Devlin-like approach to a spin-1 transverse XY model with biquadratic exchange and single-ion anisotropy

The effect of the biquadratic exchange interaction on the phase diagram of a d-dimensional spin-1 transverse XY model with easy-axis single-ion anisotropy is studied by employing the Devlin-like two-time Green functions framework. The chain of equations of motion is closed adopting the random phase approximation for the exchange higher order Green functions and treating exactly the crystal-field anisotropy terms. For short-range interactions and d > 2, analytical estimates and numerical calculations predict a reentrant behavior of the critical lines close to the magnetic-field-induced quantum critical point for appropriate values of the single-ion anisotropy parameter and suitable combinations of the bilinear and biquadratic exchange couplings. Remarkably, increasing the biquadratic exchange reduces or destroies the reentrant character of the quantum critical lines, in qualitative agreement with the findings of the Anderson-Callen-like strategy. In our formalism, the easy-plane anisotropy case can be studied similarly but the phase diagram and the quantum critical scenario do not present any reentrant phenomena. Graphical abstract