Using the model of the DM interaction in
XY antiferromagnetic spin chain with spin-phonon coupling, the nonadiabatic effects on the DM interaction have been studied by developing a nonadiabatic analytical approach. The results show that in the nonadiabatic case, to a certain finite phonon frequency, there exists a finite critical value of spin-phonon coupling. As the spin-phonon coupling decreases to the critical value, the system becomes gapless and the spin dimerization is destroyed. The DM interaction leads the system to have a finite critical value of spin-phonon coupling even in the adiabatic limit. The increase in staggered DM interaction will decrease the critical value of spin-phonon coupling, but increase that of phonon frequency, therefore, favors the dimerization. The nonadiabaticity plays an important role in suppressing the enhancement effects of the DM interaction to the dimerization. The dimerized ground state when DM interaction is present can be destroyed by the quantum lattice fluctuations. The threshold value of $\ensuremath{\beta}$ changing the effect of the DM interaction on the dimerization from suppression to promotion is not simply a constant but a crossover. For appropriate fixed values of spin-phonon coupling, phonon frequency and $\ensuremath{\beta}$, as
D increases, the system undergoes a reentry of phase between spin-dimer state and gapless state.
The low-energy properties of the S = 1/2 antiferromagnetic XY model with quasiperiodic modulation are investigated for the consideration of spin–phonon coupling. It is found that the lattice distortion at every site displays a self-similar character, which is induced by quasiperiodic modulation. As quasiperiodic modulation is strengthened, the average lattice distortion may decrease and increase for strong and weak enough spin–phonon coupling, respectively. Correspondingly, the scaling behavior of the energy gap in this system is also strongly dependent on the spin–phonon coupling.