Efficient magnetic superstructure optimization with $\Theta\Phi$

Simulating the incommensurate spin density waves (ISDW) states is not a simple task within the standard \emph{ab-initio} methods. Moreover, in the context of new material discovery, there is a need for fast and reliable tool capable to scan and optimize the total energy as a function of the pitch vector, thus allowing to automatize the search for new materials. In this paper we show how the ISDW can be efficiently obtained within the recently released $\Theta\Phi$ program. We illustrate this on an example of the single orbital Hubbard model and of $\gamma$-Fe, where the ISDW emerge within the mean-field approximation and by using the twisted boundary conditions. We show the excellent agreement of the $\Theta\Phi$ with the previously published ones and discuss possible extensions. Finally, we generalize the previously given framework for spin quantization axis rotation to the most general case of spin-dependent hopping matrix elements.