Article

Communications in Mathematical Physics

, Volume 276, Issue 2, pp 437-472

First online:

A Multi-Dimensional Lieb-Schultz-Mattis Theorem

  • Bruno NachtergaeleAffiliated withDepartment of Mathematics, University of California at Davis Email author 
  • , Robert SimsAffiliated withDepartment of Mathematics, University of California at Davis

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Abstract

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].