Semiclassical Analysis of Spin Systems near Critical Energies.pdfVIP

Semi-classical Analysis of Spin Systems near Critical Energies Pedro Ribeiro1 and Thierry Paul2 1Laboratoire de Physique Th´eorique de la Mati`ere Condens´ee, CNRS UMR 7600, Universit´e Pierre et Marie Curie, 4 Place Jussieu, 75252 Paris Cedex 05, France 2 CNRS, DMA, Ecole Normale Sup´erieure, 45 rue d’Ulm, 75230 Paris Cedex 05, France The spectral properties of su(2) Hamiltonians are studied for energies near the critical classical energy εc for which the corresponding classical dynamics presents hyperbolic points (HP). A general method leading to an algebraic relation for eigenvalues in the vicinity of εc is obtained in the thermodynamic limit, when the semi-classical parameter n−1 = (2s)−1 goes to zero (where s is the total spin of the system). Two applications of this method are given and compared with numerics. Matrix elements of observables, computed between states with energy near ε , are also computed 8 and shown to be in agreement with the numerical results. c 0 0 PACS numbers: 03.65.Sq, 05.30.-d, 21.60.Ev 2 n u Introduction. Bohr-Sommerfeld (B-S) quantization We derive and test numerically spectral analytical ex- J formulæ for nonregular values of the energy parameter pressions for two different situations arising as a LMG- 9 have been set up in [1, 2] in the case of quantum Hamilto- like model (LMG plus a cubic term) wh