We have studied the phase diagrams of the one-dimensional spin-1 Blume-Capel model with anisotropy constant $D$, in which equivalent-neighbor ferromagnetic interactions of strength $-J$ are superimposed on nearest-neighbor antiferromagnetic interactions of strength $K$. A rich critical behavior is found due to the competing interactions. At zero temperature two ordered phases exist in the $D/J-K/J$ plane, namely the ferromagnetic $\left(\mathbf{F}\right)$ and the antiferromagnetic one $\left(\mathbf{AF}\right)$. For lower values of $D/J(D/J<0.25)$ these two ordered phases are separated by the point $K_c=0.25J$. For $0.25<D/J\le 0.50$, the paramagnetic phase $\mathbf{P}$ emerges in a region separated between the lines determined by $D/J=0.5-K/J$ and $D/J=K/J$. For $D/J>0.5$, only phases $\mathbf{AF}$ and $\mathbf{F}$ exist and are separated by a line given by $D/J=K/J$. At finite temperatures, we found that the ferromagnetic region of the phase diagram in the $k_{B}T/J-D/J$ plane is enriched by another ferromagnetic phase ${\mathbf{F}}^{{}^{\prime }}$ above a first-order line for $0.195<K/J<0.250$. This first-order line, which separates phases $\mathbf{F}$ and ${\mathbf{F}}^{{}^{\prime }}$, begins at a coexistence point, where phases $\mathbf{F},{\mathbf{F}}^{{}^{\prime }}$, and $\mathbf{P}$ coexist, and ends at an ordered critical point. Similarly, we found that the phase ${\mathbf{F}}^{{}^{\prime }}$ is present in the phase diagram in the $k_{B}T/J-K/J$ plane for $0.228<D/J<0.286$.

• Received 9 April 2015
• Revised 24 August 2015

DOI:https://doi.org/10.1103/PhysRevE.92.032120