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/content/aip/journal/jmp/54/8/10.1063/1.4817865
2013-08-20
2016-09-28

Abstract

We describe random loop models and their relations to a family of quantum spin systems on finite graphs. The family includes spin Heisenberg models with possibly anisotropic spin interactions and certain spin 1 models with SU(2)-invariance. Quantum spin correlations are given by loop correlations. Decay of correlations is proved in 2D-like graphs, and occurrence of macroscopic loops is proved in the cubic lattice in dimensions 3 and higher. As a consequence, a magnetic long-range order is rigorously established for the spin 1 model, thus confirming the presence of a nematic phase.

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