Spin-Peierls transition in the Heisenberg chain with finite-frequency phonons

We study the spin-Peierls transition in a Heisenberg spin chain coupled to optical bond phonons. Quantum Monte Carlo results for systems with up to N = 256 spins show unambiguously that the transition occurs only when the spin-phonon coupling a exceeds a critical value alpha(c). Using sum rules, we show that the phonon spectral function has divergent (for N --> infinity) weight extending to zero frequency for alpha < alpha(c). The phonon correlations decay with distance r as 1/r. This behavior is characteristic for all 0 < alpha < alpha(c) and the q = pi phonon does not soften (to zero frequency) at alpha = alpha(c).