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

•We study the two-dimensional single-ion anisotropic ferromagnetic model with long-range interactions.•We show the quantum phase transition associated with the single-ion anisotropic constant.•We investigate the influence of the power-law exponent in the phase transitions.•We obtain a thermal phase transition similar to the BKT transition.

In the present work, we investigate the effects of long-range interactions on the phase transitions of a two-dimensional Heisenberg model with single-ion anisotropy at zero and finite temperatures. The Hamiltonian is given by H=∑i≠jJij(SixSjx+SiySjy+λSizSjz)+D∑i(Siz)2, where Jij=−J|rj−ri|−pJij=−J|rj−ri|−p(p≥3)(p≥3) is a long-range ferromagnetic interaction (J>0)(J>0), 0≤λ≤10≤λ≤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 mz≠0mz≠0, while in the large-D phase mz=0 because a state with 〈Sz〉≠0〈Sz〉≠0 is energetically unfavorable. Therefore a phase transition takes a place in some critical value Dc due to quantum fluctuations. For systems with short-range interaction Dc≈6 J (depending of λ 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.