Topology of atomically thin soft ferroelectric membranes at finite temperature

One account of two-dimensional (2D) structural transformations in 2D ferroelectrics predicts an evolution from a structure with Pnm2$_1$ symmetry into a structure with square P4/nmm symmetry and is consistent with experimental evidence, while another argues for a transformation into a structure with rectangular Pnmm symmetry. An analysis of the assumptions made in these models is provided here, and six fundamental results concerning these transformations are contributed as follows: (i) Softened phonon modes produce rotational modes in these materials. (ii) The transformation to a structure with P4/nmm symmetry occurs at the lowest critical temperature $T_c$. (iii) The hypothesis that one unidirectional optical vibrational mode underpins the 2D transformation is unwarranted. (iv) Being successively more constrained, a succession of critical temperatures ($T_c<T_c<T_c$) occurs in going from molecular dynamics calculations with the NPT and NVT ensembles onto the model with unidirectional oscillations. (v) The choice of exchange-correlation functional impacts the estimate of the critical temperature. (vi) Crucially, the correct physical picture of these transformations is one in which rotational modes confer a topological character to the 2D transformation via the proliferation of vortices.