General teleparallel geometric theory of defects

We revisit the geometric theory of defects. In the differential-geometric models of defects that have been adopted since the 1950s, dislocations have been associated with torsion, disclinations with the full curvature, and point defects with the first kind trace of non-metricity. The mainstream formulation exhibits several conceptual and technical shortcomings, most notably a hierarchy inconsistency, the non-exictence of a genuine metric formulation, and the potential emergence of Ostrogradsky-type instabilities. These issues have motivated us to develop a new framework, namely a generalized teleparallel geometric theory of defects. In our model, dislocations are identified with the trace of torsion, disclinations with the second kind trace of the non-metricity, and point defects with the first kind trace of the non-metricity. In addition, we retain the scalar part torsion as a free parameter for describing some possible unknown degrees of freedom in the theory of defects. The proposed geometric theory of defects is free from all of the aforementioned drawbacks and is therefore worthy of further investigation. To ensure the coherence and completeness of the discussion, we begin our analysis with elastic deformations, then summarize the existing metric-affine geometric theory of defects, and finally proceed to our original contribution, namely the new theory introduced here. We formulate the entire theory in Eulerian coordinates. Naturally, all results can be reformulated in Lagrangian coordinates as well. All analyses and formulae are expressed in the language of exterior algebra and are carried out in coordinate-independent orthonormal frames.

💡 Research Summary

The paper revisits the geometric description of crystal defects and identifies three fundamental shortcomings of the prevailing metric‑affine (or metric‑Cartan) framework: (1) a hierarchy inconsistency in which dislocations (torsion) and disclinations (curvature) are interdependent in a way that contradicts the physical reality where dislocations can exist without disclinations; (2) the lack of a genuine metric formulation because the full affine connection cannot be expressed solely in terms of the orthonormal co‑frame; and (3) the presence of curvature‑squared terms in the Lagrangian, which lead to fourth‑order field equations and Ostrogradsky‑type instabilities.

To overcome these issues, the authors propose a generalized teleparallel geometry in which the total curvature vanishes while both torsion and non‑metricity are allowed to be non‑zero. Within this setting, they re‑assign the defect densities: the trace of torsion corresponds to dislocations, the second‑kind trace of non‑metricity to disclinations, and the first‑kind trace of non‑metricity to point defects. A scalar part of torsion is retained as a free parameter, potentially encoding unknown degrees of freedom such as internal spin or extra matter fields.

All constructions are carried out in Eulerian coordinates using exterior algebra and orthonormal frames, guaranteeing coordinate‑independence. The Cartan structure equations are rewritten under the teleparallel condition (zero curvature), and a Yang–Mills‑type Lagrangian is built from torsion‑squared, non‑metricity‑squared, and mixed terms only. Consequently, the resulting field equations are second order, eliminating Ostrogradsky instabilities and providing a well‑posed initial‑value problem.

The paper also clarifies the physical interpretation: a smooth displacement field yields purely Riemannian elastic deformations, while discontinuities generate non‑Riemannian torsion and non‑metricity that obstruct a diffeomorphism between reference and current configurations. The torsion trace reproduces the Burgers vector, the non‑metricity traces reproduce the Frank vector, and the first‑kind non‑metricity trace captures volumetric point defects such as vacancies or interstitials.

In summary, the generalized teleparallel geometric theory offers a mathematically consistent, physically transparent, and dynamically stable framework for describing all three classes of crystal defects. It resolves the hierarchy, metric, and stability problems of earlier models and opens avenues for coupling defect physics with teleparallel gravity theories, numerical simulations, and experimental validation.