Quantum spin Hall effect in a three-orbital tight-binding Hamiltonian

We consider the quantum spin hall (QSH) state in a three-orbital model, with a certain spin loop current order which is induced by spin-dependent interactions. This type of order is motivated by the loop current model, which was proposed long ago to describe the pseudogap phase of cuprates. It is shown that this model has nontrivial Chern parity by directly counting the zeros of the Pfaffian of the time reversal operator. By connecting to the second Chern number, we explicitly show the property of singularities of the wave-functions, and also their dependance on the gauge choices. Also it is shown that the Berry phase of the QSH state can be mapped to a non-Abelian instanton.