Effective spin models for spin-phonon chains by flow equations

We investigate the anti-adiabatic limit of an anti-ferromagnetic S = 1/2 He isenberg chain coupled to Einstein phonons. The flow equation method is use d to decouple the spin and the phonon part of the Hamiltonian. In the effec tive spin model long range spin-spin interactions are generated. We determi ne the phase transition from a gapless state to a gapped (dimerized) phase, which occurs at a non-zero value of the spin-phonon coupling. In the effec tive phonon sector a phonon hardening is observed.