Berry curvature dipole senses topological transition in a moir\'e superlattice

Topological aspects of electron wavefunction play a crucial role in determining the physical properties of materials. Berry curvature and Chern number are used to define the topological structure of electronic bands. While Berry curvature and its effects in materials have been studied, detecting changes in the topological invariant, Chern number, is challenging. In this regard, twisted double bilayer graphene (TDBG) has emerged as a promising platform to gain electrical control over the Berry curvature hotspots and the valley Chern numbers of its flat bands. In addition, strain induced breaking of the three-fold rotation (C3) symmetry in TDBG, leads to a non-zero first moment of Berry curvature called the Berry curvature dipole (BCD), which can be sensed using nonlinear Hall (NLH) effect. We reveal, using TDBG, that the BCD detects topological transitions in the bands and changes its sign. In TDBG, the perpendicular electric field tunes the valley Chern number and the BCD simultaneously allowing us a tunable system to probe the physics of topological transitions. Furthermore, we find hysteresis of longitudinal and NLH responses with electric field that can be attributed to switching of electric polarization in moir\'e systems. Such a hysteretic response holds promise for next-generation Berry curvature-based memory devices. Probing topological transitions, as we show, can be emulated in other 3D topological systems.