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Progress of Theoretical Physics

Existence of Long-Range Magnetic Order in the Ground State of Two-Dimensional Spin-1/2 Heisenberg Antiferromagnets

Martin Indergand, Manfred Sigrist

Published Jan 1, 2007• 2 citations

Abstract

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.

DOI: 10.1143/ptp.117.1

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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.

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 →∞.

The method of Kennedy, Lieb, and Shastry is applied to the antiferromagnetic

\mathrm{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=\frac{1}{2}

. The existence of LRO in the ground state of the

S=\frac{1}{2} \mathrm{XY}

model ($\ensuremath{\Delta}=0$), therefore, has been confirmed, in agreement with the conjecture by Nishimori et al. The ground state of the

S=\frac{1}{2}

antiferromagnetic Heisenberg model in two dimensions still remains to be clarified.

The existence of long-range order is proved under certain conditions for the antiferromagnetic XYZ model on the simple cubic or the square lattice. In particular, the spin-1/2 XXZ model on the square lattice is shown to have ground-state long-range order if the exchange anisotropy Δ satisfies 0≦ Δ <0.20 or Δ >1.72, which improves the result of Kubo and Kishi. The existence of long-range order of the z -component of the spin operator is proved for the XXZ model with XY -like anisotropy (0≦ Δ ≦1) under certain conditions. A similar result is shown to hold for the long-range order in the x -direction for the Ising-like model ( Δ ≧1). The XXZ model on the two-dimensional hexagonal lattice is proved to have finite ground-state long-range order for any value of Δ (≧0) if S ≧1 and for Δ >2.55 when S =1/2.

We study the ground-state and finite-temperature properties of the one-dimensional isotropic spin-1/2 Heisenberg magnet with ferromagnetic nearest-neighbor and antiferromagnetic next-nearest-neighbor interactions which compete with each other. Extrapolating the exact results for finite-size systems of up to 20 spins, we estimate the ground-state energy and the ground-state two-spin correlation function in the infinite-size limit. Using the cluster transfer matrix method, we also calculate the temperature dependence of the internal energy and the specific heat and that of the inverse correlation length and the wave number, by both of which the asymptotic behavior of the two-spin correlation function in the long-distance limit is characterized. We compare the results of these calculations with those obtained previously for the case where the nearest-neighbor interaction is antiferromagnetic.

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.