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Journal of the Physical Society of Japan

Ground-State Long-Range Order in the Two-DimensionalXXZMode

Hidetoshi Nishimori, Yukiyasu Ozeki

Published Mar 15, 1989• 42 citations
Publisher

Abstract

We prove the existence of Néel-type long-range order in the ground state of the spin-1/2 XXZ model with Δ (exchange anisotropy)>1.67 on the square lattice. We further show the existence of long-range order for Δ >1.10 and 0≦ Δ <0.59 by assuming monotonicity of nearest neighbor correlations as functions of the system size. The assumption of monotonicity is supported by numerical calculations.

DOI
10.1143/jpsj.58.1027

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We prove the existence of Néel-type long-range order in the ground state of the spin-1/2 XXZ model with Δ (exchange anisotropy)>1.67 on the square lattice. We further show the existence of long-range order for Δ >1.10 and 0≦ Δ <0.59 by assuming monotonicity of nearest neighbor correlations as functions of the system size. The assumption of monotonicity is supported by numerical calculations.
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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.
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.
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 →∞.