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

TL;DR

This paper investigates how ambiguities in defining operators affect Berry curvature, revealing that even Hamiltonians without explicit parameter dependence can have nonzero Berry curvature, emphasizing its relation to eigenvectors.

Contribution

It extends the conventional Berry curvature formulation to problematic operator domains, showing their impact on Berry curvature and clarifying its dependence on eigenvectors rather than Hamiltonians.

Findings

Operator domain issues can alter Berry curvature calculations.

Hamiltonians without explicit parameter dependence may still have nonzero Berry curvature.

Berry curvature is fundamentally linked to eigenvectors, not just Hamiltonians.

Abstract

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.