Corrections to diffusion in interacting quantum systems

TL;DR

This work develops and tests an effective field theory (EFT) for diffusion to predict universal power-law and logarithmic corrections to diffusive transport in interacting classical and quantum systems. It derives full analytical scaling functions for the dynamical structure factor in 1D with single and multiple conserved densities, including Abelian and non-Abelian cases, and validates these predictions with large-scale classical simulations and quantum tensor-network simulations. The results demonstrate precise agreement in classical models and qualitative-to-quantitative agreement in quantum settings, with EFT-based fitting strategies significantly improving estimates of transport parameters at intermediate times. The framework generalizes to multiple charges and nonlinear response, offering a robust benchmark for fluctuating hydrodynamics and guiding future explorations of diffusion, subdiffusion, and nonlinear transport in complex quantum systems.

Abstract

Transport and the approach to equilibrium in interacting classical and quantum systems is a challenging problem of both theoretical and experimental interest. One useful organizing principle characterizing equilibration is the dissipative universality class, the most prevalent one being diffusion. In this paper, we use the effective field theory (EFT) of diffusion to systematically obtain universal power-law corrections to diffusion. We then employ large-scale simulations of classical and quantum systems to explore their validity. In particular, we find universal scaling functions for the corrections to the dynamical structure factor $\langle n(x,t)n\rangle$, in the presence of a single $U(1)$ or $SU(2)$ charge in systems with and without particle-hole symmetry, and present the framework to generalize the calculation to multiple charges. Classical simulations show remarkable agreement with EFT predictions for subleading corrections, pushing precision tests of effective theories for thermalizing systems to an unprecedented level. Moving to quantum systems, we perform large-scale tensor-network simulations in unitary and noisy 1d Floquet systems with conserved magnetization. We find a qualitative agreement with EFT which becomes quantitative in the case of noisy systems. Additionally, we show how the knowledge of EFT corrections allows for fitting methods, which can improve the estimation of transport parameters at the intermediate times accessible by simulations and experiments. Finally, we explore non-linear response in quantum systems and find that EFT provides an accurate prediction for its behavior. Our results provide a basis for a better understanding of the non-linear phenomena present in thermalizing systems.

Figures (9)

• Figure 1: (A) Nonlinear fluctuations of conserved densities are present in generic many-body systems, and (B) lead to universal corrections to hydrodynamics at late times. (C) In the case of a single diffusive density, the leading correction is positive, and can cause a diffusive system (with autocorrelation function illustrated in green) to appear superdiffusive (yellow, $z=3/2$ is shown above) at intermediate times. The EFT of diffusion predicts the coefficient of this correction $\tau = \frac{\chi^2 D'^2}{16\pi D^4}$, together with a universal scaling function of $x/\sqrt{Dt}$, see Eqs. \ref{['eq_nn_noph']} and \ref{['eq_F10']}.
• Figure 2: Left: 1-loop correction to diffusion. Right: 2-loop correction to diffusion at half-filling. The propagators $\langle nn\rangle_0(\omega,q)$ and $\langle n\phi_a\rangle_0(\omega,q)$ from Eq. \ref{['eq_propagators']} correspond to the solid lines, and half-solid half-dashed lines respectively.
• Figure 3: (A): Profile of the dynamical structure factor for the KLS model with parameters $\delta = \rho = 0.9, \epsilon = 0$. Different coloured curves denote different times $t \in \{200,2000\}$ with smaller times corresponding to darker colours. The red dashed curve is the diffusive prediction $\frac{\chi}{\sqrt{4\pi D}}F_{0,0}(y)$. Inset: Autocorrelation function ($y=0$). Diffusive predictions with (black), and without (red), leading order corrections, Eq. (\ref{['eq:autocor']}) are shown. (B): Comparison between the correction to diffusion from simulation data $\Delta n \equiv \sqrt{t}\left[\langle n(x,t)n\rangle - \frac{\chi}{\sqrt{4\pi D t}} F_{0,0}(y)\right]$ and the EFT prediction $F_{1,0}(y)$ (black, dashed), Eq. (\ref{['eq_nn_noph']}). Inset: Absolute area between the finite time curves and the analytic expression, $\mathcal{F}= \int^3_0 dy|\Delta n (t)- F_{1,0}|/\int^3_0 dy |F_{1,0}|$.
• Figure 4: Tensor network simulations of driven XXZ chain with (A,B) decoherence, $\gamma = 0.1$, and (C,D) staggered field, $g = 0.4$. (A,C) Diffusivity as a function of equilibrium magnetization. Top: different bond dimensions, $d$. Bottom: different simulated times $T$, using the same bond dimension $d = 256$. Different colours denote fitting methods which take into account an increasing amount of corrections to leading order diffusion, I$\rightarrow$ II$\rightarrow$III, see Appendix (\ref{['sec:fit']}). (B,D) Corrections to diffusion for $d = 256$, evaluated at different times $t = (40,80,\ldots,400)$, denoted by dark blue$\rightarrow$yellow colors. Black dashed line denotes linear corrections F$_{0,1}$. Red dashed line denotes F$_{1,0}$ in (D), and F$_{0,1}+\sqrt{T}$F$_{1,0}$ with $T= 400$ in (B). The brown dashed line in (D) denotes the combined effect of 2-loop correction and linear corrections F$_{0,2}+$F$_{1,0}$, where the logarithmic in time component of F$_{0,2}$ is evaluated at $T= 400$.
• Figure 5: Diagrams not contributing to transport corrections to diffusion: (a) Diagrams where the external momentum does not flow through a loop cannot produce new IR singularities; they only renormalize tree-level transport parameters. (b) The 1-loop contribution proportional to $\sigma'^2$ vanishes due to the latest time condition \ref{['eq_latesttime']}. (c) The two 1-loop contributions proportional to $\sigma' D'$ cancel. (d) A similar cancellation happens at 2-loop for the $\sigma" D"$ contribution.