Full counting statistics of spin superdiffusion in the Heisenberg chain
In plain words
In the spin-$1/2$ Heisenberg chain at high temperature, magnetization spreads faster than ordinary diffusion, with distance growing as time to the power $2/3$. The spreading has the scaling of a classical surface-growth model called KPZ (Kardar-Parisi-Zhang), but the full statistics does not match it, and the correct description is unknown.
Precise statement
For the isotropic spin-$1/2$ Heisenberg chain at infinite temperature and zero net magnetization, spin transport has dynamical exponent $z = 3/2$ and the spin structure factor matches the KPZ scaling function. The full counting statistics of magnetization transferred across a cut is known not to be KPZ (Rosenberg et al. 2024; Gopalakrishnan et al. 2023). Derive the exact asymptotic distribution of transferred magnetization and its scaling with t, and identify the universality class it defines. Answer: the limiting distribution, or its cumulant ratios as $t \to \infty$.
What would settle it
An exact theory (for example nonlinear fluctuating hydrodynamics of the giant quasiparticles) of the transferred-magnetization distribution, matched by numerics or quantum simulation at times long enough for cumulant ratios to converge.
Status in the literature
Unverified note
Non-KPZ statistics were established by a 2024 superconducting-processor experiment (Rosenberg et al., Science 2024, arXiv:2306.09333) and by theory; the limiting distribution is not derived for the quantum chain.