Linear–quadratic order parameter coupling and multiferroic phase transitions

The coupling between order parameters in systems with several instabilities has been analysed within Landau theory. The dominant term considered in this paper is linear in one order parameter and quadratic in the second order parameter ~QP²; other coupling terms have been treated previously. Typical examples for Q are proper or pseudo-proper ferroelastic instabilities, while P might be octahedral tilting in a perovskite, (anti-)ferromagnetic ordering or (anti-)ferroelectric soft modes. Coupling of this type is common in fluorites, Verwey transitions, Jahn-Teller systems, pnictide superconductors, etc. Analytical solutions and characteristic phase diagrams of the stable configurations are compiled. The coupling can lead to stepwise phase transitions even when the uncoupled systems would show continuous transitions. Mixed phases are common, so that many 'intermediate phases' described in the literature may be the result of this linear-quadratic coupling.