Phase transitions in the two-dimensional single-ion anisotropic ferromagnetic with long-range interactions

In the present work, we investigate the effects of long-range interactions on the phase transitions of two-dimensional ferromagnetic models with single-ion anisotropy at zero and finite temperatures. The Hamiltonian is given by $H=\sum_{i\neq j} J_{ij}(S_i^xS_j^x+S_i^yS_j^y+\lambda S_i^zS_j^z)+D\sum_{i}(S_i^z)^2$, where $J_{ij}=-J |r_j-r_i|^{-p}$ ($p\geq 3$) is a long-range ferromagnetic interaction ($J>0$), $0\leq \lambda\leq 1$ is an anisotropic constant and $D$ is the single-ion anisotropic constant. It is well-known that the single-ion anisotropy $D$ creates a competition between an ordered state (favored by the exchange interaction) and a disordered state, even at zero temperature. For small values of $D$, the system has a spontaneous magnetization $m_z\neq 0$, while in the large-D phase $m_z=0$ because a state with $\langle S^z\rangle\neq 0$ is energetically unfavorable. Therefore, a phase transition due to quantum fluctuations occurs in some critical value $D_c$. For systems with short-range interaction $D_c\approx 6J$, depending of $\lambda$ constant, but in our model we have found larger values of $D$ due to the higher cost to flip a spin. Since low-dimensional magnetic systems with long range interaction can be ordered at finite temperature, we also have analyzed the thermal phase transitions (similar to the BKT transition). The model has been studied by using a Schwinger boson formalism as well as the Self-consistent Harmonic Approximation (SCHA) and both methods provide according results.