Path integral approach to the truncated Wigner approximation of driven-dissipative spins

Driven-dissipative spin systems, such as those found in Rydberg atom arrays, are central to modern quantum many-body physics. While the Truncated Wigner Approximation (TWA) is a powerful tool for simulating such systems, its application to spin-1/2 models is often uncontrolled because the phase-space corrections (the star product) are of the same order as the leading-order terms. This paper addresses the need for a systematic, field-theoretic foundation for dissipative spin TWA that properly incorporates these corrections.

The authors construct a path-integral formulation for open spin-1/2 systems using the continuous SU(2) phase space. By mapping operator products from the Hilbert space onto the curved Bloch sphere, they derive the SU(2) star product, which encodes the non-commutativity of spin operators. They show that the standard approach of replacing this star product with an ordinary product is inconsistent for spin-1/2. Instead, they derive differential operators that represent the action of the Lindbladian on the phase-space kernel, leading to corrected stochastic equations of motion.

The study establishes that the SU(2) star product generates additional O(1) contributions to the dissipative sector of the equations of motion. These terms are absent in previous field-theoretic treatments of dissipative spin systems. The authors demonstrate that by incorporating these corrections, the resulting stochastic equations successfully reproduce the exact dynamics of a single dissipative spin, whereas the uncorrected version fails to reach the correct steady state. This provides a rigorous starting point for applying TWA to complex, interacting many-body spin systems.