Possible extinction of Berezinskii–Kosterlitz–Thouless transition by diagonal interactions in the checkerboard lattice

Highlights: • We developed the continuous model of the anisotropic AF on the checkerboard lattice. • We used Monte Carlo simulation to obtain the BKT temperature. • The Group Renormalization was used to obtain the vortex energy and BKT temperature. • The influence of both D and $J^{\prime }$ interaction are considered in the BKT temperature. - Abstract: We study the thermodynamics of the classical anisotropic antiferromagnetic Heisenberg model in a checkerboard lattice. The checkerboard lattice is distinguished from the antiferromagnetic square lattice (with coupling constant J) by the presence of a diagonal crossing (coupling constant $J^{\prime }$) in half of the sites. This lattice model is the direct analog of the three-dimensional pyrochlore lattice on a two-dimensional surface. Besides, we considered a single-ion anisotropy D that breaks the $O\left(3\right)$ symmetry and contributes to planar spin fields. Since the model is two-dimensional endowed with an $O\left(2\right)$ symmetry, a Berezinskii–Kosterlitz–Thouless (BKT) transition is expected to take place. We also investigated the BKT temperature as a function of the coupling constants $J^{\prime }$ and D. The problem is developed through a continuous representation given by the $O\left(3\right)$ Nonlinear Sigma Model (NLSM). Computer simulations were also carried out, and the results were in accordance with the analytical model.