Length--Velocity Gauge Equivalence of Quantum Geometric Nonlinear Conductivity

Abstract: Nonlinear transport has emerged as a sensitive probe of quantum geometry beyond the Berry-curvature physics of linear response. However, the intrinsic second-order dc response remains conceptually subtle: different quantum and semiclassical formulations can appear to give different static limits, with different assignments of Fermi sea and Fermi surface contributions. Here we resolve this ambiguity by developing a gauge-consistent density-matrix theory of intrinsic nonlinear conductivity in both the length gauge, where the electric field couples through the position operator, and the velocity gauge, where it enters through the vector potential. We show that the two gauges give the same adiabatic dc response when the same retarded continuation is used for all external frequencies and when the velocity gauge current includes all field-dependent vertices. The apparent Fermi sea terms cancel in the full expression, leaving a Fermi surface quantum geometric contribution determined by the band-normalized quantum metric. This result implies that a fully gapped insulator has no residual dc nonlinear Hall current in the adiabatic clean limit. The reactive part of the Fermi surface term agrees with the original semiclassical Berry-connection-polarizability response, while the dissipative Ohmic sector requires a more careful treatment of relaxation and impurity scattering. Our work establishes the length-velocity gauge equivalence for quantum geometric nonlinear response and provides a foundation for using nonlinear transport to probe magnetic quantum geometry, especially in PT-symmetric antiferromagnets.