Dissipative quantum geometric phase in the spin-boson system

We explore how the dissipative geometric phase evolves within the spin-boson model, focusing specifically on coupling to an Ohmic bath in the weakly coherent regime. To determine the non-unitary time evolution of the system's reduced density matrix, we employ the non-interacting blip approximation (NIBA). For the localized pure initial state used throughout, we derive a compact Bloch-sphere expression showing that the mixed-state geometric phase is a weighted azimuthal winding of the dissipative trajectory. We then map geometric-phase accumulation across various system-bath coupling strengths, temperatures, static biases, and bath cutoff frequencies. Benchmarking representative results against the numerically exact time-evolving matrix product operator (TEMPO) technique shows that NIBA reproduces the population dynamics almost indistinguishably. It also captures the geometric phase quantitatively and qualitatively, although TEMPO reveals a clearer trend in the stationary coherence. Our results show that quantum dissipation suppresses the geometric phase through two complementary mechanisms: thermal noise reduces the state's purity, while static bias localizes the dynamics and reduces the accessible geometric area.