Dissipation-Shaped Quantum Geometry in Nonlinear Transport

Abstract

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 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 $Γ^0$ conductivity decomposes into two parts: (i) a geometric contribution, $σ^{\text{geo}}$, whose form recovers the intraband quantum metric dipole ($\sim\partial_k g$), providing an exact derivation that clarifies inconsistencies in the literature, and (ii) a novel, purely kinetic contribution, $σ^{\text{kin}} \propto v^3 f^{(4)}_0$, which is absent when dissipation is modeled by white-noise disorder (e.g., a constant-$Γ$ Green's function model). The discrepancy in $σ^{\text{kin}}$ between these distinct physical mechanisms is a proof that the $Γ^0$ nonlinear conductivity is not a unique property of the Bloch Hamiltonian, but is contingent on the physical system-bath coupling.

Figures (2)

• Figure 1: Non-universality of the $\Gamma^0$ nonlinear conductivity. The non-equilibrium steady state (NESS) governing DC transport depends on the specific dissipation mechanism, leading to mechanism-dependent "intrinsic" $(\Gamma^0)$ conductivities. Illustrated environments mediating relaxation: $(\Gamma_A)$ fermionic bath (e.g., electronic leads); $(\Gamma_B)$ electron-phonon interactions; $(\Gamma_C)$ static disorder; $(\Gamma_D$) radiative processes.
• Figure 2: Competition between $\Gamma^0$ geometric and kinetic nonlinear conductivity in the ${\cal P}{\cal T}$-symmetric model [Eq. \ref{['eq:model_H']}]. (a) Calculated $\Gamma^0$ contributions $\sigma_{xyy}^{\text{geo}}$ (blue dashed) and $\sigma_{xyy}^{\text{kin}}$ (red dotted), along with the total $\Gamma^0$ response $\sigma_{xyy}^{(0)}$ (black solid), as a function of chemical potential $\mu$. The kinetic term is comparable in magnitude to the geometric term. [Parameters used: $b/v=1$, $m/v=-3.9$, $t/v=0.9$, $\beta/v = 100$; characteristic second order conductivity $\sigma_2 = a (\hbar / v) \cdot e^3 / \hbar^2$]. (b) Color map of the ratio $|\sigma_{xyy}^{\text{kin}} / \sigma_{xyy}^{\text{geo}}|$ as a function of tilt parameter $t$ and chemical potential $\mu$. The kinetic term [Eq. (\ref{['eq:sigma_kinetic']})] dominates (red regions) over the geometric term [Eq. (\ref{['eq:sigma_geometric']})] in significant portions of the parameter space, underscoring its non-negligible role in the $\Gamma^0$ nonlinear response when the system is coupled to a fermionic bath.