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, $\Gamma^0$). We clarify the origin of this ambiguity by demonstrating that the "intrinsic" response is not universal, but is inextricably linked to the dissipation mechanism that establishes the non-equilibrium steady state (NESS). We establish a benchmark by solving the exact NESS density matrix for a generic Bloch system coupled to a featureless fermionic bath. Our exact $\Gamma^0$ conductivity decomposes into two parts: (i) a geometric contribution, $\sigma^{\text{geo}}$, whose form recovers the intraband quantum metric contribution ($\sim\partial_k g$), providing an exact derivation that clarifies inconsistencies in the literature, and (ii) a novel, purely kinetic contribution, $\sigma^{\text{kin}} \propto v^3 f^{(4)}_0$, which is absent when dissipation is modeled by white-noise disorder (e.g., a constant-$\Gamma$ Green's function model). The discrepancy in $\sigma^{\text{kin}}$ between these distinct physical mechanisms is a proof that the $\Gamma^0$ nonlinear conductivity is not a unique property of the Bloch Hamiltonian, but is contingent on the physical system-bath coupling.