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 ($\mathrm{NM/FMI/NM}$) 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 $g_{\uparrow\downarrow}$ across all spin regimes. We demonstrate that $g_{\uparrow\downarrow}$ uniquely decomposes into a coherent condensed component ($g_{\mathrm{cond}}$), driven by the macroscopic phase of the spin superfluid, and an incoherent fluctuation-driven term ($g_{\mathrm{fluct}}$) mediated by stochastic thermal magnons. Crucially, in the extreme quantum limit of $S = 1/2$, $g_{\mathrm{cond}}$ drops steeply and vanishes at a finite coherence temperature $T_{\mathrm{coh}}$. Conversely, the fluctuation-driven term $g_{\mathrm{fluct}}$ vanishes at $T = 0$, exhibits a characteristic $T^2$ 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.