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Physical Review B

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

Guang-Shan Tian, Hai-Qing Lin

Published Sep 21, 2004• 13 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.70.104412

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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.
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.
The ground-state properties of the spin-(1/2, 1) mixed-spin XXZ chain with single-ion anisotropy (D) are investigated by the infinite time-evolving block decimation (iTEBD) method. A ground-state phase diagram including three phases, i.e., a fully polarized phase, an XY phase and a ferrimagnetic phase, is obtained. The ferrimagnetic phase is found to extend to the regions with (Δ > 1, D > 0) and (Δ < 1, D < 0), where Δ denotes the coupling anisotropy between the localized spins. By the discontinuous behavior of bipartite entanglement, quantum phase transitions (QPTs) between the XY phase and the other two phases are verified to be of the first-order. Furthermore, two constant spontaneous magnetization values (M z = 3/2 and 1/2) are observed in the fully polarized and the ferrimagnetic phases, respectively. In both cases of Δ → +∞ and D → -∞, the ground state tends to the Ising limit. In addition, both the long-range ferromagnetic and antiferromagnetic orders are found to coexist in the whole ferrimagnetic phase.
In the present paper, we investigate the existence of the antiferromagnetic long-range order in a S=1S=1S=1 spatially anisotropic antiferromagnetic Heisenberg model with a single-ion energy term. By applying the reflection-positivity method developed by Dyson, Lieb, and Simon [J. Stat. Phys. 53, 1019 (1988)], we are able to determine the parameter regions where either the longitudinal or the transverse antiferromagnetic long-range order exist. We find that the single-ion energy anisotropy has a strong influence on the critical value of the spin coupling constant between the one-dimensional integer-spin chains. Qualitatively, our results agree with the previous ones obtained by the mean-field theory and hence, reestablish the conclusions on a mathematically rigorous basis.
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.
The effect of uniaxial single-ion anisotropy on quantum entanglement is rigorously quantified using negativity in a mixed spin-(1,1/2,1) Heisenberg trimer, accounting for different exchange coupling constants between identical and distinct spins. Bipartite negativities between the single-spin entity and the remaining spin dimer are analyzed alongside the global tripartite negativity (gTN) of the whole trimer under the effect of an external magnetic field and both easy-axis and easy-plane types of single-ion anisotropy. Interestingly, the single-ion anisotropy significantly influences the degree of entanglement by altering the stability regions of energetically preferred ground states and it may also introduce additional phases in the overall ground-state phase diagram. Moreover, even an arbitrarily small single-ion anisotropy is sufficient to generate global tripartite entanglement in a state that remains biseparable in the isotropic limit. Finally, the thermal stability of entanglement is discussed in detail, including the emergence of a peculiar local minimum at finite temperatures. The obtained theoretical results may offer deeper insights into bipartite and tripartite entanglement in trimetallic Ni2+-Cu2+-Ni2+molecular compounds.
We use the mean-field approximation of Dyson–Maleev representation to study an XXZ Heisenberg ferrimagnetic spin chain with single-ion anisotropy. By solving the self-consistent equations with different anisotropies, λ and D respectively, the energy spectrums, internal energy, static susceptibility and specific heat are calculated. Especially, the quantum phase transition of the magnetization plateau induced by single-ion anisotropy D is obtained in the model of the ferrimagnetic spin chain by using Dyson–Maleev mean-field theory.