Consistent perturbative disorder treatment beyond constant scattering rate for NLHE

Develop a full perturbative disorder theory for the intrinsic second-order DC nonlinear Hall conductivity that consistently incorporates momentum-dependent self-energies, vertex corrections, and crossing diagrams—going beyond a constant, band-independent relaxation rate—and determine the impact of these processes on the quantum geometric decomposition into nonlinear Drude, Berry curvature dipole, interband quantum metric dipole, and intraband quantum metric dipole terms.

Background

The main text and Supplemental Material employ a constant, band-independent scattering rate Γ to regularize the DC limit, enabling a clean separation of band structure and quantum geometry. The authors show that introducing band-dependent (non-constant) Γ directly in the band basis breaks gauge-consistent identities (e.g., Ward identity) and fails to recover known linear-response limits like the TKNN formula without a proper disorder treatment.

They explain that a consistent inclusion of non-constant Γ requires a full perturbative disorder framework that accounts for momentum-dependent self-energies, vertex corrections, and possibly crossing diagrams. Such a framework is necessary to assess how realistic scattering processes modify the identified quantum geometric contributions to NLHE.