Abstract
We consider the general spin-1 SU(2) invariant Heisenberg model with a two-body interaction. A random loop model is introduced and relation to quantum spin systems is proved. Using this relation it is shown that for dimensions 3 and above Néel order occurs for a large range of values of the relative strength of the bilinear (−J 1) and biquadratic (−J 2) interaction terms. The proof uses the method of reflection positivity and infrared bounds. Links between spin correlations and loop correlations are proved.
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Lees, B. Existence of Néel Order in the S=1 Bilinear-Biquadratic Heisenberg Model via Random Loops. Commun. Math. Phys. 347, 83–101 (2016). https://doi.org/10.1007/s00220-016-2656-1
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DOI: https://doi.org/10.1007/s00220-016-2656-1