A mean field study of quantum transitions in a spin-1/2 XY chain with a transverse long-range interaction

We study the anisotropic one-dimensional XY model s=12 with uniform long-range interaction between the transverse components of the spins. The solution of the model was obtained by using the Jordan-Wigner transformation, and by treating the four fermion term within the mean field approximation. The proposed approximation reproduces the known exact results for two limiting cases of the model, namely, the isotropic model with long-range interaction and the anisotropic model without long-range interaction. Explicit expressions are obtained for the Helmholtz free energy, the induced magnetization and the isothermal susceptibility at arbitrary temperatures. Special attention is given to the study of quantum critical behaviour at T=0, by determining the phase diagram for the quantum phase transitions, and it is shown that the system presents a critical behaviour analogous to the one presented by isotropic model with long-range interaction. The spontaneous magnetization is also determined, at T=0, and we show that our results obtained by mean field approximation are in good agreement with those obtained by exact diagonalization of a finite chain; a comparison is also made with already known results.