Bruno Nachtergaele
Published 2006-03-06, updated 2006-03-08Version 2
In their 1978 paper, Dyson, Lieb, and Simon (DLS) proved the existence of Ne'el order at positive temperature for the spin-S Heisenberg antiferromagnet on the d-dimensional hypercubic lattice when either S >= 1 and d >= 3 or S=1/2 and d is sufficiently large. This was the first proof of spontaneous breaking of a continuous symmetry in a quantum model at finite temperature. Since then the ideas of DLS have been extended and adapted to a variety of other problems. In this paper I will present an overview of the most important developments in the study of the Heisenberg model and related quantum lattice systems since 1978, including but not restricted to those directly related to the paper by DLS.
Comments: Dedicated to Barry Simon and to appear in a festschrift on the occasion of his 60th birthday. v2: corrected typos and references
Journal: in "Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon's 60th Birthday", Fritz Gesztesy et al. (Eds), Proceedings of Symposia in Pure Mathematics, Vol 76, part 1, pp 47--68, AMS, 2007.
Quantum Spin Systems
Reflection positivity and infrared bounds for quantum spin systems
Recent Progress in Quantum Spin Systems
