Quantum Heisenberg models and random loop representations
Abstract
We review random loop representations for the spin1/2 quantum Heisenberg models, that are due to Toth (ferromagnet) and AizenmanNachtergaele (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 longrange 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.
 Publication:

arXiv eprints
 Pub Date:
 November 2012
 arXiv:
 arXiv:1211.4141
 Bibcode:
 2012arXiv1211.4141U
 Keywords:

 Mathematical Physics;
 Mathematics  Probability
 EPrint:
 Contribution to the proceedings of ICMP12, Aalborg, Denmark, August 611, 2012