Dissipation-Shaped Quantum Geometry in Nonlinear Transport

The theory of the intrinsic nonlinear Hall effect, a key probe of quantum geometry, is plagued by conflicting expressions for the conductivity that is independent of the dissipation strength (rate, Γ^{0}). We clarify the origin of this ambiguity by demonstrating that the "intrinsic" response is not universal, but is inextricably linked to how the dissipation mechanism shapes the nonequilibrium steady state (NESS) density matrix. We establish a benchmark by solving the exact NESS density matrix for a generic Bloch system coupled to a featureless fermionic bath. Our exact Γ^{0} conductivity decomposes into two parts: (i) a geometric contribution, σ^{geo}, which establishes the definitive structure of the quantum metric contribution (including the intraband ∼∂_{k}g term), clarifying inconsistencies in the literature, and (ii) a novel, purely kinetic contribution, σ^{kin}∝v^{3}f_{0}^{(4)}, arising from mechanism-specific modifications to the occupation functions, which is absent in approaches that postulate, rather than derive, the relaxation dynamics. The discrepancies in both σ^{geo} and σ^{kin} between these distinct physical mechanisms prove that the Γ^{0} nonlinear conductivity is not a unique property of the Bloch Hamiltonian, but is contingent on the physical system-bath coupling.