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Progress of Theoretical Physics

Long-Range Order in Ideal Ferromagnets

Masuo Suzuki

Published Nov 1, 1969• 12 citations
Publisher

Abstract

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.

DOI
10.1143/ptp.42.1086

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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.
We give a rigorous proof of the existence of long-range antiferromagnetic order in the ground state of several two-dimensional spin-1/2 Heisenberg systems.Three types of systems are considered.The first type consists of an even number, N , of coupled square lattices with antiferromagnetic nearest-neighbor Heisenberg interactions.In this case, we prove the existence of long-range order (LRO) in the ground state for an interplane to inplane coupling ratio r in the range 0.16 ≤ r ≤ 2.1 for N ≥ 4. Further, we prove that the antiferromagnetic bilayer with ferromagnetic next-nearest-neighbor (nnn) interplane couplings possesses LRO in the ground state for 0.21 ≤ r < 1/4, where r is the absolute value of the ratio of the ferromagnetic and antiferromagnetic couplings.The final example consists of two antiferromagnetic spin-1/2 square lattices that are coupled via an antiferromagnetic nnn interplane coupling r.For r = 1, this system is effectively a spin-1 square lattice and has an ordered ground state.We show that LRO exists in the region 0.85 ≤ r ≤ 1. * ) Note that E r q in the notation of KLS corresponds to 2 + r -E q in our notation.
We discuss new possibilities for off-diagonal long range order (ODLRO) in spin chains involving operators which add or delete sites from the chain. For the one-dimensional, periodic Heisenberg, inverse square exchange, and XY\mathrm{XY}XY models we give numerical evidence for the hidden ODLRO conjectured by Anderson. A connection to the singlet pair correlations in one-dimensional models of interacting electrons is made and briefly discussed.
The existence of long range order (LRO) is examined in the antiferromagnetic Heisenberg model on the square or the simple cubic lattice with the next nearest neighbor interactions which are taken to be λ times the nearest neighbor ones (λ≧0). It is rigorously shown that Néel LRO exists at a low temperature on the simple cubic lattice and in the ground state on the square lattice for λ smaller than a critical value λ c . On the square lattice λ c increases with S starting from 0.072 for S =1 and approaches 0.5 in the limit S →∞. On the simple cubic lattice λ c =0.002 for S =1/2 and approaches 0.25 in the limit S →∞.
For a strongly interacting boson system, it has been proposed that the concept of off-diagonal long-range ordering (ODLRO) is equivalent to Bose-Einstein condensation for a free boson system. Therefore, the existence of ODLRO in the ground state implies superfluidity. The author checks the validity of this proposition from another point of view. He rigorously shows that, for a hard-core lattice-boson system with a short-ranged interaction ODLRO is suppressed when a charged excitation gap develops and, hence, the boson system becomes a Mott-insulator. Finally, by using his theorem, he shows some interesting properties of the antiferromagnetic Heisenberg model, which can be taken as a hard-core lattice-boson system.
The method of Kennedy, Lieb, and Shastry is applied to the antiferromagnetic XXZ\mathrm{XXZ}XXZ model to examine the existence of long-range order (LRO). On the 3D simple-cubic lattice LRO is proved to exist for any $S\ensuremath{\geqq}\frac{1}{2}$ and $\ensuremath{\Delta}\ensuremath{\geqq}0$ at low temperature. On the 2D square lattice LRO exists in the ground state for any $\ensuremath{\Delta}\ensuremath{\leqq}0$ if $S\ensuremath{\geqq}1$ and for $0\ensuremath{\leqq}\ensuremath{\Delta}<0.13$ and $\ensuremath{\Delta}=0$ if S=12S=\frac{1}{2}S=21​. The existence of LRO in the ground state of the S=12XYS=\frac{1}{2} \mathrm{XY}S=21​XY model ($\ensuremath{\Delta}=0$), therefore, has been confirmed, in agreement with the conjecture by Nishimori et al. The ground state of the S=12S=\frac{1}{2}S=21​ antiferromagnetic Heisenberg model in two dimensions still remains to be clarified.
The Mermin-Wagner theorem is strengthened so as to rule out magnetic long-range order at T>0T>0T>0 in one- or two-dimensional Heisenberg and XY systems with long-range interactions decreasing as ${R}^{\ensuremath{-}\ensuremath{\alpha}}$ with a sufficiently large exponent $\ensuremath{\alpha}$. For oscillatory interactions, ferromagnetic long-range order at T>0T>0T>0 is ruled out if $\ensuremath{\alpha}\ensuremath{\ge}1(D\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1)$ or $\ensuremath{\alpha}>5/2(D\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2)$. For systems with monotonically decreasing interactions, ferro- or antiferromagnetic long-range order at T>0T>0T>0 is ruled out if $\ensuremath{\alpha}\ensuremath{\ge}2D$.