Paradoxical coexistence of superconductivity and magnetism, and explaining unexpected preferred domain orientations

The classical Ginzburg-Landau model has long provided the foundation for modeling superconductivity, yet it does not fully capture the rich diversity of unconventional superconducting phenomena observed experimentally. Examples include magnetic superconductors, re-entrant superconductivity, and the cuprates-materials whose behaviors are not adequately explained by traditional Ginzburg-Landau and/or Bardeen-Cooper-Schrieffer (BCS) theories. In this work, we develop a thermodynamically consistent reformulation of the Ginzburg-Landau theory expressed in gauge-invariant variables. This framework, when extended to anisotropic settings, naturally predicts preferential domain orientations and, crucially, reveals a previously overlooked symmetry-allowed term that breaks time-reversal symmetry. Such a term is essential for capturing the behavior of magnetic superconductors, particularly antiferromagnetic systems where superconductivity and magnetism coexist. We further demonstrate that stabilization of antiferromagnetic superconducting states necessitates a loss of convexity in the free energy. Together, these results unify disparate phenomena within a single-component order parameter and offer a systematic route for understanding, modeling, and controlling unconventional superconducting states.