Shape descriptors of equilibrium states in a quantum lattice model with local multi-well potentials: A geometric analysis near the phase transitions in Sn$_2$P$_2$S$_6$ ferroelectric crystals
S. Özüm, T. Akkurt, R. Erdem, N. Güçlü
TL;DR
This work analyzes equilibrium states of a quantum lattice model with local multi-well potentials for $Sn_2P_2S_6$ ferroelectrics by embedding the free energy surface in differential geometry via mean curvature $H$, Gaussian curvature $K$, curvedness $C$, and shape index $S$. Using a mean-field formulation, the Gibbs free energy per site $φ$ is derived and self-consistent equations for the order parameter $η$ and deformation $u$ are solved as functions of energy gap $ε$, pressure $p$, and temperature $θ$, revealing FE and PE phases and a tricritical point. The authors show that $H$, $C$, and $S$ exhibit cusp singularities at criticality while $K→0$ near critical and tricritical points, indicating valley-like free energy surfaces; $K$'s vanishing behavior provides a practical route to sketch phase boundaries. The results offer a geometric lens on phase transitions in ferroelectric crystals and can extend to multi-order-parameter systems, connecting surface morphology to metastable behavior and Ising-like descriptions in related ferroelectric/ferrielectric materials.
Abstract
We analyze the equilibrium states of quantum lattice model with local multi-well potentials for Sn$_2$P$_2$S$_6$ ferroelectric crystals using the mean and Gaussian curvatures ($H$, $K$), curvedness ($C$) and shape index ($S$). From the energy gap, pressure and temperature variations of $H$, $K$, $C$ and $S$, we have reported the geometric construction of the free energy surfaces for the ferroelectric and paraelectric phases. Their behaviors are explicitly observed near the ferroelectric-paraelectric phase transitions. It is found that $H$, $C$ and $S$ display a cusp singularity at the criticality while $K$ converges to zero on both sides of the critical and tricritical points.