Exact real-time dynamics of quantum spin systems using the positive-P representation

We discuss a scheme for simulating the real-time quantum quench dynamics of interacting quantum spin systems within the positive-P formalism. As model systems we study the transverse field Ising model as well as the Heisenberg model undergoing a quench away from the classical ferromagnetic ordered state and the antiferromagnetic Néel state, depending on the sign of the Heisenberg exchange interaction. The connection to the positive-P formalism as it is used in quantum optics is established by mapping the spin operators on to Schwinger bosons. In doing so, the dynamics of the interacting quantum spin system is mapped onto a set of Ito stochastic differential equations the number of which scales linearly with the number of spins, N, compared to an exact solution through diagonalization that, in the case of the Heisenberg model, would require matrices exponentially large in N. This mapping is exact and can be extended to higher dimensional interacting systems as well as to systems with an explicit coupling to the environment.