Phase diagram and critical properties of the spin-1 Heisenberg model with single-ion anisotropy

We study the spin-1 bilinear-biquadratic Heisenberg model with the single-ion anisotropic term H=J∑〈ij〉[cosθSi⋅Sj+sinθ(Si⋅Sj)2]+D∑i(Siz)2 to show that the phase transition and critical behaviors from the large-D phase to other phases in two and three dimensions. The phase boundary of the spin-1 Heisenberg model with biquadratic exchange and single-ion anisotropy D is presented. When the bilinear coupling is antiferromagnetic, it shows that a positive biquadratic coupling constant tends to collaborate with the bilinear term to break the large-D   phase while a negative one has the opposite contribution. For the case of θ=π/4θ=π/4, all spins occupy the state of |sz=0〉|sz=0〉 and a second-order phase transition occurs at this point when the system is in the large-D phase.