Singular geometry and eigenframe topology in local rank-2 tensor observables

Abstract

Many observables are symmetric second-rank tensors, reported through magnitude-ordered principal values and axes. This chart folds tensor space: the parameters develop cusps and exchange labels where the tensor is smooth. It also hides a global effect: an arrow carried along a principal axis around a loop encircling a degeneracy can return reversed, defining a binary return parity, invariant under smooth deformations of the loop avoiding degeneracy. For tensor fields, this structure is long established, but the parameters are coordinates of the domain on which the field is defined, and the degeneracies are a feature of that particular field. Here we show that the electric field gradient (EFG) carries the same structure in a control space: a traceless observable at a single probe site, steered through its five-dimensional tensor space by symmetry-adapted strain, with the encircling loop applied rather than found. First-principles calculations reveal an isolated degeneracy with nontrivial parity in rutile TiO$_2$, point- or line-like degeneracies in SnO$_2$ depending on the control slice, and access to all five EFG components in cubic MgO. Thus, return parity becomes accessible for a local observable, and strain-tuned, orientation-resolved hyperfine spectroscopy offers a route to reconstruct it.