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Ground-state degeneracies in quantum antiferromagnets - Lieb-Schultz-Mattis theorem in dimension 2.

With my former advisor C. Lhuillier ( LPTL, Univ. Paris 6) and her collaborators, we have been working on the issue of ground-state degeneracies in quantum disordered phase of two-dimensional antiferromagnets (spin liquids).

The Lieb-Schultz-Mattis (LSM) theorem (1961) states that the ground-state of a quantum spin chain (or ladder) with a half-integer spin in the unit cell is ether degenerate or supports gapless excitations. Since the argument is readily extended to spin ladders of arbitrary finite width, it is a natural to consider the validity of this theorem in higher dimension. We analysed its validity in dimension 2 in various cases. Systems which spontaneously breaks a continuous (spin rotation) or discrete (lattice) symmetry will have degenerate ground-states but we found examples where the corresponding momenta are not those imposed by the LSM theorem. On the other hand, spin models which do not break any symmetry (spin liquids) are also observed to have degenerate ground-states. This phenomenon is related to the use of boundary conditions and has a topological origin. This is best understood in the language of short-range resonating valence-bond (RVB) states (Rokhsar and Kivelson, Phys. Rev. Lett. (1988).) where the degeneracy is a consequence of the disconnected topological sectors in the space of dimer coverings of the lattice. We have illustrated this aspect by several numerical calculations of exact spectra and have determined the quantum numbers associated to the ground-state multiplet as a function of the lattice geometry. Here again we found example where the momenta differ from the one-dimensional situation.

Reference: G. Misguich,C. Lhuillier, M. Mambrini, P. Sindzingre. Eur. Phys. J. B.26, 167 (2002) .