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investigate the anti-adiabatic limit of an anti-ferromagnetic S=1/2 chain coupled to Einstein phonons. The flow equation method is used decouple the spin and the phonon part of the Hamiltonian. the effective spin model long range spin-spin interactions are generated. determine the phase transition from a gapless state to a gapped (dimerized), which occurs at a non-zero value of the spin-phonon coupling. the effective phonon sector a phonon hardening is observed.
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Raas et al. (2001) studied this question.