A Clarification on Quantum-Metric-Induced Nonlinear Transport

Over the years, Berry curvature, which is associated with the imaginary part of the quantum geometric tensor, has profoundly impacted many branches of physics. Recently, quantum metric, the real part of the quantum geometric tensor, has been recognized as indispensable in comprehensively characterizing the intrinsic properties of condensed matter systems. The intrinsic second-order nonlinear conductivity induced by the quantum metric has attracted significant recent interest. However, its expression varies across the literature. Here, we reconcile this discrepancy by systematically examining the nonlinear conductivity using the standard perturbation theory, the wave packet dynamics, and the Luttinger-Kohn approach. Moreover, inspired by the Dirac model, we propose a toy model that suppresses the Berry-curvature-induced nonlinear transport, making it suitable for studying the quantum-metric-induced nonlinear conductivity. This work provides a clearer and more unified understanding of the quantum-metric contributions to nonlinear transport. It also establishes a solid foundation for future theoretical developments and experimental explorations in this highly active and rapidly evolving field.