Anomalous Diffusion and Superdiffusion in Integrable Spin Chains via a Hard-Rod Gas Mapping

Abstract

We introduce a multi-species generalization of the hard-rod gas in which each species has a distinct effective length, and the repulsive scattering shift is set by the smaller of the two colliding rods. We argue that this model shares key quasiparticle and scattering features with the XXZ spin chain. We show that fixing only the functional decay of bare velocities with rod length is sufficient to reproduce the XXZ spin-transport phase diagram: diffusion (with anomalous fluctuations) in the anisotropic regime and superdiffusion at the isotropic point. We then demonstrate that the statistics of charge transfer differs qualitatively from that of particle trajectories. For long rods, trajectories are Gaussian in the diffusive regime and appear to exhibit KPZ statistics at the isotropic point, providing a direct microscopic signature of KPZ physics in integrable quasiparticle motion. In contrast, charge-transfer fluctuations are anomalous in the anisotropic regime, while they cross over to Gaussian statistics at late times at the isotropic point, reconciling non-Gaussian trajectory fluctuations with Gaussian charge-transfer statistics. Our results establish classical hard-rod dynamics as a minimal yet quantitatively faithful framework for anomalous spin and charge transport in integrable systems, and offer new insight into the origin of KPZ fluctuations in isotropic integrable models.

Figures (9)

• Figure 1: Particle trajectories for $\Delta = 8$ and $\Delta = 1$ for different sizes $s$. Due to the distinct bare-velocity behavior, the motion of large particles in the $\Delta = 8$ case is dominated by collisions with small particles with $s=1$, resulting in random-walk-like behavior. In the case of $\Delta = 1$, large particles of size $s$ experience jumps $O(s)$ caused by scattering with other large ones, leading to non-linear, long-time tails.
• Figure 2: Plot of the distribution of fluctuations $\mathcal{P}(\delta x_{s})$ in Eq. \ref{['eq:displacement']} for right-moving particles $\sigma=1$ (also averaged with the left-movers with $\xi \to -\xi$) and type $s =15 - 30$ for $\Delta =1$ and $s = 5-10$ for $\Delta = 8$. The distribution is compared to a normalized Gaussian distribution and to a skewed KPZ distributions (Baik--Rains). All distributions are rescaled to have unit variance and zero mean. The right panel compares the approach to the asymptotic diffusion $t\Delta D_{s,t} = t(D_s(t) - D_s)$ (see Eq. \ref{['eq:finitetimediff']}) rescaled with particle type and plotted against rescaled time $t/s^3$ and unscaled time $t$ for the $\Delta = 1$ and $\Delta = 8$ cases, respectively (see also Fig. \ref{['fig:dxdx_no_subtraction']} and \ref{['fig:smallStringPDF']}).
• Figure 3: Behavior of the magnetization transfer for each species (see Eq. \ref{['eq:dm-def']}), $\langle \delta m _s^2 \rangle$ (left) and summed off-diagonal elements $C_s = \sum_{s' \neq s} \langle \delta m_s \delta m_{s'} \rangle / s^2$ (right) for both $\Delta = 1$ and $\Delta = 8$, demonstrating the presence of a transition from superdiffusive to ballistic behavior for $\Delta=1$ (top) and from diffusive to ballistic for $\Delta>1$ (bottom).
• Figure 4: Plots of rescaled distributions of magnetic fluctuations for specific string species $\mathcal{P}_t(\delta m_s)$ and the total magnetic fluctuations due to all particle species $\mathcal{P}_t(\delta m)$, see Eq. \ref{['eq:dm-def']} (right-most column) for both $\Delta = 1$ (top) and $\Delta = 8$ (bottom). These are compared with both the anomalous non-Gaussian distribution of Eq. \ref{['eq:anomalous']} (with $\mathcal{P}_t$ the KPZ scaling function $f_{\rm KPZ}$ for $\Delta=1$ and the normal distribution for $\Delta=8$) and a normal distribution. All distributions are rescaled to have unit variance. The insets show the variance as a function of $t$.
• Figure 5: Rescaled correlation function $C(x,t) = \langle m(x,t) m(0,0) \rangle_{\rm c}$ for both $\Delta = 8$ and $\Delta = 1$. In either anisotropy, the scaling is significantly different between the anisotropies. For $\Delta = 8$ the evolution is well described by convolving the initial correlation with the diffusion kernel (continuous line). For $\Delta=1$ the correct convolution is not known, and we show the disagreement with the (naive) distribution $f_{\rm KPZ}$. The insets show the value of the equilibrium correlation at $t=0$ showing that it is not a $\delta(x)$ function.