Quantum Spin-$1$ Anisotropic Ferromagnetic Heisenberg Model in a Crystal Field: A Variational Approach

摘要

A variational approach based on Bogoliubov inequality for the free energy isemployed in order to treat the quantum spin-$1$ anisotropic ferromagneticHeisenberg model in the presence of a crystal field. Within the Bogoliubovscheme an improved pair approximation has been used. The temperature dependentthermodynamic functions have been obtained and provide much better results thanthe previous simple mean-field scheme. In one dimension, which is stillnon-integrable for quantum spin-$1$, we get the exact results in the classicallimit, or near-exact results in the quantum case, for the free energy,magnetization and quadrupole moment, as well for the transition temperature. Intwo and three dimensions the corresponding global phase diagrams have beenobtained as a function of the parameters of the Hamiltonian. First-ordertransition lines, second-order transition lines, tricritical and tetracriticalpoints, and critical endpoints have been located through the analysis of theminimum of the Helmholtz free energy and a Landau like expansion in theapproximated free energy. Only first-order quantum transitions have been foundat zero temperature. Limiting cases, such as isotropic Heisenberg, Blume-Capeland Ising models have been analyzed and compared to previous results obtainedfrom other analytical approaches as well as from Monte Carlo simulations.