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arXiv:2307.03807 · Physical review. B./Physical review. B

Family-Vicsek dynamical scaling and Kardar-Parisi-Zhang-like superdiffusive growth of surface roughness in a driven one-dimensional quasiperiodic model

Sreemayee Aditya, Nilanjan Roy

Published Jan 30, 2024• 21 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.109.035164

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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.
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.
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 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.
We construct a family of chaotic brickwork quantum circuits exhibiting superdiffusive transport with dynamical exponent z=3/2z=3/2z=3/2 in the full Kardar--Parisi--Zhang (KPZ) universality class. This is enabled by the equilibrium current associated with a inhomogeneous conserved density having a nontrivial dependence on the chemical potential. Using nonlinear fluctuating hydrodynamics, we derive the propagation velocity, broadening scale, and universal scaling form of the charge correlation function from microscopic equilibrium properties. Tensor-network simulations of a two-site spin-111 (qutrit) circuit reproduce these parameter-free predictions, including the full stationary KPZ profile. We further construct a three-site spin-12\frac1221​ (qubit) circuit, where transport varies from diffusive to superdiffusive with the chemical potential, while long-lived coherent quasiparticles delay the convergence of the profile to the KPZ scaling form. Our results establish that KPZ superdiffusion in quantum systems is not restricted to fine-tuned integrable models.
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.
The one-dimensional Kardar-Parisi-Zhang (KPZ) equation is becoming an overarching paradigm for the scaling of nonequilibrium, spatially extended, classical and quantum systems with strong correlations. Recent analytical solutions have uncovered a rich structure regarding its scaling exponents and fluctuation statistics. However, the zero surface tension or zero viscosity case eludes such analytical solutions and has remained ill-understood. Using numerical simulations, we elucidate a well-defined universality class for this case that differs from that of the viscous case, featuring intrinsically anomalous kinetic roughening (despite previous expectations for systems with local interactions and time-dependent noise) and ballistic dynamics. The latter may be relevant to recent quantum spin chain experiments which measure KPZ and ballistic relaxation under different conditions. We identify the ensuing set of scaling exponents in previous discrete interface growth models related with isotropic percolation, and show it to describe the fluctuations of additional continuum systems related with the noisy Korteweg-de Vries equation. Along this process, we additionally elucidate the universality class of the related inviscid stochastic Burgers equation.