Effects of Ill-Defined Domain of Definitions of the Parameter Operator on Berry Curvature and the Adiabatic Theorem

We present a comprehensive analytical study that extends the conventional formulation of Berry curvature, highlighting its derivation in the context of problematic domains of definition of the operators. Our analysis reveals that handling these domains carefully can have a substantial impact on Berry curvature, demonstrating that even Hamiltonians without explicit parameter dependence may exhibit nonzero Berry curvature. This finding emphasizes that Berry curvature is intrinsically related to the eigenvectors rather than the Hamiltonian itself. Our approach utilizes the standard Bloch (k-space) framework for spatially periodic systems, illustrating these effects from first principles and discussing potential implications for solid-state systems.