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

Abstract 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 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.