Adiabatic-antiadiabatic crossover in a spin-Peierls chain

We consider an XXZ spin-1/2 chain coupled to optical phonons with non-zero frequency $\omega_0$. In the adiabatic limit (small $\omega_0$), the chain is expected to spontaneously dimerize and open a spin gap, while the phonons become static. In the antiadiabatic limit (large $\omega_0$), phonons are expected to give rise to frustration, so that dimerization and formation of spin-gap are obtained only when the spin-phonon interaction is large enough. We study this crossover using bosonization technique. The effective action is solved both by the Self Consistent Harmonic Approximation (SCHA)and by Renormalization Group (RG) approach starting from a bosonized description. The SCHA allows to analyze the lowfrequency regime and determine the coupling constant associated with the spin-Peierls transition. However, it fails to describe the SU(2) invariant limit. This limit is tackled by the RG. Three regimes are found. For $\omega_0\ll\Delta_s$, where $\Delta_s$ is the gap in the static limit $\omega_0\to 0$, the system is in the adiabatic regime, and the gap remains of order $\Delta_s$. For $\omega_0>\Delta_s$, the system enters the antiadiabatic regime, and the gap decreases rapidly as $\omega_0$ increases. Finally, for $\omega_0>\omega_{BKT}$, where $\omega_{BKT}$ is an increasing function of the spin phonon coupling, the spin gap vanishes via a Berezinskii-Kosterlitz-Thouless transition. Our results are discussed in relation with numerical and experimental studies of spin-Peierls systems.