Comprehensive understanding of disorder- and interaction-induced NLHE

Establish a comprehensive theoretical understanding of disorder- and interaction-induced contributions to the intrinsic second-order DC nonlinear Hall effect in multiband dispersive systems, beyond minimal models such as the two-dimensional Dirac point, by determining how impurity and electron–phonon scattering interplay with nontrivial quantum geometry to generate and modify the nonlinear Hall response.

Background

The paper decomposes the intrinsic second-order nonlinear Hall effect (NLHE) into quantum geometric contributions—nonlinear Drude, Berry curvature dipole, interband quantum metric dipole, and a newly identified intraband quantum metric dipole—within a fully gauge-invariant projector formalism. While these intrinsic mechanisms are clarified, the authors note that combining disorder (impurity scattering) and interactions (e.g., electron–phonon coupling) with quantum geometry can produce additional nonlinear Hall responses that are not yet fully understood.

They emphasize that numerical progress exists for minimal models (such as the 2D Dirac point), but achieving a complete, general theoretical framework for disorder- and interaction-induced NLHE in realistic multiband systems remains open. This would connect microscopic scattering processes with the presented quantum geometric expansion and help explain discrepancies among semiclassical and quantum approaches.

References

While numerical progress has been made for minimal models, such as the 2D Dirac point, a complete understanding remains an open challenge.

— Quantum Geometric Origin of the Intrinsic Nonlinear Hall Effect  (2506.17386 - Ulrich et al., 20 Jun 2025) in Discussion (Main Text)

Whether current-induced quadrupole accumulation can thus serve as a nonequilibrium probe of Bloch-state quantum geometry, complementing nonlinear-transport and spectroscopic approaches, is an intriguing open question.

— Observation of current-induced orbital quadrupole accumulation  (2608.22838 - Lee et al., 24 Aug 2026) in Section “Discussion and outlook”

The study of the effects of disorder and interactions on the quantum geometric nonlinear Zitterbewegung currents is left for future work.

— Multistate Torsion in Zitterbewegung  (2609.37767 - Jankowski, 29 Sep 2026) in Discussion and conclusion, third paragraph

At present, the microscopic origin of the Hall anomaly in Tb$_2$IrAl$_4$Ge$_2$ remains unclear.

— Complex magnetic phase diagrams in Tb$_2$IrAl$_4$Ge$_2$ and Er$_2$IrAl$_4$Ge$_2$  (2610.01714 - Gornicka et al., 1 Oct 2026) in Results and discussion, subsection “Electronic transport”