We introduce a novel tensor-network approach to extract the time-dependent cumulants of the full counting statistics up to unprecedently long times [1], which we apply to the investigation of spin-transfer in quantum spin chains. The archetypal S=1/2 anisotropic Heisenberg model exhibits ballistic and diffusive transport regimes, separated (at the isotropic point) by a superdiffusive regime — which has proven to be ubiquitous in integrable quantum and classical models with non-abelian symmetries. Surprisingly, aspects of the superdiffusive spin dynamics are reminiscent of classical interface growth phenomena. On the one hand, spatio-temporal spin correlations display a scaling behavior with a characteristic dynamical exponent (z=3/2), and superdiffusive spin transfer was conjectured to fall within the Kardar-Parisi-Zhang (KPZ) universality class [2] despite lacking key features of KPZ physics. The hypothesis has been falsified by recent experiments on Google's Sycamore quantum processor [3]. Our results are in quantitative agreement with the experimental data and extend to timescales far beyond the coherence time of the superconducting qubits architecture, thus providing unambiguous evidence that spin transfer in integrable quantum spin chains is incompatible with KPZ universality.  On the other hand, we show that, for the quantum analogue of surface roughness, the subsystem and temporal fluctuation are well-described by the self-similar Family-Vicsek scaling behavior. We verify that the relation z = \alpha / \beta between the roughness (\alpha), growth (\beta), and dynamical exponents holds in all spin transport regimes and across models with SU(N) symmetry [4]. Our results shed light on how classical universal scaling laws extend to the quantum many-body realm.

 

[1] A. Valli, C. P. Moca, M. A. Werner, M. Kormos, Ž. Krajnik, T. Prosen, and G. Zaránd, Phys. Rev. Lett. 135, 100401 (2025)

[2] M. Ljubotina, M. Žnidarič, and T. Prosen, Phys. Rev. Lett. 122, 210602 (2019)

[3] E. Rosenberg et al., Science 384, 48-53 (2024)

[4] C. P. Moca, B. Dóra, D. Sticlet, A. Valli, T. Prosen, and G. Zaránd, arXiv:2503.21454 (2024)