An attempt is made to investigate the relation between (diagonal) long-range order (LRO or spontaneous magnetization in spin systems) and off-diagonal long-range order (ODLRO or long-range correlation in spin systems). It is shown that ODLRO cannot exceed LRO in ideal ferromagnets under some plausible conditions. Several kinds of long-range order and their mutual relations are also discussed in ideal ferromagnets whose total spin commutes with the Hamiltonian. It is also argued that neither LRO or ordinary ODLRO exists in the two-dimensional ferromagnetic Heisenberg model. However, a possibility is proposed that other kinds of ODLRO exist in the case of higher spin for such a system. The Heisenberg model with long-range interaction is exemplified, and it is pointed out that the total spin, instead of the total magnetization, should be taken as an order parameter in phenomenological arguments of the Heisenberg model.An attempt is made to investigate the relation between (diagonal) long-range order (LRO or spontaneous magnetization in spin systems) and off-diagonal long-range order (ODLRO or long-range correlation in spin systems). It is shown that ODLRO cannot exceed LRO in ideal ferromagnets under some plausible conditions. Several kinds of long-range order and their mutual relations are also discussed in ideal ferromagnets whose total spin commutes with the Hamiltonian. It is also argued that neither LRO or ordinary ODLRO exists in the two-dimensional ferromagnetic Heisenberg model. However, a possibility is proposed that other kinds of ODLRO exist in the case of higher spin for such a system. The Heisenberg model with long-range interaction is exemplified, and it is pointed out that the total spin, instead of the total magnetization, should be taken as an order parameter in phenomenological arguments of the Heisenberg model.

Progress of Theoretical PhysicsNov 1, 196912 citations

Bifurcation of ground states and ferrimagnetic long-range order in anisotropic mixed-spin systems

Guang-Shan Tian, Hai-Qing Lin

In this paper, we study a generalized anisotropic mixed-spin ferrimagnetic Heisenberg model on an arbitrary bipartite lattice. We prove rigorously that, under some very general conditions, this model has a unique ground state with ${S}_{z}=0$ in the $\mathrm{XY}$ regime and two degenerate ground states with nonzero magnetization in the Ising regime, respectively. Therefore, the isotropic point of this model is a bifurcation point for its ground states. Furthermore, we also show that, if magnetization of the ground states in the Ising regime is a macroscopic quantity in the thermodynamic limit, then these states have both ferromagnetic and antiferromagnetic long-range order. In other words, they are ferrimagnetically ordered. These conclusions confirm the previous results derived by numerical calculations on small one-dimensional mixed-spin chains.In this paper, we study a generalized anisotropic mixed-spin ferrimagnetic Heisenberg model on an arbitrary bipartite lattice. We prove rigorously that, under some very general conditions, this model has a unique ground state with ${S}_{z}=0$ in the $\mathrm{XY}$ regime and two degenerate ground states with nonzero magnetization in the Ising regime, respectively. Therefore, the isotropic point of this model is a bifurcation point for its ground states. Furthermore, we also show that, if magnetization of the ground states in the Ising regime is a macroscopic quantity in the thermodynamic limit, then these states have both ferromagnetic and antiferromagnetic long-range order. In other words, they are ferrimagnetically ordered. These conclusions confirm the previous results derived by numerical calculations on small one-dimensional mixed-spin chains.

Physical review. B, Condensed matterDec 16, 20027 citations

{S}_{z}=0

in the

\mathrm{XY}

regime and two degenerate ground states with nonzero magnetization in the Ising regime, respectively. Therefore, the isotropic point of this model is a bifurcation point for its ground states. Furthermore, we also show that, if magnetization of the ground states in the Ising regime is a macroscopic quantity in the thermodynamic limit, then these states have both ferromagnetic and antiferromagnetic long-range order. In other words, they are ferrimagnetically ordered. These conclusions confirm the previous results derived by numerical calculations on small one-dimensional mixed-spin chains.