Geometric phase of a central spin coupled to an antiferromagnetic environment

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.