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arXiv · 1301.0811

Random loop representations for quantum spin systems

Daniel Ueltschi
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Abstract

We describe random loop models and their relations to a family of quantum spin systems on finite graphs. The family includes spin 1/2 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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Daniel Ueltschi. 2013-01-04. Random loop representations for quantum spin systems. https://doi.org/10.1063/1.4817865

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