We study the finite-size properties of the infinitely coordinated XY model in a transverse field, also known as the Lipkin-Meshkov-Glick model. In the thermodynamical limit, this model exhibits a Quantum Phase Transition when varying the transverse field, whose behaviour is mean-field. Because of its simplicity, this model was introduced already fourty years ago, but has only recently been proved to be solvable by algebraic Bethe ansatz (Links et al, 2003). From numerical calculations, its finite-size scaling exponents for the spectrum and correlation functions have been conjectured to be simple fractions (Botet et al, 1982), but had never been given any analytical support. By computing the 1/N expansion for the spectrum and correlation functions thanks to the Continuous Unitary Transformation technique, and by using a scaling argument, we have been able to determine these exponents (S. Dusuel and J. Vidal, cond-mat/0408624, to be published in PRL, December 2004). Thanks to these results, we also have also computed a two-spin entanglement property known as the concurrence, which becomes maximum at the phase transition (as has been conjectured for generic Quantum Phase Transition by Osborne and Nielsen, 2000 and Osterloh et al, 2000).

Université de Cologne