Abstract:

“Recent advances in materials science have established Moiré materials as a new highly
tunable and versatile form of quantum matter. When two dimensional atomic layers are
brought into proximity, a tiny relative twist or a slight lattice mismatch produces Moiré
patterns manifested in a superlattice structure with a lattice constant that is much larger than
the lattice constants of the constituent layers. The new length scale has dramatic consequences
for the underlying properties. A particular distinctive feature of Moiré materials is the
emergence of nearly flat bands upon tuning external parameters such as the twist angle or the
applied gate voltage. In a flat band, the kinetic energy is quenched, and interactions are
enhanced bringing us to the realm of strongly correlated systems. A prime example of Moiré
materials is twisted bilayer graphene, formed by taking two graphene layers and twisting them
relative to each other.
On the other hand, a famous class of interaction-induced phases of matter are fractional
quantum Hall states and their lattice analogues known as fractional Chern insulators. These
topologically ordered phases represent a departure from the conventional Landau symmetry
breaking classification of matter, seen in the absence of local order parameters and the
presence of global topological properties insensitive to local perturbations. Identifying and
manufacturing materials that could host fractional Chern insulator states has a great potential
for technological use.
In this thesis, we provide the necessary background required for understanding the results of
the accompanying papers [Phys. Rev. Lett. 124, 106803 & Phys. Rev. Lett. 126, 026801].
The theory of fractional Chern insulators is discussed followed by an introduction to the
Moiré models used. In the two accompanying papers, we theoretically study a number of flat
band Moiré materials aiming at identifying the possible phases that occur at fractional band
fillings using a combination of analytical and numerical techniques. By reformulating the
problem in terms of holes instead of electrons, it’s possible to identify a variety of emergent
weakly interacting Fermi liquids from an initial strongly interacting problem. In addition, our findings
also include several high temperature fractional Chern insulator states at different fillings without external magnetic field.