Modern imaging techniques can map physical properties across a material with very high spatial resolution. However, understanding materials often requires more than identifying where a signal is strong or weak. Equally important is understanding how neighboring regions are connected, especially when strain, stress, and electromechanical responses vary from place to place.

In this study, we investigated stress-induced ferroelectric SrTiO₃ using birefringence imaging. This technique measures both the strength of the optical response and a local optical direction at each pixel. The optical direction is influenced by how strain and stress are arranged locally, but examining individual pixels provides only local information. To understand the overall organization of the material, it is necessary to study how these directions are connected across the entire image.

One way to do this is to follow the optical direction from pixel to pixel along a closed loop. If the directional changes cancel around the loop, the optical direction returns to its original orientation. However, in practice, many small directional changes can accumulate, producing a residual rotation. Consequently, the final orientation depends on the path taken through the image. This path-dependent residual rotation is known as holonomy.

Holonomy is important because it cannot be determined from a single pixel. If the optical directions represented only smooth local distortions, the directional changes would cancel after completing a loop. A nonzero holonomy indicates that the optical directions form a more complex geometric structure extending across the image. Thus, holonomy reflects a collective property of the directional pattern over a finite region of the image.

Applying this approach to stress-induced ferroelectric SrTiO₃ revealed localized regions with unusually large holonomy. These regions were found near areas of concentrated stress and were associated with an enhanced ferroelectric transition temperature. This suggests that large holonomy identifies regions where strain-related electromechanical fields are organized in an unusual manner. Such regions may also contain nonuniform polarization and associated bound-charge effects. Although holonomy does not directly image electric charge, it provides a geometric method for locating regions where hidden electromechanical activity may occur.

More broadly, this work demonstrates that experimental images contain not only local information but also geometric information encoded in the connections between neighboring directions. Holonomy provides a way to extract this hidden information and may become a useful tool for studying complex materials with spatially varying strain, order, and anisotropy.

(Written by Hirotaka MANAKA on behalf of all authors.)

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