  Our theoretical study unfolds the topological phase transitions (within bands
of the Moir\'e super-lattice) in small angle twisted double bilayer graphene
(tDBLG) under the influence of external gate voltage and intrinsic spin-orbit
coupling (SOC) for both AB-AB and AB-BA stacking configurations. Utilizing a
low-energy continuum model, we investigate the band structure and perform a
comprehensive topological characterization of the system by analysing the
direct band gap closing as well as various Chern numbers. In the absence of
SOC, the tDBLG exhibits characteristics of a valley Hall insulator. However, in
the presence of SOC, we observe a transition to a quantum spin Hall insulator
state and band topology emerges in the parameter spaces of non-topological
regime without SOC. Furthermore, we conduct a comparative analysis between
untwisted double bilayer graphene and tDBLG to assess the impact of twisting on
the system's properties. Our findings reveal the construction of topological
phase diagrams that showcase distinct phases arising from changes in the twist
angle compared to the untwisted case. These phase diagrams provide valuable
insights into the diverse topological phases achievable in tDBLG with SOC. Our
findings contribute to the understanding of the interplay between small twist
angle, SOC, and external electric field on the topological band properties of
twisted multilayer graphene systems.
