Quantum Heisenberg models and random loop representations

TL;DR

This paper reviews random loop representations for quantum Heisenberg models, exploring their relation to spin order, and discusses conjectures on macroscopic loop distributions and symmetry breaking.

Contribution

It extends known loop representations to interpolating models like the quantum XY model and analyzes their implications for long-range order and symmetry breaking.

Findings

Loop representations relate to spin long-range order.

Extensions to XY model are discussed.

Conjectures on macroscopic loops and symmetry breaking are presented.

Abstract

We review random loop representations for the spin-1/2 quantum Heisenberg models, that are due to Toth (ferromagnet) and Aizenman-Nachtergaele (antiferromagnet). These representations can be extended to models that interpolate between the two Heisenberg models, such as the quantum XY model. We discuss the relations between long-range order of the quantum spins and the size of the loops. Finally, we describe conjectures about the joint distribution of the lengths of macroscopic loops, and of symmetry breaking.