Uhlmann phase of a thermal spin-1 system with zero field splitting

We study the Uhlmann geometric phase of a spin-1 particle subjected to zero-field splitting (ZFS) interaction, modulated by a dimensionless parameter $\alpha$, under the effect of an external magnetic field with a tilting angle $\theta$. We show that the ZFS term induces a transition in the geometrical phase behavior, characterized by a critical parameter value, $\alpha ={\alpha }_c$. For $\alpha <{\alpha }_c$, this phase displays two critical temperatures at $\theta =\pi /2$, similar to spin-1 systems without ZFS, but with a separation that varies with $\alpha$. In contrast, for $\alpha >{\alpha }_c$, the phase exhibits two singularities at a single critical temperature but at different field orientations $\theta \ne \pi /2$. The phase disappears for significantly large $\left|\alpha \right|$, regardless of the values of the Hamiltonian parameters. This behavior clearly departs from the usual thermal Uhlmann phase observed in SU(2) systems. In addition, we analytically calculate the heat capacity, which, for $\theta =\pi /2$ and nearby values, displays two different regimes according to the sign of $\alpha$. For $\alpha <0$, it develops two peaks associated to the multilevel nature of the system, while for $\alpha >0$ only a single Schottky-anomaly like peak appears as in two level systems. Interestingly, when $\theta =\pi /2$, the temperature centroids of the Uhlmann phase and the heat capacity coincide in the region between critical temperatures for a given value of $\alpha <{\alpha }_c$. Furthermore, we demonstrate that when $\alpha =0$, the Uhlmann phase, a global topological property of the system, can be expressed as a function of the thermal component of the Bures metric, a local geometric property related to the heat capacity.