Adiabatic-antiadiabatic crossover in a spin-Peierls chain

Abstract

We consider an $XXZ$ spin-$1∕2$ chain coupled to optical phonons with nonzero frequency ${\ensuremath{\omega}}_{0}$. In the adiabatic limit (small ${\ensuremath{\omega}}_{0}$), the chain is expected to spontaneously dimerize and open a spin gap, while the phonons become static. In the antiadiabatic limit (large ${\ensuremath{\omega}}_{0}$), 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 low-frequency 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 ${\ensuremath{\omega}}_{0}⪡{\ensuremath{\Delta}}_{s}$, where ${\ensuremath{\Delta}}_{s}$ is the gap in the static limit ${\ensuremath{\omega}}_{0}\ensuremath{\rightarrow}0$, the system is in the adiabatic regime, and the gap remains of order ${\ensuremath{\Delta}}_{s}$. For ${\ensuremath{\omega}}_{0}>{\ensuremath{\Delta}}_{s}$, the system enters the antiadiabatic regime, and the gap decreases rapidly as ${\ensuremath{\omega}}_{0}$ increases. Finally, for ${\ensuremath{\omega}}_{0}>{\ensuremath{\omega}}_{\mathit{BKT}}$, where ${\ensuremath{\omega}}_{\mathit{BKT}}$ 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.