We will discuss two examples in which symmetry breaking and topology give rise to exotic ground states. In the first part of this talk, we will discuss the few- and many-body states in flat topological insulator bands, inspired by Magic Angle Twisted Bilayer Graphene (MATBG). In these bands charge neutral excitons behave effectively as charged particles in ordinary Landau levels. Due to this, Laughlin states of excitons can win over the default valley ferromagnet. Remarkably, these excitonic Laughlin states feature valley number fractionalization, but no charge fractionalization, and a quantized charge Hall conductivity identical to the Ising magnet, 𝜎_{xy}= 𝑒^2⁄h.
In the second part of the talk, we will discuss the physics in the unexplored higher Landau levels of graphene. To this date, a model that captures its valley dependent symmetry breaking interactions is lacking. We develop systematically such a model and show that this model can lead to qualitatively new ground states relative to the N=0 Landau level, such as ground states with entangled spin and valley degrees of freedom. Moreover, at half-filling we have found a new phase that is absent in the N=0 Landau level which combines the characteristics of a valence bond solid and an antiferromagnet. We discuss the estimation of parameters of this model based on recent compressibility experiments.
