Structural instabilities and sequence of phase transitions in SrBi$_2$Ta$_2$O$_9$ and SrBi$_2$Nb$_2$O$_9$ from first principles and Monte Carlo simulations

Despite their structural similarities, SrBi$_2$Ta$_2$O$_9$ (SBT) and SrBi$_2$Nb$_2$O$_9$ (SBN) undergo a different sequence of phase transitions. The phase diagram of SBT as a function of the temperature includes an intermediate phase between the high-temperature phase and the ferroelectric ground state, while in the Niobium compound the intermediate phase is suppressed and a single transition between the high- and low temperature structures is observed. We present \emph{ab initio} calculations that reveal the relevance of a trilinear coupling between three symmetry-adapted modes to stabilize the ground sate in both compounds, being this coupling much stronger in SBN. Within the framework of the phenomenological Landau theory, it is shown that by solely increasing the strength of the trilinear coupling the topology of the phase diagram of SBT can change up to suppress the intermediate phase. Monte Carlo simulations on an idealized $\phi^4$ Hamiltonian confirm that the trilinear coupling is the key parameter that determines the sequence of phase transitions and that for higher dimensionality of the order parameters the stability region of the intermediate phase is narrower.