhlmann phase exhibits finite-temperature topological phase transitions characterized by abrupt jumps. The IGP for the same state similarly shows discontinuous jumps as the temperature varies. In the case of the $j = 1$ one-axis SSS, both Uhlmann phase and IGP display discrete finite-temperature jumps. By contrast, the $j = 1$ two-axis SSS shows no such transitions because the Uhlmann phase and IGP both vary smoothly with temperature. We also briefly discuss potential realizations and simulations related to these phenomena in spin systems.