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

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