Exploring the quantum Hall landscape with duality

It is an important open problem to understand the landscape of exotic fractional quantum Hall (FQH) phases accessible to physically motivated theories of composite particles. Of particular interest are (i) non-Abelian FQH phases and (ii) strongly coupled quantum critical states of particles strongly interacting with emergent gauge fields. I will show that progress can be made using the recently proposed family of boson-fermion dualities in two spatial dimensions. In the quantum Hall context, these dualities connect the dynamics of the ordinary, Abelian composite particles to dual degrees of freedom coupled to (Abelian or non-Abelian) gauge fields. In the dual variables, exotic states may be more transparently accessible, for example through pairing, filling of Landau levels, or behavior in periodic electric or magnetic fields. In this talk, I will consider the particular example of the Fibonacci FQH state of bosons at filling ν = 2, which is salient as the simplest platform for a universal topological quantum computer. Starting with a parent trilayer of Abelian FQH states, I will present a construction of the Fibonacci state using duality with a theory of bosonic "composite vortices" coupled to an emergent U(2) gauge field, which cluster between the layers to form the Fibonacci state. Beyond non-Abelian phases, I will also discuss a proposal using duality for how strongly interacting quantum critical states of Dirac fermions coupled to gauge fields can emerge in Dirac materials with spatially periodic magnetic fields.