Schwinger-Keldysh semionic approach for quantum spin systems

We derive a path-integral Schwinger-Keldysh approach for quantum spin systems. This is achieved by means of a semionic representation of spins as fermions with imaginary chemical potential. The major simplifying feature in comparison with other representations (Holstein-Primakoff, Dyson-Maleev, slave bosons/fermions etc) is that the local constraint is taken into account exactly. As a result, the standard diagram technique with usual Feynman codex is constructed. We illustrate the application of this technique for the N\'eel and spin-liquid states of the AFM Heisenberg model.