Spin-1 Blume–Capel model in a transverse crystal field: An improved variational approach

• Two tricritical points are found at finite temperature.
• Two symmetrical quantum critical points are determined at absolute zero.
• Quantum tricritical points are found at zero temperature.
• Quadrupole moment turns out to be important to specify the order of the transitions.

We examine the phase diagram of the spin-1 Blume–Capel model in a transverse crystal field. Using an improved pair approximation within the framework of Bogoliubov variational approach for the free energy, we go beyond the simple mean-field approximation (MFA) and effective field theory (EFT) at investigating the phase diagram of this model. The present results may clarify important points with respect to the multicritical phenomena in the phase diagram, which are not yet fully resolved, since MFA and EFT yield conflicting results. Phase diagrams in one, two and three dimensions are investigated, and compared with previous ones obtained by employing MFA and EFT. We show how the nature of the phase transition changes as both longitudinal and transverse crystal fields are varied. In addition, the spin-1 Ising model in a transverse crystal field is obtained as a limiting case of zero longitudinal crystal field, and its phase diagram is also analysed carefully. We show analytical expressions for the critical equation at finite as well as at absolute zero temperatures. By means of a Landau expansion of the free energy, we locate tricritical points besides critical ones. Remarkably, quantum tricritical points are found in two and three dimensions. We also study the behaviour of the thermodynamic quantities, namely magnetisation and quadrupole moment, which turn out to be important to define the order and nature of the transitions.