Phase transition and ferrimagnetic long-range order in the mixed-spin Heisenberg model with single-ion anisotropy

In the present paper, we study the quantum phase transition in the mixed-spin Heisenberg model with the single-ion anisotropy on a bipartite lattice. We prove rigorously that, when the single-ion anisotropy energy D is positive, the model has a unique ground state with the total spin-z component S-z=0. On the other hand, when the single-ion anisotropy energy is negative and favors the longitudinal spin direction, the global ground state of the system becomes doubly degenerate. Therefore, D=0 is the bifurcation point for the global ground state of the system. Furthermore, we show also that, in the latter case, the global ground state of the mixed-spin Heisenberg chain has the ferrimagnetic long-range order. Our conclusions confirm and generalize the previous results derived by numerical calculations on small size samples.