Two-dimensional quantum spin-1/2 Heisenberg model with competing interactions

We study the quantum spin-1/2 Heisenberg model in two dimensions, interacting through a nearest-neighbor antiferromagnetic exchange ($J$) and a ferromagnetic dipolar-like interaction ($J_d$), using double-time Green's function, decoupled within the random phase approximation (RPA). We obtain the dependence of $k_B T_c/J_d$ as a function of frustration parameter $\delta$, where $T_c$ is the ferromagnetic (F) transition temperature and $\delta$ is the ratio between the strengths of the exchange and dipolar interaction (i.e., $\delta = J/J_d$). The transition temperature between the F and paramagnetic phases decreases with $\delta$, as expected, but goes to zero at a finite value of this parameter, namely $\delta = \delta_c = \pi /8$. At T=0 (quantum phase transition), we analyze the critical parameter $\delta_c(p)$ for the general case of an exchange interaction in the form $J_{ij}=J_d/r_{ij}^{p}$, where ferromagnetic and antiferromagnetic phases are present.