Coherent and incoherent interfacial spin transport: Quantum-to-classical crossover in spin superfluids

We investigate the thermodynamics of interfacial spin transport within a normal metal/ferromagnetic insulator/normal metal (NM/FMI/NM) trilayer heterostructure, where the central magnetic layer is described by the anisotropic quantum XXZ model. By employing the self-consistent harmonic approximation (SCHA) combined with a microscopic linear response formulation, we evaluate the interfacial spin-mixing conductance g↑↓ across all spin regimes. We demonstrate that g↑↓ uniquely decomposes into a coherent condensed component (gcond), driven by the macroscopic phase of the spin superfluid, and an incoherent fluctuation-driven term (gfluct) mediated by stochastic thermal magnons. Crucially, in the extreme quantum limit of $S = 1/2$, gcond drops steeply and vanishes at a finite coherence temperature Tcoh. Conversely, the fluctuation-driven term gfluct vanishes at $T = 0$, exhibits a characteristic T² quadratic scaling at low temperatures, and undergoes a systematic $1/S$ amplitude suppression as the macroscopic magnetization becomes robust. Our microscopic insights bridge the gap between quantum many-body fluctuations and macroscopic spin-superfluid hydrodynamics, providing clear foundational principles for optimizing long-range coherent transport in quantum spintronic devices.