The Hamiltonian of the XY model -- when is it called the XX model?

[LagrangeEuler]

Hamiltonian of XY model is defined by
$H=J\sum_i (\sigma_i^x \sigma_{i+1}^x+\sigma_i^y \sigma_{i+1}^y)$
and because it is isotropic it is sometimes called XX model. If we do some unitary transformation, and get hamiltonian
$H=J\sum_i (\sigma_i^x \sigma_{i+1}^x-\sigma_i^y \sigma_{i+1}^y)$
what is name of that model? Is it like X-X model?

 

[MathematicalPhysicist]

You have written the same Hamiltonian!