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arXiv:2603.09060

Universal Family-Vicsek scaling in quantum gases far from equilibrium

Kiryang Kwon, Kazuya Fujimoto, Junhyeok Hur, Byungjin Lee, Samgyu Hwang, Sumin Kim, Ryusuke Hamazaki, Yuki Kawaguchi, Jae-yoon Choi

cond-mat.quant-gascond-mat.stat-mechquant-phSubmitted Mar 10, 2026• Announced Mar 11, 2026• 0 citations
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Abstract

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.

Comments
9 pages, 5 figures, and supplemental materials

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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.
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)SU(N)SU(N) spin chains, focusing on the well-known SU(2)SU(2)SU(2) XXZ model and the SU(3)SU(3)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)SU(N)SU(N) symmetry and integrability, we identify distinct transport regimes characterized by the dynamical exponent zzz: (i) ballistic transport with z=1z=1z=1, (ii) superdiffusive transport with the Kardar-Parisi-Zhang exponent z=3/2z=3/2z=3/2, and (iii) diffusive transport with the Edwards-Wilkinson exponent z=2z=2z=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)SU(N)SU(N) symmetry.
Family-Vicsek scaling is one of the most essential scale-invariant laws emerging in surface-roughness growth of classical systems. In this Letter, we theoretically elucidate the emergence of the Family-Vicsek scaling even in a strongly interacting quantum bosonic system by introducing a surface-height operator. This operator is comprised of a summation of local particle-number operators at a simultaneous time, and thus the observation of the surface roughness in the quantum many-body system and its scaling behavior are accessible to current experiments of ultracold atoms.
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.
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.
The investigation of the dynamical universality classes of quantum systems is an important, and rather less explored aspect of nonequilibrium physics. In this paper, considering the out-of-equilibrium dynamics of spinless fermions in a one-dimensional quasiperiodic model with and without periodic driving, we report the existence of the dynamical one-parameter Family-Vicsek (FV) scaling of the ``quantum surface roughness'' associated with the particle-number fluctuations. In the absence of periodic driving, the model is interestingly shown to host a subdiffusive critical phase with anomalous FV scaling exponents, separated by two subdiffusive critical lines and a triple point from other phases. Our analysis on the fate of the critical phase in the presence of (interphase) driving indicates that the critical phase is quite fragile, and has a tendency to get absorbed into the delocalized or localized regimes depending on the driving parameters, especially in the slow driving limit. Interestingly, periodic driving can also conspire to show quantum Kardar-Parisi-Zhang (KPZ)-like superdiffusive dynamical behavior, which seems to have no classical counterpart. We further construct an effective Floquet Hamiltonian, which qualitatively captures this feature occurring in the driven model.
Abstract Understanding the relaxation dynamics of isolated quantum systems remains a central challenge in nonequilibrium physics. In this short review, we discuss recent experimental and theoretical advances in universal dynamics of turbulent Bose gases, with particular emphasis on nonthermal fixed points and wave turbulence in our experimental system at the São Carlos Institute of Physics of the University of São Paulo. We highlight the emergence of self-similar scaling behavior and its connection to particle and energy transport across momentum scales. In addition, we discuss an approach in which a differential equation has a universal scaling solution, providing a practical framework for extracting universal exponents from limited regions of the momentum distribution. Furthermore, we recap experimental observations of direct and inverse cascades in a three–dimensional trapped Bose–Einstein condensate, revealing distinct relaxation stages governed by universal scaling laws. These results demonstrate that turbulence plays a key role in far–from–equilibrium dynamics, offering a unified perspective on transport processes and thermalization in quantum many–body systems.