Finite-temperature spin diffusion in the gapped spin-1 2 XXZ chain: analytical benchmarks, intermediate-temperature approximation, and crossover structure

Abstract We present a systematic analytical study of finite-temperature spin diffusion in the gapped spin-$\tfrac{1}{2}$ XXZ chain with easy-axis anisotropy $\Delta>1$ at zero magnetic field. While the finiteness of the spin-diffusion constant $D_s(T,\Delta)$ for all $T>0$ and $\Delta>1$ has recently been established analytically \cite{R01Carmelo2025a,R02Carmelo2025b}, its precise values away from the asymptotic limits remain an outstanding open problem \cite{R03Carmelo2026BJP}. We provide a unified treatment covering three complementary analytical regimes. First, we formulate the two-sided bound $D_s^\infty(\Delta)\leq D_s(T,\Delta)\leq D_s^{\rm LT}(T,\Delta)$ between the exact high-temperature Bethe-ansatz value and the leading-order low-temperature activated formula; this bound is a conjecture supported by both asymptotic limits and the available numerical data, and it provides qualitative constraints at every temperature. Second, we exploit the high-temperature approximation of Carmelo and Sacramento \cite{R01Carmelo2025a} to obtain quantitative values of $D_s$ throughout the window $\beta J\in[0,1]$, tabulated for $\Delta\in\{1.25,1.5,1.75,2.0\}$. Third, we identify and compute the crossover temperature $T^*(\Delta)$ at which the high-temperature approximation meets the low-temperature activated formula; since both expressions are used outside their strict domains of validity at that point, $T^*$ is a heuristic indicator of the window in which neither closed-form expression is controlled, rather than a sharp regime boundary. We find that $T^*(\Delta)$ increases monotonically with anisotropy, ranging from $T^*/J\approx0.27$ at $\Delta=1.25$ to $T^*/J\approx0.60$ at $\Delta=2.0$, and give an explicit expression for it. Finally, we show that the high-temperature series can be resummed exactly into the form $D_s^\infty=(4J/3\pi)\sinh\eta\sum_{m\geq1}m/\sinh(m\eta)$ ($\Delta=\cosh\eta$), from which the near-isotropic divergence $D_s^\infty(\Delta)\simeq C_0(\Delta-1)^{-1/2}$ follows with the exact amplitude $C_0=\pi J/(3\sqrt{2})$. These results provide a quantitative analytical picture of spin diffusion over the temperature range accessible to closed-form expressions. The exact high-temperature floor $D_s^\infty(\Delta)$ and the crossover temperature $T^*(\Delta)$ serve as reference values against which future tensor-network and generalized-hydrodynamics calculations may be compared.