A continuous symmetry breaking measure for finite clusters using Jensen-Shannon divergence

TL;DR

This work tackles the challenge of quantifying local symmetry breaking in finite clusters by introducing a continuous symmetry breaking measure (SBM) based on the Jensen–Shannon divergence applied to a normalized electron-weighted density $\mu(\mathbf{x})$ and its transformed density under a symmetry operation $(T_\alpha)_\#\mu$. The SBM is implemented as two divergences, $\mathscr{S}_{T_\alpha}^{KL}[\mu]$ and $\mathscr{S}_{T_\alpha}^{JS}[\mu]$, with JS offering a bounded, comparable metric on $[0,1]$ and facilitating cross-cluster comparisons; the framework also defines operator SBMs and a single-atom analytic result for KL. The authors validate the approach on finite nickel clusters and perovskite tilt systems, compare KL and JS behavior, and provide open-source Monte Carlo tools for estimating SBMs, highlighting potential uses in structure refinement and ML-guided symmetry analysis. Overall, the continuous SBM enables nuanced, quantitative assessments of where and how symmetries are violated locally, offering a practical metric to study distortions and their structural consequences. The work paves the way for integrating SBMs into refinement workflows and machine learning pipelines as principled, bounded descriptors of local symmetry breaking.

Abstract

A quantitative measure of symmetry breaking is introduced that allows the quantification of which symmetries are most strongly broken due to the introduction of some kind of defect in a perfect structure. The method uses a statistical approach based on the Jensen-Shannon divergence. The measure is calculated by comparing the transformed atomic density function with its original. Software code is presented that carries the calculations out numerically using Monte Carlo methods. The behavior of this symmetry breaking measure is tested for various cases including finite size crystallites (where the surfaces break the crystallographic symmetry), atomic displacements from high symmetry positions, and collective motions of atoms due to rotations of rigid octahedra. The approach provides a powerful tool for assessing local symmetry breaking and offers new insights that can help researchers understand how different structural distortions affect different symmetry operations.

Figures (8)

• Figure 1: (a) The JS-SBM of the translational symmetry when an atom is displaced by $\bm{d}$ from itself, plotted for atoms with different $U_\text{iso}$. The star on each curve represents the point at which the JS-SBM reaches $0.1$. (b) The value of $d^*$ where the JS-SBM first reaches the $0.1$ threshold for different $U_\text{iso}$.
• Figure 2: (a and b) The KL-SBM $\mathscr{S}^{KL}_{R_\alpha}[\mu]$ and (c and d) the JL-SBM $\mathscr{S}^{JS}_{R_\alpha}[\mu]$, extracted from finite clusters of a nickel crystal structure. In subplots (a and c), spherical (blue) and cubic (red) shapes are used for cutouts, while spheroidal (blue) and rectangular (red) shapes are demonstrated in subplots (b and d). These curves depict the variation of $\mathscr{S}_{R_\alpha}[\mu]$ as the respective cluster is rotated by angle $\alpha$ (measured in degrees). This rotation is about an axis aligned with the crystallographic $c$-axis and intersects the cluster's center of mass.
• Figure 3: JS–SBM $\mathscr{S}^{JS}_{C_4}[\mu_{\textbf{d}}]$ vs. displacement magnitude $d=|\textbf{d}|$. Panel a (left): displacing a top/bottom face atom on the $C_4$ axis. Colors encode displacement direction: blue (along $a$-$b$ plane), red ($45^{\circ}$), yellow (along $c$). Panel b (right): displacing a side-face atom off-axis (same color coding). The $C_4$ axis is fixed through the cluster center.
• Figure 4: The SBM $\mathscr{S}^{JS}_{\sigma_h}[\mu_{\textbf{d}}]$ of a cubic cutout from the Nickel structure when one of its in-plane atoms (darker blue atoms) is shifted parallel to the plane (blue), deviated from the plane by 45 degrees (red), or perpendicular to the plane (yellow). The SBM is calculated for the reflection operation $\sigma_h$, whose mirror plane passes through the center of the finite cluster, with its normal vector along the positive $z$-axis. The figure is plotted as a function of the length of the atom displacement $d$.
• Figure 5: The SBM $\mathscr{S}^{JS}_{\sigma_h}[\mu_{\textbf{d}}]$ of a cubic cutout from the Nickel structure calculated for the reflection operation $\sigma_h$ (mirror plane normal to positive $c$-axis). The measure is plotted against displacement magnitude $d$ for shifts parallel to the plane (blue), perpendicular to the plane (yellow), and at 45 degrees (red). The yellow curve (left) represents the perpendicular displacement. It coincides with the trajectory obtained by scaling the $d$-axis of the 45-degree red curve (right) by a factor of $\cos(45^\circ) = 1/\sqrt{2}$, confirming that the scaled angled displacement is equivalent to the perpendicular displacement.