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Physical Review Letters

Existence of Long-Range Order in the XXZ Model

Kenn Kubo, Tatsuya Kishi

Published Nov 28, 1988• 106 citations
Publisher

Abstract

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.

DOI
10.1103/physrevlett.61.2585

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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≧12S\ensuremath{\geqq}\frac{1}{2}S\ensuremath≧21​ and \ensuremathΔ\ensuremath≧0\ensuremath{\Delta}\ensuremath{\geqq}0\ensuremathΔ\ensuremath≧0 at low temperature. On the 2D square lattice LRO exists in the ground state for any \ensuremathΔ\ensuremath≦0\ensuremath{\Delta}\ensuremath{\leqq}0\ensuremathΔ\ensuremath≦0 if S\ensuremath≧1S\ensuremath{\geqq}1S\ensuremath≧1 and for 0\ensuremath≦\ensuremathΔ<0.130\ensuremath{\leqq}\ensuremath{\Delta}<0.130\ensuremath≦\ensuremathΔ<0.13 and \ensuremathΔ=0\ensuremath{\Delta}=0\ensuremathΔ=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 (\ensuremathΔ=0\ensuremath{\Delta}=0\ensuremathΔ=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 existence of long-range order (LRO) at low temperature in the isotropic XY\mathrm{XY}XY model with spin $S\ensuremath{\geqq}1$ on simple cubic lattice in three or more dimensions is shown. This completes the proof of the existence of LRO at low temperature in the isotropic XY\mathrm{XY}XY model with arbitrary spin size in three or more dimensions, as the proof for S=12S=\frac{1}{2}S=21​ was given earlier by Dyson, Lieb, and Simon. The existence of LRO in the ground state is shown for $S\ensuremath{\geqq}\frac{3}{2}$ in two dimensions.
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 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 prove the existence of long-range order (LRO) in the ground state of the 2D quantum spin-1/2 XXZ-model for all anisotropies 0≤Δ leq0.220 \leq \Delta\ leq 0.220≤Δ leq0.22 and Δ≥1.47. We achieve this extension of previous proofs by developing an improved variant of the Dyson-Lieb-Simon proof of LRO in quantum spin systems and by deriving improved upper and lower bounds for the ground state energy of the XXZ-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 apply the method of infrared bounds with the technique of Kennedy, Lieb and Shastry to the isotropic spin-1 Hamiltonian with bilinear (- J ) and biquadratic (- J ') exchange interactions to examine the existence of antiferro-dipole long-range order (DLRO) and ferro-quadrupole long-range order (QLRO). We prove that DLRO exists at zero temperature in two and three dimensions for 0⩽ J '<-0.188 J and 0⩽ J '<-1.954 J , respectively. In three dimensions we also prove the existence of QLRO in the ground state for 0<2 J ⩽ J '<2.664 J .