The phase diagrams and the order parameters of the transverse spin-1 Ising model with a longitudinal crystal-field

The transverse spin-1 Ising model with a longitudinal crystal field presents a rich variety of critical phenomena. Using the effective field theory with a probability distribution technique that accounts for the self-spin correlation functions, the phase diagrams and the tricritical points are investigated for the square lattice (N=4). The results show that the tricritical points exist for certain values of the strength of the transverse field and the strength of the longitudinal crystal field. The longitudinal and the transverse magnetizations as well as the longitudinal and the transverse quadrupolar moments are also examined. These quantities as functions of the temperature, the strength of the longitudinal crystal field and the strength of the applied transverse field are calculated numerically and some interesting results are obtained.