Exact Real Time Dynamics of Quantum Spin Systems Using the Positive-P Representation

TL;DR

This paper introduces an exact, scalable method using the positive-P representation to simulate real-time quantum spin dynamics, enabling efficient analysis of large interacting systems and their quenches.

Contribution

It develops a novel mapping of quantum spin dynamics onto stochastic differential equations via the positive-P formalism, extending quantum optics techniques to spin systems.

Findings

Efficient simulation of quantum quenches in spin models.

Linear scaling of stochastic equations with system size.

Potential extension to higher dimensions and open systems.

Abstract

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 antiferromagnetic Neel 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 (SDEs) 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.