Microscopic Studies of the Geometric Aspects in Fractional Quantum Hall Effect
The importance of quantum geometry in strongly correlated topological systems is increasingly appreciated recently, for its roles in the incompressibility gap, phase transitions and emergent classical geometric effects. In this talk I focus on the fractional quantum Hall effect (FQHE), which can serve as the prototypical case for other related systems. I start with a brief overview of the geometric aspects in FQHE, together with the well-established pseudopotential formalism. The latter is very successful in characterising the rotationally invariant effective two-body interactions and in the construction of model Hamiltonians for some prominent FQH states. This will be followed by a discussion of our recent work in extending this formalism for systems where rotational symmetry is broken, with the construction of the generalized pseudopotentials that are projection operators for the intra-Landau level dynamics and at the same time form a complete basis for arbitrary interactions. I will then illustrate the interplay between geometry and topological orders of the FQH with the generalised pseudopotentials in some common experimental systems, with the focus on how effective interactions can be tuned experimentally within different materials. I will also discuss how various metrics can emerge from microscopic anisotropic Hamiltonians in both compressible and incompressible FQH systems.