Exact realization of integer and fractional quantum Hall phases in U(1)×U(1)U(1)×U(1) models in (2+1)d(2+1)d

• We present a set of models of two species of bosons which realize topological phases.
• The models exist on a lattice, and can be realized by local Hamiltonians.
• These models exhibit both integer and fractional bosonic Hall effects.
• The models can be studied in sign-free Monte Carlo.

In this work we present a set of microscopic U(1)×U(1)U(1)×U(1) models which realize insulating phases with a quantized Hall conductivity σxyσxy. The models are defined in terms of physical degrees of freedom, and can be realized by local Hamiltonians. For one set of these models, we find that σxyσxy is quantized to be an even integer. The origin of this effect is a condensation of objects made up of bosons of one species bound to a single vortex of the other species. For other models, the Hall conductivity can be quantized as a rational number times two. For these systems, the condensed objects contain bosons of one species bound to multiple vortices of the other species. These systems have excitations carrying fractional charges and non-trivial mutual statistics. We present sign-free reformulations of these models which can be studied in Monte Carlo, and we use such reformulations to numerically detect a gapless boundary between the quantum Hall and trivial insulator states. We also present the broader phase diagrams of the models.