The Berry phase, rooted in the wave functions of the Bloch electrons, leads to a number of remarkable phenomena such as the quantum anomalous Hall effect. In recent years, it has also been shown that the dipole moment of the Berry curvatures plays an essential role in the nonlinear Hall effect. In this thesis, we study the nonlinear and quantum anomalous Hall effects induced by Berry phase and its multipole moments, in spin-orbit coupled and moiré materials. We first show that strong nonlinear Hall effect exists in strained 2H-structure polar transition metal dichalcogenides and strained twisted bilayer graphene (TBG). In particular, we find that the strain can strongly affect the band structure of TBG due to its flatness, which can lead to a topological band inversion that can be detec...[ Read more ]

The Berry phase, rooted in the wave functions of the Bloch electrons, leads to a number of remarkable phenomena such as the quantum anomalous Hall effect. In recent years, it has also been shown that the dipole moment of the Berry curvatures plays an essential role in the nonlinear Hall effect. In this thesis, we study the nonlinear and quantum anomalous Hall effects induced by Berry phase and its multipole moments, in spin-orbit coupled and moiré materials. We first show that strong nonlinear Hall effect exists in strained 2H-structure polar transition metal dichalcogenides and strained twisted bilayer graphene (TBG). In particular, we find that the strain can strongly affect the band structure of TBG due to its flatness, which can lead to a topological band inversion that can be detected with the nonlinear Hall effect. Then, we present a general theory for the higher-order nonlinear anomalous Hall effects induced by Berry curvature multipoles. Importantly, by analyzing the symmetry properties of the Berry curvature quadrupole, we point out the third-order nonlinear anomalous Hall effect exists in all the 66 piezomagnetic groups. Furthermore, we find that the quadrupole, hexapole and even higher Berry curvature moments can cause the leading-order nonlinear anomalous Hall effects in certain magnetic point groups. Finally, we turn to the quantum anomalous Hall effect recently observed in AB-stacked moiré MoTe₂/WSe₂ heterobilayers. We propose it to be a topological p_x+ip_y inter-valley coherent state, where the spin-polarized moiré bands from both the MoTe₂ and the WSe₂ layers are involved. We show that the massive Dirac nature of the MoTe₂ moiré bands together with the Coulomb interactions could favor the p_x+ip_y pairing. Furthermore, our theory predicts that the pairing of electrons and holes from two pockets with a definite momentum mismatch would generate a supercell, which is three times the size of the original moiré unit cell and could be verified in experiments.