Dynamic scaling and Family-Vicsek universality in $SU(N)$ quantum spin chains

The Family-Vicsek scaling is a fundamental framework for understanding surface growth in non-equilibrium classical systems, providing a universal description of temporal surface roughness evolution. While universal scaling laws are well established in quantum systems, the applicability of Family-Vicsek scaling in quantum many-body dynamics remains largely unexplored. Motivated by this, we investigate the infinite-temperature dynamics of one-dimensional $SU(N)$ spin chains, focusing on the well-known $SU(2)$ XXZ model and the $SU(3)$ Izergin-Korepin model. We compute the quantum analogue of classical surface roughness using the second cumulant of spin fluctuations and demonstrate universal scaling with respect to time and subsystem size. By systematically breaking global $SU(N)$ symmetry and integrability, we identify distinct transport regimes characterized by the dynamical exponent $z$: (i) ballistic transport with $z=1$, (ii) superdiffusive transport with the Kardar-Parisi-Zhang exponent $z=3/2$, and (iii) diffusive transport with the Edwards-Wilkinson exponent $z=2$. Notably, breaking integrability always drives the system into the diffusive regime. Our results demonstrate that Family-Vicsek scaling extends beyond classical systems, holding universally across quantum many-body models with $SU(N)$ symmetry.

💡 Research Summary

The paper investigates whether the Family‑Vicsek (FV) scaling, a cornerstone of classical non‑equilibrium surface growth theory, also governs the dynamics of quantum many‑body systems. The authors focus on one‑dimensional spin chains with SU(N) symmetry at infinite temperature, studying two representative models: the SU(2) spin‑½ XXZ chain and the SU(3) spin‑1 Izergin‑Korepin (IK) chain.

To define a quantum analogue of surface roughness, they consider the second cumulant κ₂,Sz(l,t) of the total Sᶻ magnetization within a contiguous subsystem of length l. The “roughness” is taken as W_Sz(l,t)=√κ₂,Sz(l,t). Using the quantum generating function (QGF) method, they compute κ₂,Sz directly from the λ‑derivative of the generating function G(λ,t)=Tr