Flavour Hund's Coupling, Correlated Chern Gaps, and Diffusivity in Moir\'e Flat Bands

Interaction-driven spontaneous symmetry breaking lies at the heart of many quantum phases of matter. In moir\'e systems, broken spin/valley 'flavour' symmetry in flat bands underlies the parent state out of which ultimately correlated and topological ground states emerge. However, the microscopic mechanism of such flavour symmetry breaking and its connection to the low-temperature many-body phases remain to be understood. Here, we investigate the symmetry-broken many-body ground state of magic angle twisted bilayer graphene (MATBG) and its nontrivial topology using simultaneous thermodynamic and transport measurements. We directly observe flavour symmetry breaking as a pinning of the chemical potential $\mu$ at all integer fillings of the moir\'e superlattice, highlighting the importance of flavour Hund's coupling in the many-body ground state. The topological nature of the underlying flat bands is manifested upon breaking time-reversal symmetry, where we measure energy gaps corresponding to Chern insulator states with Chern numbers $C=3,2,1$ at filling factors $\nu=1,2,3$, respectively, consistent with flavour symmetry breaking in the Hofstadter's butterfly spectrum of MATBG. Moreover, our concurrent measurements of resistivity and chemical potential allow us to obtain the temperature dependence of the charge diffusivity of MATBG in the strange metal regime, a quantity previously explored only in ultracold atom systems. Our results bring us one step closer to a unified framework for understanding interactions in the topological bands of MATBG, both in the presence and absence of a magnetic field.