  Using the spin-wave approximation, we study the geometric phase (GP) of a
central spin (signal qubit) coupled to an antiferromagnetic (AF) environment
under the application of an external global magnetic field. The external
magnetic field affects the GP of the qubit directly and also indirectly through
its effect on the AF environment. We find that when the applied magnetic field
is increased to the critical magnetic field point, the AF environment undergoes
a spin-flop transition, a first-order phase transition, and at the same time
the GP of the qubit changes abruptly to zero. This sensitive change of the GP
of a signal qubit to the parameter change of a many-body environment near its
critical point may serve as another efficient tool or witness to study the
many-body phase transition. The influences of the AF environment temperature
and crystal anisotropy field on the GP are also investigated.
