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We perform a symmetry analysis of the zero-temperature instabilities of thetetragonal phase of Ca3Mn2O7 and Ca3Ti2O7 which is stable at high temperature.We introduce order parameters to characterize each of the possible latticedistortions in order to construct a Landau free energy which elucidates theproposed group-subgroup relations for structural transitions in these systems.We include the coupling between the unstable distortion modes and themacroscopic strain tensor. We also analyze the symmetry of the dominantlyantiferromagnetic ordering which allows weak ferromagnetism. We show that inthis phase the weak ferromagnetic moment and the spontaneous ferroelectricpolarization are coupled, so that rotating one of these ordering by applying anexternal electric or magnetic field one can rotate the other ordering. Wediscuss the number of different domains (including phase domains) which existin each of the phases and indicate how these may be observed.

Symmetry Analysis for the Ruddlesden-Popper Systems, Ca3Mn2O7 and Ca3Ti2O7