Anomalous diffusion from hydrodynamic recoupling in particle-hole symmetric fluids

TL;DR

This work shows that in particle-hole symmetric fluids, such as the Dirac fluid in graphene, charge diffusion remains diffusive due to symmetry protection even as sound modes propagate ballistically. A three-mode hydrodynamic framework including energy, momentum, and charge reveals a singular zero-noise limit for the diffusion constant, with $D(\gamma_e,\gamma_p) \sim D_{\rm Fick}+D_{\rm conv} (\gamma_e/\gamma_p)^{1/2}$, and a divergence when momentum is conserved ($\gamma_p\to 0$, $\gamma_e>0$). An exact quasiparticle representation of the linear fluctuating hydrodynamics and a minimal stochastic gas model confirm this hydrodynamic recoupling, showing nonmonotonic and nonuniversal diffusion behavior depending on the relative rates of energy and momentum relaxation. These results imply that zero-noise extrapolation methods can fail for transport coefficients in such systems and motivate experimental tests in graphene ribbons and related multicomponent fluids. The findings also bridge linear fluctuating hydrodynamics with nonlinear coupled Burgers/KPZ dynamics in the weak-dissipation regime.

Abstract

In charged fluids obeying particle-hole symmetry, such as the Dirac fluid in graphene, charge transport is diffusive despite the presence of ballistically propagating sound waves: sound waves "hydrodynamically decouple" from the slower charge fluctuations. For quasi-one-dimensional fluids, we show that this symmetry-protected charge diffusion is not smoothly connected to the normal diffusion that arises when momentum conservation is broken by noise (or static impurities). Instead, the charge diffusion constant is a discontinuous function of noise, which (in the weak-noise limit) depends only on the ratio of momentum and energy relaxation rates. In the special limit of momentum-conserving noise (e.g., spatially uniform fluctuations of the Hamiltonian), the diffusion constant diverges in the presence of noise. We describe the resulting superdiffusion in terms of coupled Burgers equations. We present a general mechanism--hydrodynamic recoupling--by which weak noise can induce singular changes in transport coefficients. Our results highlight the limits of zero-noise extrapolation for predicting dynamical quantities like diffusion constants.

Figures (4)

• Figure 1: Singular behavior of the diffusion constant of quasi-one-dimensional two-component fluids as momentum and energy relaxation rates ($\gamma_e$ and $\gamma_p$ respectively) are tuned to zero. At point $\mathfrak{D}$, momentum and energy are conserved. As this point is approached along a ray in the $(\gamma_p, \gamma_e)$ plane, the diffusion constant converges to a ray-dependent value (Eq. \ref{['eq2']}), which diverges as $\gamma_p \to 0$. When $\gamma_p = 0$, charge transport is superdiffusive. The right panel shows the mechanism for a discontinuous diffusion constant: the autocorrelation function of the current develops a long timescale, due to repeated collisions of a charge element (marked in black) with sound waves. At the point "$\mathfrak{D}$" sound waves are ballistic and only collide once with each charge element (the resulting current autocorrelator is shown in gray). Energy (momentum) relaxation causes repeated collisions, giving rise to positive (negative) correlations. These correlations dominate the diffusion constant, which is proportional to the integrated current autocorrelation function.
• Figure 2: Singular charge diffusion at weak symmetry breaking. Numerical results from the stochastic gas model in the regime of weak energy and momentum relaxation. (Left) Time-dependent charge diffusion constant $D(t)$ extracted from equilibrium current fluctuations (see suppmat), shown versus rescaled time $t/\min(\gamma_e,\gamma_p)$ for two fixed ratios $r\equiv\gamma_e/\gamma_p$: for $r=4.25$ (red) we show $\gamma_e = 0.006, 0.009, 0.012$ (from dark to light); for $r=0.04$ (blue) we show $\gamma_e = 0.015, 0.0, 0.03$ (dark to light). The black curve corresponds to the fully conserved case $\gamma_e=\gamma_p=0$ (for comparison to the weak dissipation data, we rescale time axis by $0.02\,t$). The plateau value depends nontrivially on the approach direction to $(\gamma_e,\gamma_p)=(0,0)$, and can be either enhanced or suppressed relative to the fully conserved case. These simulations used systems of length $L=2000$. (Upper right) Total-current autocorrelator $L^{-1}\langle J(t)J(0)\rangle$ for $r=2$ (red), $r=0.02$ (blue), and fully conserved case $\gamma_e=\gamma_p=0$ (black), illustrating how the sign structure of current correlations (correlated versus anticorrelated) depends on the relative strength of the two relaxation channels. These simulations used $L=1000$. (Lower right) Weak-dissipation diffusion constant $D(r)$ obtained from the long-time plateau, plotted as a function of $r=\sqrt{\gamma_e/\gamma_p}$, showing agreement with the predicted square-root singularity in Eq. (9). These simulations used $L=2000$ and simulation times of $t=1000$.
• Figure 3: Pure momentum breaking ($\gamma_e=0$). (Main panel) Equilibrium charge-transfer variance ${\rm Var}(\Delta Q(t))$ (plotted as ${\rm Var}(Q_{\rm eq}(t))$) with charge-label swaps disallowed ($p_{\rm swap}=0$, so $D_{\rm Fick}=0$). For $\gamma_p=0$ the transfer is diffusive at long times, while any nonzero $\gamma_p$ produces a crossover on the timescale $\gamma_p^{-1}$ to subdiffusive growth ${\rm Var}(\Delta Q)\sim t^{1/4}$ (guide lines). These simulations used systems length $L=1500$. (Inset) Restoring charge-label swaps ($p_{\rm swap}=0.05$) yields a time-dependent diffusion constant $D(t)$ with two clear plateaus: an early-time plateau corresponding to the sum of the Fickian and convective contributions, and a late-time plateau set by the bare Fickian diffusion constant $D_{\rm Fick}$. This simulation used $L=6000$.
• Figure 4: Pure number/energy breaking ($\gamma_p=0$). (Left) Variances of equilibrium charge (black) and momentum transfer (red), $\langle \Delta Q(t)^2\rangle$ and $\langle \Delta P(t)^2\rangle$, in the number-breaking, momentum-conserving regime (here $\gamma_e=0.1$). Over the accessible time window the data show effective power-law growth with exponents $\alpha\approx 0.69$ for charge transfer and $\alpha\approx 0.62$ for momentum transfer (guide lines), consistent with an intermediate-time regime between the linear hydrodynamics estimate ($\Delta Q \sim t^{0.75}$, $\Delta P \sim t^{0.5}$), and the expected KPZ scaling ($\Delta Q\sim \Delta P \sim t^{2/3}$). (Right) Total-current autocorrelator $L^{-1}\langle J(t)J(0)\rangle$, which decays approximately as $t^{-1/2}$, indicating nonintegrable long-time tails and superdiffusive charge transport in this regime. These simulations used $L=1000$.