Quantum correction to the BKT transition for 2D easy-plane antiferromagnets

We consider the quantum antiferromagnet with easy-plane exchange anisotropy, namely, the antiferromagnetic XXZ model, on the square lattice. Its classical counterpart, compared to the planar model shows a reduction of the critical temperature T_BKT of the Berezinskii-Kosterlitz-Thouless phase transition, that is a consequence of the thermal out-of-plane fluctuations. For the quantum system we use the pure-quantum self-consistent harmonic approximation to calculate how much the effective exchange interaction is weakened as an effect of the pure-quantum part of the fluctuations. One can then predict the further reduction of T_BKT with respect to the corresponding classical system. The theory works well in a wide range of values of the easy-plane anisotropy. In the extreme case of the spin-1/2 model, the result is compatible with the estimate of T_BKT obtained by previous quantum Monte Carlo simulations. When the anisotropy is weak the theory leads to an unphysical ''isotropization'' due to the use of the Villain spin-boson transformation.