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Journal of Physics A Mathematical and General

Coexistence of the ferromagnetic and antiferromagnetic long-range orders in the generalized antiferromagnetic Heisenberg model on a bipartite lattice

Guang-Shan Tian

Published Apr 7, 1994• 22 citations
Publisher

Abstract

The concept of ferrimagnetism was first proposed by Neel to explain why some materials have a macroscopic magnetization but no ferromagnetic long-range order, when the temperature T is lower than a phase transition temperature T c . In this article, based on a theorem of Lieb and Mattis, we show in a mathematically rigorous way that the global ground states of the generalized antiferromagnetic Heisenberg model on a bipartite lattice with unequal sublattice points have both ferromagnetic and antiferromagnetic long-range orders with the latter being predominant. Our rigorous results conform to Neel's theory.

DOI
10.1088/0305-4470/27/7/012

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The concept of ferrimagnetism was first proposed by Neel to explain why some materials have a macroscopic magnetization but no ferromagnetic long-range order, when the temperature T is lower than a phase transition temperature T c . In this article, based on a theorem of Lieb and Mattis, we show in a mathematically rigorous way that the global ground states of the generalized antiferromagnetic Heisenberg model on a bipartite lattice with unequal sublattice points have both ferromagnetic and antiferromagnetic long-range orders with the latter being predominant. Our rigorous results conform to Neel's theory.
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 Sz=0{S}_{z}=0Sz​=0 in the XY\mathrm{XY}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 show rigorously that there exists the ferrimagnetic long-range order in the ground state of the positive-U Hubbard model at half filling on some bipartite lattices. When NA{\mathit{N}}_{\mathit{A}}NA​>NB{\mathit{N}}_{\mathit{B}}NB​(NA{\mathit{N}}_{\mathit{A}}NA​ and NB{\mathit{N}}_{\mathit{B}}NB​ are the total site numbers of two sublattices A and B), except for the ferromagnetism which was found by Lieb [Phys. Rev. Lett. 62, 1201 (1989)], there also exists the antiferromagnetic long-range order in the ground state. This result only requires U>0 and is independent of the dimension of the lattices.
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 DDD is positive, the model has a unique ground state with the total spin-zzz component Sz=0{S}_{z}=0Sz​=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=0D=0D=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.
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