Adiabatic-antiadiabatic crossover in a spin-Peierls chain

We consider an $XXZ$ spin-$1∕2$ chain coupled to optical phonons with nonzero 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 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 ${\omega }_0⪡{\Delta }_s$, 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 ${\omega }_0>{\omega }_{\mathit{BKT}}$, where ${\omega }_{\mathit{BKT}}$ 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.

Figure 1

The graph of the function $f\left(x\right)$ (solid line) defined in Eq. (47) with $K=1∕3$. Two regimes are visible. For $\Delta ⪢{\omega }_0$, $f\left(x\right)\sim x^{2-2K}$ (dashed curve). In that regime, the gap is given by the adiabatic formula Eq. (44). For $\Delta ⪡{\omega }_0$, $f\left(x\right)\sim x^{2-4K}$ (dotted curve) and the gap is given by the antiadiabatic formula Eq. (39). The crossover regime is observed for $0.3<\Delta ∕{\omega }_0<3$.

Figure 2

The behavior of the gap $\Delta$ as a function of the spin-Peierls coupling constant $G$ in the case $y\left(0\right)=0$. When $G⪢\alpha {\omega }_0∕u$, the system is in the adiabatic regime, and the gap $\Delta$ varies linearly with $G$. When $G⪡\alpha {\omega }_0∕u$, the gap decreases very rapidly with an essential singularity for $G\left(0\right)=0$ described by Eq. (65).

Figure 3

The behavior of the gap Δ as a function of phonon frequency ${\omega }_0$ for a fixed value of $G\left(0\right)$ and $y\left(0\right)=0$. In the adiabatic regime ${\omega }_0⪡uG\left(0\right)∕\alpha$, the gap is independent of ${\omega }_0$ and is equal to ${\Delta }_0=uG\left(0\right)∕\alpha$. In the antiadiabatic regime, the gap is a rapidly decreasing function of phonon frequency given by Eq. (65).

Figure 4

The phase diagram in the $(G,{\omega }_0)$ plane, where $G$ is the spin-Peierls coupling constant and ${\omega }_0$ the phonon frequency. The solid line corresponds to the BKT transition between the gapped and the gapless phase described by Eq. (64). The dash-dotted line is a crossover line between the antiadiabatic regime $uG\left(0\right)∕\alpha ⪡{\omega }_0$, and the adiabatic regime $uG\left(0\right)⪢{\omega }_0$.

Figure 5

Comparison of our prediction for ${\Delta }^{01}$ with the results of exact diagonalization (Ref. 18). In both cases, the gap is an increasing function of the spin-phonon coupling constant. Our results are roughly twice the exact diagonalization result.

Figure 6

Various compounds exhibiting a spin-Peierls transition plotted on a gap $\Delta ∕{\omega }_0$ vs Cross-Fisher gap ${\Delta }_{MF}∕{\omega }_0$. The data for the gap $\Delta$, the phonon frequency ${\omega }_0$ and the Cross-Fisher gap ${\Delta }_{MF}$ are extracted from Table 1 of Ref. 12. Note that the experimental determination of the gap is to be taken with a grain of salt given the differences between the various determinations from various measurements (neutrons, thermodynamics, etc.) The full line is Eq. (E20), which is the dependence of the gap on frequency that we expect when logarithmic corrections to scaling is neglected.