In delocalized systems, particle number fluctuations, also known as quantum surface roughness, and the mean-square displacement exhibit a temporal power law growth followed by a saturation to a system size--dependent value. We use simple scaling arguments to show that these quantities satisfy the Family-Vicsek scaling law and derive a dynamic scaling relation between the dynamical exponents, assuming that the saturation times of both quantities scale similarly with the system size. This relation clarifies the mechanism behind quantum surface roughness growth and suggests that diffusive quantum many-body systems belong to the Edwards-Wilkinson universality class. Moreover, it provides a convenient way to assess quantum transport in cold-atoms experiments. We numerically verify our results by studying two noninteracting models and one interacting model having regimes with distinct dynamical exponents.
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.
Fluctuations in the growing surfaces of classical systems can exhibit universal scaling behavior, known as Family-Vicsek (FV) scaling. Although this phenomenon was originally discovered in classical stochastic models, recent theoretical studies have demonstrated the presence of FV scaling in quantum many-body systems as well. Here, we observe the universal FV scaling in a one-dimensional Bose gas in an optical lattice. By monitoring the fluctuations of particle number in half of the system, which corresponds to the surface roughness, we extract all scaling exponents and demonstrate that the entire relaxation-from the growth of quantum fluctuations to their saturation-is captured by a single universal scaling function. Our results demonstrate that universal scaling laws of classical surface growth extend to quantum many-body systems, establishing a unified framework for nonequilibrium universality across classical and quantum systems.
Localization is one of the most fundamental interference phenomena caused by randomness, and its universal aspects have been extensively explored from the perspective of one-parameter scaling mainly for static properties. We numerically study dynamics of fermions on disordered one-dimensional potentials exhibiting localization and find dynamical one-parameter scaling for surface roughness, which represents particle-number fluctuations at a given length scale, and for entanglement entropy when the system is in delocalized phases. This dynamical scaling corresponds to the Family-Vicsek scaling originally developed in classical surface growth, and the associated scaling exponents depend on the type of disorder. Notably, we find that partially localized states in the delocalized phase of the random-dimer model lead to anomalous scaling, where destructive interference unique to quantum systems leads to exponents unknown for classical systems and clean systems.
Recent theoretical and experimental works have explored universal dynamics related to surface growth physics in isolated quantum systems. In this Letter, we theoretically elucidate that dissipation drastically alters universal particle-number-fluctuation dynamics associated with surface-roughness growth in one-dimensional free fermions and bosons. In a system under dephasing that causes loss of spatial coherence, we numerically find that a universality class of surface-roughness dynamics changes from the ballistic class to a class with the Edwards-Wilkinson scaling exponents and an unconventional scaling function. We provide the analytical derivation of the diffusion equation from the dephasing Lindblad equation via a renormalization-group technique and succeed in explaining the drastic change. Furthermore, we numerically find the same change of the universality class under a more nontrivial dissipation, i.e., symmetric incoherent hopping.
We study Family-Vicsek scaling of charge, spin, and energy fluctuations in the one-dimensional Hubbard model at infinite temperature. Using a quantum generating function approach, we compute time-dependent cumulants of transferred conserved quantities and analyze how the corresponding roughness depends on subsystem size and time. We start by focusing on a single interface at half the chain and determine the transport exponents. Then we turn to fluctuations of a small finite interval and study the Family-Vicsek universality of fluctuations over an extended timescale. We find that the long-time scaling behavior is controlled by integrability. In the free limit, charge, spin, and energy all display ballistic transport. In the interacting integrable Hubbard chain, charge and spin cross over to a KPZ scaling regime, while the energy sector remains ballistic. Once integrability is broken by a next-nearest-neighbor density interaction, the long-time dynamics becomes diffusive in all sectors. In every case we also observe a short-time microscopic regime with apparently universal ballistic growth before the hydrodynamic scaling window sets in. The Family-Vicsek setup allows us to determine the growth, the saturation as well as the dynamical exponents.
Localization is one of the most fundamental interference phenomena caused by randomness, and its universal aspects have been extensively explored from the perspective of one-parameter scaling mainly in equilibrium states. We theoretically study dynamics of fermions on disordered one-dimensional potentials exhibiting localization, and find that surface roughness and entanglement entropy show dynamical one-parameter-scaling in delocalized phases. The scaling of the roughness corresponds to the Family-Vicsek scaling in classical surface growth, and the associated universal scaling exponents depend on the type of disorder. Particularly, we find that partially localized states in the delocalized phase of the random-dimer model lead to anomalous scaling exponents, which are absent in classical systems and clean systems. Furthermore, we show that the surface roughness is approximately proportional to the square root of the von Neumann entanglement entropy, and then demonstrate that even the entanglement entropy obeys the Family-Vicsek-type scaling. This finding suggests that the surface roughness becomes a reliable measure for the entanglement dynamics.