The nonlinear Hall effect (NHE), a recently discovered phenomenon, represents a novel member of the Hall effect family. Characterized by a nonlinear transverse voltage response to two perpendicular alternating currents, the NHE stands out due to its existence even in the presence of time-reversal symmetry, unlike its linear counterpart. This unconventional behavior hinges on the breaking of inversion and other discrete crystal symmetries, making it a powerful tool for probing phase transitions driven by spontaneous symmetry breaking, such as those observed in ferroelectric materials or systems exhibiting hidden order transitions. Its potential applications extend to the study of quantum critical points and the characterization of Néel vector orientations in antiferromagnets. Related phenomena, like the gyrotropic Hall effect, Magnus Hall effect, and nonlinear Nernst effect, further highlight the significance of this area. The quantum nature of NHE is deeply linked to the Berry curvature dipole, a concept that describes the dipole moment of the Berry curvature in momentum space. The Berry curvature itself represents a magnetic field in parameter space, reflecting the geometric structure of quantum eigenstates. Furthermore, the low-frequency nature of experimental NHE measurements places it firmly within the realm of quantum transport. Existing theoretical approaches primarily rely on semiclassical Boltzmann equations under the relaxation time approximation, lacking a comprehensive quantum description that fully captures the influence of disorder. This paper aims to address this gap by developing a systematic quantum theory of the nonlinear Hall effect, explicitly incorporating disorder effects.

Previous studies have primarily relied on semiclassical theories based on Boltzmann equations and the relaxation time approximation to understand the nonlinear Hall effect. These models, while offering insights, fall short in fully capturing the quantum nature of the phenomenon, particularly the role of disorder. While some attempts have been made towards a more quantum mechanical description, a comprehensive framework that explicitly addresses disorder effects has been lacking. Existing work has highlighted the connection between the nonlinear Hall effect and the Berry curvature dipole, providing a semiclassical understanding of the intrinsic contribution. However, the influence of disorder and its interaction with the intrinsic Berry curvature contributions have not been fully explored within a quantum mechanical framework.

This research employs the diagrammatic technique to construct a quantum theory of the nonlinear Hall effect. Unlike the simpler bubble diagrams used in linear response theory, quadratic responses necessitate the use of more complex triangular and two-photon diagrams to represent two inputs and one output. The authors identified and analyzed 69 Feynman diagrams contributing to the leading nonlinear responses in the weak-disorder limit. These diagrams incorporate intrinsic, side-jump, and skew-scattering contributions. Unlike nonlinear optics, where disorder plays a less significant role, the authors demonstrate that the majority (64 out of 69) of these diagrams reflect the crucial impact of disorder on electronic transport. The theoretical framework is developed for a generic two-band model and then applied to a disordered 2D tilted Dirac model – a minimal model exhibiting both strong Berry curvature and broken inversion symmetry, making it ideal for studying the NHE. The general formulas derived from the diagrammatic calculations are suitable for direct application in first-principles calculations. A detailed symmetry analysis of the nonlinear Hall response tensor is performed for all 32 point groups in 2D and 3D, utilizing the symmetry properties of the diagrams. The quantum calculations of the nonlinear Hall conductivity are performed within the non-crossing approximation, considering the intrinsic, side-jump, and skew-scattering contributions. A key aspect of the methodology is the careful classification of the Feynman diagrams based on their underlying physical mechanisms (intrinsic, side-jump, and skew-scattering), and their dependence on the impurity concentration. The authors introduce simplified diagrammatic representations to enhance understanding and to highlight the differences between quantum and semiclassical descriptions. The disordered system is modeled using delta-function scatters, allowing for a systematic analysis of disorder effects on the nonlinear Hall response.

The study's key findings include: 1. **Comprehensive Quantum Theory:** The authors present a complete quantum theory of the NHE using Feynman diagrams, explicitly including disorder effects, which are critical for electronic transport. This contrasts with previous semiclassical approaches. 2. **Diagrammatic Analysis:** A total of 69 Feynman diagrams contributing to the leading nonlinear responses are identified and classified into intrinsic, side-jump, and skew-scattering contributions. 3. **Disorder Enhancement:** Application to a 2D tilted Dirac model reveals that disorder significantly enhances the nonlinear Hall conductivity, while preserving the sign of the conductivity. This contradicts certain aspects of previous semiclassical predictions. 4. **Symmetry Analysis:** A comprehensive symmetry analysis reveals the conditions under which a purely disorder-induced nonlinear Hall effect can occur. This analysis was carried out for all 32 point groups in 2D and several in 3D, identifying specific symmetries where the intrinsic contribution vanishes, leaving only the disorder-induced component. 5. **Qualitative Differences from Semiclassical Theory:** The quantum calculations show qualitative differences compared to semiclassical results, particularly regarding the signs of the side-jump and skew-scattering contributions and their overall impact on the total conductivity. These differences highlight limitations of purely semiclassical descriptions. 6. **Scaling Behavior:** The quantum theory does not alter the scaling laws determined by the order of disorder dependence. 7. **Simplified Diagrammatics:** The authors introduce a simplified representation of the Feynman diagrams, reducing complexity and facilitating a deeper understanding of the underlying physics. These simplified diagrams highlight qualitative differences between the nonlinear Hall effect diagrams and those of the anomalous Hall effect, emphasizing the unique features of the nonlinear response. 8. **Pure Disorder-Induced NHE:** The study predicts the existence of a pure disorder-induced nonlinear Hall effect in specific crystal symmetry point groups (e.g., C3, C3h, C3v, D3h, D3 in 2D, and Td, T, C3h, D3h in 3D), providing experimental guidance for identifying systems where this effect dominates.

This work provides the first comprehensive quantum description of the nonlinear Hall effect, incorporating disorder effects which were previously neglected or treated inadequately. The findings significantly enhance our understanding of this phenomenon. The demonstration that disorder enhances the NHE, while maintaining its sign, clarifies the interplay between intrinsic and extrinsic contributions. The detailed symmetry analysis provides valuable criteria for identifying materials exhibiting a dominant disorder-induced NHE. The discrepancies between quantum and semiclassical calculations highlight the limitations of semiclassical models and the necessity of a full quantum treatment, especially when disorder is prominent. This study significantly advances our understanding of the NHE and offers new avenues for materials design and experimental investigations. The theoretical framework developed is robust and applicable beyond the specific 2D tilted Dirac model used in the study. The findings are also significant for future research in topological materials, where understanding nonlinear responses is crucial.

This paper presents a comprehensive quantum theory of the nonlinear Hall effect using a diagrammatic technique. The theory explicitly incorporates disorder effects, revealing significant enhancements to the nonlinear Hall conductivity and predicting the existence of a purely disorder-induced nonlinear Hall effect in certain crystal symmetries. The results highlight qualitative differences between quantum and semiclassical descriptions, underscoring the importance of a fully quantum mechanical treatment for accurate understanding and prediction. Future work could focus on applying this theoretical framework to a broader range of materials and exploring the influence of different types and strengths of disorder.

The study primarily focuses on the weak-disorder limit within the non-crossing approximation. The applicability of these findings to strongly disordered systems requires further investigation. The analysis utilizes a simplified model (2D tilted Dirac model) for quantitative calculations. While this model captures essential features of the NHE, the results may not be universally applicable to all materials exhibiting the effect. Future work could explore more complex band structures and investigate the impact of interactions beyond the non-crossing approximation.