Eur. Phys. J. B 26, 167-183 (2002)
https://doi.org/10.1140/epjb/e20020078

Degeneracy of the ground-state of antiferromagnetic spin-1/2 Hamiltonians

¹ Service de Physique Théorique (URA 2306 of CNRS) , Orme des Merisiers CEA-Saclay, 91191 Gif-sur-Yvette Cedex France
² Laboratoire de Physique Théorique des Liquides (UMR 7600 of CNRS) Université P. et M. Curie, case 121, 4 Place Jussieu, 75252 Paris Cedex France

Received: 19 December 2001
Published online: 15 March 2002

Abstract

In the first part of this paper, the extension of the Lieb-Schultz-Mattis theorem to dimensions larger than one is discussed. A counter example to the original formulation of Lieb-Schultz-Mattis and Affleck is exhibited and a more precise statement is formulated. The degeneracy of the ground-state in symmetry breaking phases with long-range order is analyzed. The second and third parts of the paper concern resonating valence-bond (RVB) spin liquids. In these phases the relationship between various authors approaches: Laughlin-Oshikawa, Sutherland, Rokhsar and Kivelson, Read and Chakraborty and the Lieb-Schultz-Mattis-Affleck proposal is studied. The deep physical relation between the degeneracy property and the absence of stiffness is explained and illustrated numerically. A new conjecture is formed concerning the absolute absence of sensitivity of the spin liquid ground-states to any twist of the boundary conditions (thermodynamic limit). In the third part of the paper the relations between the quantum numbers of the degenerate multiplets of the spin liquid phases are obtained exactly. Their relationship with a topological property of the wave functions of the low lying levels of this spin liquid phase is emphasized. In spite of the degeneracy of the ground-state, we explain why these phases cannot exhibit spontaneous symmetry breaking.