I want to ask about the argument in the Kalmeyer-Laughlin paper that connects the Heisenberg Hamiltonian to a quantum Hall one. I know there is a related question about the same paper. My question is more about the details of the argument.
So I understand that the essence of the argument is that we can invent a magnetic field by a gauge transformation. The authors noted that the original Holstein-Primakoff bosons from the Heisenberg Hamiltonian have a "wrong" sign of hopping. They are positive, unlike free particles. I think the gauge transformation in the paper produces a uniform magnetic field that changes signs of only a subset of all hopping processes on the lattice (see fig.1 of the paper). My question is what is the logic behind using FQH wave functions when the bosons still hop differently from free particles after the gauge transformation?
I also want to write down just another way to ask the same question. The original Hamiltonian actually has time-reversal symmetry (TRS). The gauge transformation can not change that. But in the case where the mapping to the bosons hopping in a magnetic field works well, wouldn't it be saying the original problem breaks TRS?
