Emergent Ising degrees of freedom above double-stripe magnetism

Double-stripe magnetism $[\mathbf{Q}=(\pi/2, \pi/2)]$ has been proposed as the magnetic ground state for both the iron-telluride and BaTi$_2$Sb$_2$O families of superconductors. Double-stripe order is captured within a $J_1-J_2-J_3$ Heisenberg model in the regime $J_3 \gg J_2 \gg J_1$. Intriguingly, besides breaking spin-rotational symmetry, the ground state manifold has three additional Ising degrees of freedom associated with bond-ordering. Via their coupling to the lattice, they give rise to an orthorhombic distortion and to two non-uniform lattice distortions with wave-vector $(\pi, \pi)$. Because the ground state is four-fold degenerate, modulo rotations in spin space, only two of these Ising bond order parameters are independent. Here we introduce an effective field theory to treat all Ising order parameters, as well as magnetic order, and solve it within a large-$N$ limit. All three transitions, corresponding to the condensations of two Ising bond order parameters and one magnetic order parameter are simultaneous and first order in three dimensions, but lower dimensionality, or equivalently weaker interlayer coupling, and weaker magnetoelastic coupling can split the three transitions, and in some cases allows for two separate Ising phase transitions above the magnetic one.