The spin-3/2 Blume-Capel model with competing short- and long-range interactions

The phase diagrams of the spin-$3/2$ Blume-Capel model with competing short and long-range interactions were studied through the free energy density obtained by analytical methods. The competition emerges when positive short-range interactions of strength $K$ arranged in a linear chain tend to establish an anti-parallel spin order, whereas negative long-range interactions $-J$ tend to align them in parallel. Thus, no ferromagnetic order exists for $K/J > 0.25$. So, the phase-diagrams were scanned by varying the values of $K$ in this interval. As in other similar study done for the spin-1 case, the second-order frontier separating the ferromagnetic and the paramagnetic phases is transformed gradually into a first-order line, when $K/J$ is greater than a certain critical value. Accordingly, there is a subinterval of $K$, for which two tricritical points appear restricting the length of the second-order frontier. Nevertheless, for greater values of $K/J$, the ferromagnetic-paramagnetic frontier becomes wholly of first order. Also, the tipical coexistence line, which divides two different ferromagnetic phases of magnetization $m=3/2$ and $m=1/2$, becomes more complex by giving rise to another line of coexistence with a reentrant behavior that encloses a third ordered phase. In this case, the competition is such that there is a region in the phase diagram, where for each spin $i$ with $S_{i}=3/2$ ($S_{i}=-3/2$), there is another one spin $j$ with $S_{j}=-1/2$ ($S_{j}=1/2$), so the absolute value of the magnetization per spin is one.