The electric field gradient tensor as a symmetry-adapted order parameter in Landau theory

Abstract

Quadrupolar hyperfine spectroscopies, including Nuclear Quadrupole Resonance (NQR), Nuclear Magnetic Resonance (NMR), Time-Differential Perturbed Angular Correlations (TDPAC), and M\"ossbauer spectroscopy, have long used the electric field gradient (EFG) at a nuclear site as an empirical proxy for order parameters in structural and electronic phase transitions, yet the EFG has never been systematically incorporated into Landau theory. Here we provide that framework. The EFG is an exactly traceless, symmetric rank-2 tensor defined at a crystallographic site. Decomposing it under the site-symmetry group and inducing over the Wyckoff orbit determines its irreducible representation content in the parent space group. Whenever the representation of a zone-center transition is present, symmetry requires the corresponding EFG combination to vanish above the transition and grow linearly with the order parameter below it, inheriting its critical exponent, sign, and domain structure. Symmetry-orthogonal channels are quadratic, recovering the classic empirical relations. This yields a falsifiable classification of primary, secondary, and forbidden EFG responses. The framework is validated against five decades of quadrupolar experiments, reproducing critical exponents, first-order discontinuities, and a null result, and by first-principles calculations satisfying the predicted parity and zero theorems. All-electron calculations for $\alpha$-quartz confirm the orbit-selection rule: only the EFG combination transforming as the soft-mode irreducible representation varies linearly with distortion amplitude, while the orthogonal combination remains suppressed by two orders of magnitude. A proposed study of the $^{75}$As site across the nematic transition in BaFe$_2$As$_2$ provides five falsifiable predictions, including a previously unstated null result.