UC Santa Barbara

When interactions between electrons dominate over their kinetic energies, this can lead to emergent collective states with new quantum degrees of freedom described in terms of quasiparticles. These collective states can fall into two distinct categories: either the interactions lead to a breaking of an underlying symmetry, or electrons can form topologically ordered states characterized by a degenerate ground state in the zero temperature limit. The condition for strong interactions is met in two-dimensional electron systems (2DES) immersed in high magnetic fields, where electron kinetic energies quench into a ladder of dispersionless Landau levels broadened by disorder. Landau levels are an example of a topological band, characterized by a Chern number C ∈ Z. Correlated electron states then appear at partial filling of a Landau level — these are called fractional quantum Hall states.

In this thesis, I present a series of magneto-capacitance measurements of the thermo- dynamic density of states of low-disorder dual-gated graphene/boron nitride heterostructures – two-dimensional materials that can be isolated and later reassembled into stacks of designer properties. In such devices, we observe the elusive even-denominator fractional quantum Hall states at half filling of mono- and bilayer graphene. For monolayer graphene, we propose a scenario where the observed states are multicomponent states that incorporate correlations between electrons on different carbon sublattices. In the bilayer graphene case, the observed even-denominator states are single-component and

potentially host non-Abelian excitations, i.e. quasiparticles with fractional statistics, that are neither fermions or bosons.

Moreover, if we introduce a twist between the graphene and boron nitride layers, a

periodic moire potential will appear. As a result of the interplay between a magnetic field and the moire potential, new Hofstadter bands with Chern numbers C ≠ 1 can arise, in which we observe fractional Chern insulators — a generalization of fractional quantum Hall states to bands with a higher Chern index.