Nonlinear Hall effects induced by Berry curvature dipole in CuPb₉(PO₄)₆O

The nonlinear Hall effect (NLHE), an emergent response in systems with broken inversion symmetry, provides a powerful tool for probing topological transport properties. In this context, we investigate copper-substituted lead apatite (LK-99), a material that initially garnered attention for its controversial claim of room-temperature superconductivity. Despite the unresolved nature of its superconducting properties, LK-99's unique electronic structure characterized by flat bands near the Fermi level and broken inversion symmetry makes it a promising candidate for exploring Berry curvature-driven phenomena, such as the NLHE. Using first-principles density functional theory and an augmented tight-binding Hamiltonian model, we investigate LK-99's band topology and transport properties. Our calculations indicate that spin-orbit coupling in LK-99 generates multiple Weyl points near the Fermi level, thereby enhancing the Berry curvature distribution by further splitting the bands. Crucially, the absence of inversion symmetry in LK-99 leads to a net Berry curvature dipole (BCD), producing a nonlinear Hall current that scales quadratically with the applied electric field. The nonlinear Hall effect is solely due to the BCD, as the contributions from the Drude weight and quantum metric are zero due to time reversal symmetry. Moreover, we demonstrate that the NLHE in LK-99 can be tuned by varying the direction of the applied electric field, underscoring its potential as a versatile platform for exploring topological transport phenomena and designing next-generation nonlinear electronic devices.