Engineering a Quantized Integer Hall Response in Non-Interacting Bosonic Hamiltonians

The TKNN formulation of the integer quantum Hall effect states that the transverse conductivity of a two-dimensional electron system subjected to low temperatures and an external magnetic field is directly related to the Chern number characterizing the non-trivial topology of the underlying electronic band structure. We revisit the original TKNN argument for non-interacting (quadratic) bosonic Hamiltonians by relaxing the requirement that they be thermodynamically stable, while maintaining dynamical stability, and provide a protocol for engineering quantized transverse response in topologically non-trivial regimes. This is done by generalizing the Smrcka-Streda formulation of the Hall conductivity for non-interacting many-boson systems, wherein a non-thermal distribution yielding quantized transverse response of the Hamiltonian density is identified. We then explore the possibility of quantized response with this non-thermal distribution via numerical simulations of the anomalous quantum Hall effect in number non-conserving bosonic models on the honeycomb and kagome lattices. Our work provides a possible recipe for the experimental detection of topologically non-trivial phases of bosons in two-dimensional synthetic photonic materials.