Electronic bands with non-zero Chern number is understood to give rise to topological phenomena. On the other hand, strong correlation between electrons heavily modifies band structure, resulting in topological property beyond single-particle theory. Moiré material consisting of two lattice-mismatched atomically thin transition metal dichalcogenide (TMD) layers is demonstrated to host strongly correlated phenomena and topological effects, serving as a tunable platform for the study of their interplay between correlation and topology. In this talk, I will first introduce our recently developed local thermodynamic measurement using optical sensing. We employed this technique to study electric field- tuned topological phase transition between a Chern insulator and a Mott insulator in two different moiré systems with contrasting behavior. In one system, the Chern insulator stems from the intertwined moiré band, while an excitonic mechanism is involved in the other system.