Thermodynamic limit of the XXZ central spin model with an arbitrary central magnetic field

The U(1) symmetry of the XXZ central spin model with an arbitrary central magnetic field B is broken, since its total spin in the z-direction is not conserved. We obtain the exact solutions of the system by using the off-diagonal Bethe ansatz method. The thermodynamic limit is investigated based on the solutions. We find that the contribution of the inhomogeneous term in the associated T-Q relation to the ground state energy satisfies an N -1 scaling law, where N is the total number of spins. This result makes it possible to investigate the properties of the system in the thermodynamic limit. By assuming the structural form of the Bethe roots in the thermodynamic limit, we obtain the contribution of the direction of B to the ground state energy. It is shown that the contribution of the direction of the central magnetic field is a finite value in the thermodynamic limit. This is the phenomenon caused by the U(1) symmetry breaking of the system.