  We study the geometric Uhlmann phase of entangled mixed states in a composite
system made of two coupled spin-$\frac 1 2$ particles with a magnetic field
acting on one of them. Within a depolarizing channel setup, an exact analytical
expression for such a phase in each subsystem is derived. We find an explicit
connection to the concurrence of the depolarizing channel density matrix, which
allows to characterize the features of the Uhlmann phase in terms of the degree
of entanglement in the system. In the space of field direction and coupling
parameter, it exhibits a phase singularity revealing a topological transition
between orders with different winding numbers. The transition occurs for fields
lying in the equator of the sphere of directions and at critical values of the
coupling which can be controlled by tuning the depolarization strength.
Notably, under these conditions the concurrence of the composite system is
bounded to the range $[0,1/2]$. We also compare the calculated Uhlmann phase to
an interferometric phase, which has been formulated as an alternative for
density matrices. The latter does not present a phase vortex, although they
coincide in the weak entanglement regime, for vanishing depolarization (pure
states). Otherwise they behave clearly different in the strong entanglement
regime.
