We analyze the recent numerical computations made by Fabricius, L{\" o}w and Stolze to show that the long time behavior of the zz correlation function of the spin 1/2 XXZ chain at $T=\infty$ is very well fit by the formula $t^{-d}[A+Be^{-γ(t-t_0)}\cos Ω(t-t_0)]$ where $d$ is substantially greater than 1/2. This confirms the conclusion that there is no spin diffusion in this model.
No spin diffusion in the spin 1/2 XXZ chain at T=∞: Numerical asymptotics
We analyze the recent numerical computations made by Fabricius, L{" o}w and Stolze to show that the long time behavior of the zz correlation function of the spin 1/2 XXZ chain at T=∞ is very well fit by the formula t−d[A+Be−γ(t−t0)cosΩ(t−t0)] where d is substantially greater than 1/2. This confirms the conclusion that there is no spin diffusion in this model.
We study the long time behavior of the zz and xx time-dependent autocorrelation function of the spin-1/2 XXZ chain at $T=\ensuremath{\infty}$ by exact diagonalizations on a chain of 16 sites. We find that the numerical results for the zz correlation are very well fit by the formula ${t}^{\ensuremath{-}d}[{A+Be}^{\ensuremath{-}\ensuremath{\gamma}(t\ensuremath{-}{t}_{0})}\mathrm{cos}\ensuremath{\Omega}(t\ensuremath{-}{t}_{0})].$ From this we estimate d as a function of the anisotropy of the chain and study the crossover from ballistic to diffusive behavior.
Abstract We present a systematic analytical study of finite-temperature spin diffusion in the gapped spin-21 XXZ chain with easy-axis anisotropy Δ>1 at zero magnetic field. While the finiteness of the spin-diffusion constant Ds(T,Δ) for all T>0 and Δ>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 Ds∞(Δ)≤Ds(T,Δ)≤DsLT(T,Δ) 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 Ds throughout the window βJ∈[0,1], tabulated for Δ∈{1.25,1.5,1.75,2.0}. Third, we identify and compute the crossover temperature T∗(Δ) 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∗(Δ) increases monotonically with anisotropy, ranging from T∗/J≈0.27 at Δ=1.25 to T∗/J≈0.60 at Δ=2.0, and give an explicit expression for it. Finally, we show that the high-temperature series can be resummed exactly into the form Ds∞=(4J/3π)sinhη∑m≥1m/sinh(mη) (Δ=coshη), from which the near-isotropic divergence Ds∞(Δ)≃C0(Δ−1)−1/2 follows with the exact amplitude C0=πJ/(32). These results provide a quantitative analytical picture of spin diffusion over the temperature range accessible to closed-form expressions. The exact high-temperature floor Ds∞(Δ) and the crossover temperature T∗(Δ) serve as reference values against which future tensor-network and generalized-hydrodynamics calculations may be compared.
We review some recent advances in the computation of exact correlation functions of the XXZ-½ Heisenberg chain. We first give a general introduction to our method which is based on the algebraic Bethe ansatz and the resolution of the quantum inverse scattering problem, leading in particular to multiple integral representations for the correlation functions. Then we describe recently obtained compact formulas for the spin-spin correlation functions of the XXZ-½ Heisenberg chain. We outline how this leads to several explicit results including the known two point functions in the limit of free fermions, the so-called emptiness formation probability at anisotropy Δ=1/2 and its large distance asymptotic behaviour in the massless phase of the model.
Based on the ballistic macroscopic fluctuation theory, the integration of the spin correlation function (spin conductivity) is analyzed for the spin-1/2 XXZ chain in the critical regime. In the time when the magnetization of an infinite spin chain fluctuates from an initial state with a wavelength as long as the infinite length N, the equal-time two-point spin correlation function is scaled up to O(1/N). In the state where the ballistic spin transport decays at high temperature T, the diffusive transport remains on a large scale. We show that the spin conductivity is proportional to 1/T in the limit T→∞ and its high temperature proportionality constant diverges in the case where one-quasiparticle magnetization is infinitely large. This analysis informs that the superdiffusive spin transport is driven by the 1/N-scaled long-range spin correlation and sheds a light on the dynamic scaling in spin transport at the isotropic point.
Time-dependent spin-autocorrelation functions at T=\ensuremath{\infty} and (in particular) their spectral densities for the bulk spin and the boundary spin of the semi-infinite spin-1/2 XXZ model (with exchange parameters Jx=Jy\ensuremath{\equiv}J, Jz) are investigated on the basis of (i) rigorous bounds in the time domain and (ii) a continued-fraction analysis in the frequency domain. We have found strong numerical evidence for spin diffusion in quantum spin models. For Jz/J increasing from zero, the results of the short-time expansion indicate a change of the bulk-spin xx-autocorrelation function from Gaussian decay to exponential decay. The continued-fraction analysis of the same dynamic quantity signals a change from exponential decay to power-law decay as Jz/J approaches unity and back to a more rapid decay upon further increase of that parameter. By contrast, the change in symmetry at Jz/J=1 has virtually no impact on the bulk-spin zz-autocorrelation function (as expected). Similar contrasting properties are observable in the boundary-spin autocorrelation functions.