Quantum Geometric Origin of the Intrinsic Nonlinear Hall Effect

Abstract: We analyze the quantum geometric contribution to the intrinsic second-order nonlinear Hall effect (NLHE) for a general multiband Hamiltonian. The nonlinear conductivity, obtained in Green's function formalism, is decomposed into its quantum geometric constituents using a projector-based approach. In addition to the previously identified Berry curvature and interband quantum metric dipoles, we obtain a third term of quantum geometric origin, given by the momentum derivative of the $intraband$ quantum metric. This contribution, which we term the intraband quantum metric dipole, provides substantial corrections to the NLHE in topological magnets and becomes the dominant geometric term in topological antiferromagnets with gapped Dirac cones. Considering generalized 2D and 3D Weyl/Dirac Hamiltonians, describing a large class of topological band crossings with sizable quantum geometry, we derive analytical expressions of the NLHE, thereby revealing the individual contributions of the three quantum geometric terms. Combined with an exhaustive symmetry classification of all magnetic space groups, this analysis leads to the identification of several candidate materials expected to exhibit large intrinsic NLHE, including the antiferromagnets $\text{Yb}_3\text{Pt}_4$, $\text{CuMnAs}$, and $\text{CoNb}_3\text{S}_6$, as well as the nodal-plane material $\text{MnNb}_3\text{S}_6$. Finally, our projector-based approach yields a compact expression for the NLHE in terms of momentum derivatives of the Bloch Hamiltonian matrix alone, enabling efficient numerical evaluation of each contribution in the aforementioned materials.

Quantum Geometric Origin of the Intrinsic Nonlinear Hall Effect

The paper delves into the quantum geometric contributions to the intrinsic second-order nonlinear Hall effect (NLHE) within a multiband Hamiltonian framework. Utilizing Green's function formalism, the authors decompose the nonlinear conductivity into quantum geometric constituents. This decomposition is achieved by employing a projector-based method, which unveils a third term of quantum geometric origin, augmenting the previously known Berry curvature and interband quantum metric dipoles. This term, termed the intraband quantum metric dipole, offers significant corrections to the NLHE and emerges as the preeminent geometric component in topological antiferromagnets featuring gapped Dirac cones.

Main Contributions

1. Quantum Geometric Decomposition:

   • This work introduces a comprehensive quantum geometric decomposition of the NLHE's second-order effects. The nonlinear conductivity is further dissected into its geometric elements.
   • In addition to Berry curvature dipole (BCD) and interband quantum metric dipole (interQMD), a novel intraband quantum metric dipole (intraQMD) is identified, particularly influencing systems with exotic symmetry properties.

2. Symmetry Classification and Candidate Materials:

   • By applying symmetry classifications to magnetic space groups, potential materials likely to exhibit substantial intrinsic NLHE are pinpointed. These include antiferromagnets like Yb(_3)Pt(_4), CuMnAs, CoNb(_3)S(_6), and nodal-plane material MnNb(_3)S(_6).

3. Numerical Evaluation Methodology:

   • The projector-based approach leads to a concise expression for evaluating NLHE contributions numerically across different materials, focusing solely on momentum derivatives of the Bloch Hamiltonian matrix.

Core Findings

The study confirms that nodal planes and antiferromagnetic Dirac systems are promising candidates for large, quantum metric-dipole–driven NLHE. The intraQMD particularly becomes significant in topological antiferromagnets possessing gapped Dirac cones. The manuscript posits that by comprehensively understanding nodal-plane structures, it can enhance our comprehension of quantum geometry's intrinsic contributions beyond those captured by Berry curvature alone.

Implications

From a theoretical lens, this research expands our understanding of the geometric nature of second-order nonlinear responses. Practically, this enhances our capability to design materials with optimized quantum geometrical responses, potentially enriching their application in advanced electronic and spintronic devices. Furthermore, the separation of geometric contributions based on symmetry enables refined interpretations of experimental data regarding topological phenomena.

Future Directions

The results from this study set the stage for deeper exploration of nonlinear responses in multiband systems, paving the way for identifying additional quantum geometric invariants. Investigations into the roles of disorder and interactions, especially within the nonlinear regime, could offer newer insights into exotic transport phenomena in topological materials.

In conclusion, the paper thoroughly explores the intricacies of quantum geometry in nonlinear Hall effects, heralding new avenues of research in topological quantum materials. The intraband quantum metric dipole opens promising opportunities to rethink strategies used in designing and analyzing materials with unique geometrical and electronic properties.