  The 4D scanning transmission electron microscopy (STEM) method has enabled
mapping of the structure and functionality of solids on the atomic scale,
yielding information-rich data sets containing information on the interatomic
electric and magnetic fields, structural and electronic order parameters, and
other symmetry breaking distortions. A critical bottleneck on the pathway
toward harnessing 4D-STEM for materials exploration is the dearth of analytical
tools that can reduce complex 4D-STEM data sets to physically relevant
descriptors. Classical machine learning (ML) methods such as principal
component analysis and other linear unmixing techniques are limited by the
presence of multiple point-group symmetric variants, where diffractograms from
each rotationally equivalent position will form its own component. This
limitation even holds for more complex ML methods, such as convolutional neural
networks. Here, we propose and implement an approach for the systematic
exploration of symmetry breaking phenomena from 4D-STEM data sets using
rotationally invariant variational autoencoders (rrVAE), which is designed to
disentangle the general rotation of the object from other latent
representations. The implementation of purely rotational rrVAE is discussed as
are applications to simulated data for graphene and zincblende structures that
illustrate the effect of site symmetry breaking. Finally, the rrVAE analysis of
4D-STEM data of vacancies in graphene is illustrated and compared to the
classical center-of-mass (COM) analysis. This approach is universal for probing
of symmetry breaking phenomena in complex systems and can be implemented for a
broad range of diffraction methods exploring the 2D diffraction space of the
system, including X-ray ptychography, electron backscatter diffraction (EBSD),
and more complex methods.
