Communications in Mathematical Physics

, Volume 276, Issue 2, pp 437–472

A Multi-Dimensional Lieb-Schultz-Mattis Theorem


DOI: 10.1007/s00220-007-0342-z

Cite this article as:
Nachtergaele, B. & Sims, R. Commun. Math. Phys. (2007) 276: 437. doi:10.1007/s00220-007-0342-z


For a large class of finite-range quantum spin models with half-integer spins, we prove that uniqueness of the ground state implies the existence of a low-lying excited state. For systems of linear size L, with arbitrary finite dimension, we obtain an upper bound on the excitation energy (i.e., the gap above the ground state) of the form (C log L)/L. This result can be regarded as a multi-dimensional Lieb-Schultz-Mattis theorem [14] and provides a rigorous proof of the main result in [8].

Copyright information

© B. Nachtergaele and R. Sims 2007

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California at DavisDavisUSA