Spin-Peierls transition in an anisotropic two-dimensional XY model

The two-dimensional Jordan-Wigner transformation is used to investigate the zero temperature spin-Peierls transition for an anisotropic two-dimensional XY model in adiabatic limit. The phase diagram between the dimerized (D) state and uniform (U) state is shown in the parameter space of dimensionless interchain coupling $h$ $(=J_{perp}/J)$ and spin-lattice coupling $eta$. It is found that the spin-lattice coupling $eta$ must exceed some critical value $eta_c$ in order to reach the D phase for any finite $h$. The dependence of $eta_c$ on $h$ is given by $-1/ln h$ for $hto 0$ and the transition between U and D phase is of first-order for at least $h>10^{-3}$.