Phase transitions and order-by-quantum disorder for the antiferromagnetic XY model in the checkerboard lattice

•We have applied the LSWA and the SCHA to study the effects of the quantum spin-wave interactions.•We have shown the order-by-quantum disorder for the antiferromagnetic XY model in the checkerboard lattice.•The thermodynamic has been determined and a thermal phase transition similar to the BKT transition is observed.•The quantum phase transition associated with a single-ion anisotropy has also been considered.

In this work we have studied the phases of the XY   antiferromagnetic model in the checkerboard lattice (the two-dimensional analog of the pyrochlore lattice). Using the traditional linear spin–wave approximation, we obtain a transition from an order to a disordered phase in some critical value rc=J′/Jrc=J′/J. The phase transition occurs due to quantum fluctuations. However, when we consider higher order perturbations the scenario is entirely different. We have applied the Self-Consistent Harmonic Approximation method and the results show that quantum perturbations induce long-range spin–spin correlations even above the critical point rc. This is a typical feature of order-by-quantum disorder when the system chooses one of the many classical ground states to occupy. We have also determined the thermal phase transitions similar to the Berezinskii–Kosterlitz–Thouless phase transition and the action of a single-ion anisotropy, responsible for a Quantum Phase Transition at zero temperature.