Field theory for magnetic and lattice structure properties of Fe_(1+y)Te_(1-x)Se_x

We study the magnetism and lattice distortion of Fe_{1+y}Te_{1-x}Se_x with a general field theory formalism, motivated by recent neutron scattering experiments. Besides the Ising nematic order parameter which can be naturally defined in the 1111 and 122 materials, We show that the collinear order observed in $\mathrm{Fe_{1+y}Te}$ materials is stabilized by an extra Ising order parameter which doubles the generalized unit cell of the system. By tuning this Ising order parameter, one can drive a transition between collinear and noncollinear spin density wave orders, both of which are observed in the FeTe family. The nature of the quantum and classical phase transitions are also studied in this field theory formalism, with lattice strain tensor fluctuation considered.