The nonlinear Hall effect can act as a window into the hidden geometry of electrons in a crystal. In a real sample, however, defects and impurities can bend electron trajectories in ways that resemble the desired quantum signal. This overlap has made it difficult to determine how much of a measured response truly comes from quantum geometry.
A research team led by Hai-Zhou Lu at the Southern University of Science and Technology (SUSTech), with collaborators from Shenzhen Technology University, Peking University, Fudan University, the Quantum Science Center of Guangdong-Hong Kong-Macao Greater Bay Area and Hefei National Laboratory, has now developed a unified scaling framework to separate those contributions. The study was published in Science Bulletin.
The geometry in this problem is not the physical shape of a device. Quantum geometry describes how an electron's quantum state changes as its momentum moves through a crystal. Its two complementary parts, the quantum metric and Berry curvature, can generate measurable second-order electrical responses. In a second-order response, the output does not grow in direct proportion to the driving current; one common experimental signature is a transverse voltage at twice the driving frequency.
Disorder complicates the picture. When electrons encounter imperfections, they may shift sideways in a process called side jump or scatter asymmetrically in a process called skew scattering. These effects can be comparable in size to the quantum-metric and Berry-curvature signals. A measurement may therefore contain several mechanisms at once.
The researchers treated quantum geometry and disorder on an equal footing within a semiclassical theory and catalogued 10 mechanisms that can contribute to the second-order nonlinear Hall effect. They then expressed the nonlinear Hall conductivity as a fourth-order polynomial of the ordinary longitudinal conductivity. The five polynomial coefficients carry different relative weights for different mechanisms.
Those weight patterns function like fingerprints. Eight mechanisms have their own unique pattern. One pair—Berry-curvature-plus-skew-scattering and second-order side jump—shares the same pattern and cannot be separated by this weight test alone. In practice, researchers can first fit a measured conductivity curve to the polynomial and then compare the fitted coefficient weights with the fingerprint table to identify and quantify the active mechanisms.
The team tested the workflow using published measurements from three widely studied quantum materials. In MnBi 2 Te 4 thin films, the analysis indicated that the quantum metric dipole was a dominant contribution in most devices, sometimes together with a disorder-induced Drude response. In WTe 2 and MoTe 2 , the method separated the Berry curvature dipole from side-jump and skew-scattering contributions. The fits gave coefficients of determination from 0.883 to 0.994 for the fitted MnBi 2 Te 4 samples, 0.998 for WTe 2 and 0.988 for MoTe 2 .
The study also provides a symmetry guide for future measurements. Among 122 magnetic point groups, the analysis found 9 groups in two dimensions and 20 groups in three dimensions that permit the relevant nonzero second-order conductivity components. This guide can help researchers choose materials, crystal directions and measurement arrangements before an experiment begins.
By turning competing microscopic processes into fit-ready fingerprints, the framework links theory directly to experimental data analysis. It can be used to reassess existing nonlinear-transport measurements and to design new searches for quantum-geometric effects. The approach may also inform research on rectification, piezoelectric responses and memory devices that use nonlinear electrical transport.
The article is authored by Zhen-Hao Gong, Z.Z. Du, Hai-Peng Sun, Hai-Zhou Lu and X. C. Xie. Zhen-Hao Gong and Z.Z. Du contributed equally, and Hai-Zhou Lu is the corresponding author.
Science Bulletin
Computational simulation/modeling