Random loop representations for quantum spin systems

TL;DR

This paper introduces random loop models to analyze quantum spin systems, establishing correlation decay in 2D and long-range order in higher dimensions, including a nematic phase in spin 1 models.

Contribution

It develops a novel connection between random loop representations and quantum spin correlations, proving decay of correlations and long-range order in various lattice dimensions.

Findings

Decay of correlations in 2D-like graphs

Occurrence of macroscopic loops in 3D and higher

Rigorous proof of magnetic long-range order in spin 1 models

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.