Phase transitions in the two-dimensional Anisotropic Biquadratic Heisenberg Model

In this paper we study the influence of the single-ion anisotropy in the two-dimensional biquadratic Heisenberg model (ABHM) on the square lattice at zero and finite low temperatures. It is common to represent the bilinear and biquadratic terms by $J_1=J\cos\theta$ and $J_2=J\sin\theta$, respectively, and it is well documented the many phases present in the model as function of $\theta$. However we have adopted a constant value for the bilinear constant ($J_1=1$) and small values of the biquadratic term ($|J_2|<J_1$). In special, we have analyzed the quantum phase transition due to the single-ion anisotropic constant $D$. For values below a critical anisotropic constant $D_{c}$ the energy spectrum is gapless and at low finite temperatures the order parameter correlation has an algebraic decay (quasi long-range order). Moreover, in $D<D_c$ phase there are a transition temperature where the quasi long-range order (algebric decay) is lost and the decay becomes exponential, similar to the Berezinski-Kosterlitz-Thouless (BKT) transition. For $D > D_c$, the excited states are gapped and there is no spin long-range order (LRO) even at zero temperature. Using Schwinger bosonic representation and Self-Consistent Harmonic Approximation (SCHA), we have studied the quantum and thermal phase transitions as a function of the bilinear and biquadratic constants.