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arXiv:2012.12210 · Physical review. E

Kardar-Parisi-Zhang universality in two-component driven diffusive models: Symmetry and renormalization group perspectives

Pritha Dolai, Aditi Simha, Abhik Basu

Published Jun 7, 2024• 1 citation
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Abstract

We elucidate the universal spatiotemporal scaling properties of the time-dependent correlation functions in a class of two-component one-dimensional (1D) driven diffusive system that consists of two coupled asymmetric exclusion processes. By using a perturbative renormalization group framework, we show that the relevant scaling exponents have values same as those for the 1D Kardar-Parisi-Zhang (KPZ) equation. We connect these universal scaling exponents with the symmetries of the model equations. We thus establish that these models belong to the 1D KPZ universality class.

DOI
10.1103/physreve.109.064122

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We elucidate the universal spatiotemporal scaling properties of the time-dependent correlation functions in a class of two-component one-dimensional (1D) driven diffusive system that consists of two coupled asymmetric exclusion processes. By using a perturbative renormalization group framework, we show that the relevant scaling exponents have values same as those for the 1D Kardar-Parisi-Zhang (KPZ) equation. We connect these universal scaling exponents with the symmetries of the model equations. We thus establish that these models belong to the 1D KPZ universality class.
We investigate the universal behavior of the Kardar-Parisi-Zhang (KPZ) equation with temporally correlated noise. The presence of time correlations in the microscopic noise breaks the statistical tilt symmetry, or Galilean invariance, of the original KPZ equation with δ-correlated noise (denoted SR-KPZ). Thus, it is not clear whether the KPZ universality class is preserved in this case. Conflicting results exist in the literature, some advocating that it is destroyed even in the limit of infinitesimal temporal correlations, while others find that it persists up to a critical range of such correlations. Using nonperturbative and functional renormalization group techniques, we study the influence of two types of temporal correlators of the noise: a short-range one with a typical timescale τ, and a power-law one with a varying exponent θ. We show that for the short-range noise with any finite τ, the symmetries (the Galilean symmetry, and the time-reversal one in 1+1 dimension) are dynamically restored at large scales, such that the long-distance and long-time properties are governed by the SR-KPZ fixed point. In the presence of a power-law noise, we find that the SR-KPZ fixed point is still stable for θ below a critical value θ_{th}, in accordance with previous renormalization group results, while a long-range fixed point controls the critical scaling for θ>θ_{th}, and we evaluate the θ-dependent critical exponents at this long-range fixed point, in both 1+1 and 2+1 dimensions. While the results in 1+1 dimension can be compared with previous studies, no other prediction was available in 2+1 dimension. We finally report in 1+1 dimension the emergence of anomalous scaling in the long-range phase.
Isotropic integrable spin chains such as the Heisenberg model feature superdiffusive spin transport belonging to an as-yet-unidentified dynamical universality class closely related to that of Kardar, Parisi, and Zhang (KPZ). To determine whether these results extend to more generic one-dimensional models, particularly those realizable in quantum simulators, we investigate spin and energy transport in non-integrable, long-range Heisenberg models using state-of-the-art tensor network methods. Despite the lack of integrability and the asymptotic expectation of diffusion, for power-law models (with exponent 2<α<∞2 < α< \infty2<α<∞) we observe long-lived z=3/2z=3/2z=3/2 superdiffusive spin transport and two-point correlators consistent with KPZ scaling functions, up to times t∼103/Jt \sim 10^3/Jt∼103/J. We conjecture that this KPZ-like transport is due to the proximity of such power-law-interacting models to the integrable family of Inozemtsev models, which we show to also exhibit KPZ-like spin transport across all interaction ranges. Finally, we consider anisotropic spin models naturally realized in Rydberg atom arrays and ultracold polar molecules, demonstrating that a wide range of long-lived, non-diffusive transport can be observed in experimental settings.
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.
Integrable spin chains with a continuous non-Abelian symmetry, such as the one-dimensional isotropic Heisenberg model, show superdiffusive transport with little theoretical understanding. Although recent studies reported a surprising connection to the Kardar-Parisi-Zhang (KPZ) universality class in that case, this view was most recently questioned by discrepancies in full counting statistics. Here, by combining extensive numerical simulations of classical and quantum integrable isotropic spin chains with a framework developed by exact studies of the KPZ class, we characterize various two-point quantities that remain hitherto unexplored in spin chains, and find full agreement with KPZ scaling laws without adjustable parameters. This establishes the partial emergence of the KPZ class in integrable isotropic spin chains. Moreover, we reveal that the KPZ scaling laws are intact in the presence of an energy current, under the appropriate Galilean boost required by the propagation of spacetime correlation.
Anomalous transport in one-dimensional translation invariant hamiltonian systems with short range interactions is shown to belong in general to the Kardar-Parisi-Zhang universality class. Exact asymptotic forms for density-density and current-current time correlation functions and their Fourier transforms are given in terms of the Prähofer-Spohn scaling functions, obtained from their exact solution for the polynuclear growth model. The exponents of corrections to scaling are found as well, but not so the coefficients. Mode coupling theories developed previously are found to be adequate for weakly nonlinear chains but in need of corrections for strongly anharmonic interparticle potentials. A simple condition is given under which Kardar-Parisi-Zhang behavior does not apply, sound attenuation is only logarithmically superdiffusive, and heat conduction is more strongly superdiffusive than under Kardar-Parisi-Zhang behavior.