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A new approximating technique is developed so as to study the quantumferromagnetic spin-1 Blume-Capel model in the presence of a transverse crystalfield in the square lattice. Our proposal consists of approaching the spinsystem by considering islands of finite clusters whose frontiers are surroundedby non-interacting spins that are treated by the effective-field theory. Theresulting phase diagram is qualitatively correct, in contrast to mosteffective-field treatments, in which the first-order line exhibits spuriousbehavior by not being perpendicular to the anisotropy axis at low temperatures.The effect of the transverse anisotropy is also verified by the presence ofquantum phase transitions. The possibility of using larger sizes constitutes anadvantage to other approaches where the implementation of larger sizes iscostly computationally.

A new effective-field technique for the ferromagnetic spin-1 Blume-Capel model in a transverse crystal field