Anomalous hydrodynamic fluctuations in the quantum XXZ spin chain

Abstract

The quantum XXZ spin-1/2 chain features non-Gaussian spin current fluctuations in the regime of easy-axis anisotropy. Using ballistic macroscopic fluctuation theory, we derive the exact probability distribution of typical spin-current fluctuations in thermal equilibrium. The obtained nested Gaussian distribution is fully characterized by its variance which we analytically relate to the spin diffusion constant and static spin susceptibility, and compare our with numerical simulations. By unveiling how the same mechanism which leads to anomalous charge current fluctuations in single-file systems manifests in the XXZ chain, our approach establishes the universal hydrodynamic origin of the observed anomalous fluctuations.

Figures (3)

• Figure 1: Origin of anomalous fluctuations of the integrated spin current. At the hydrodynamic scale, the magnetization of microscopic spin excitations is captured by a large-scale classical fluctuating field $\delta q_0(x,t)$ (blue curves). Anomalous fluctuations stem from the nesting of two sources of Gaussian fluctuations. Transported magnetization equals the fluctuating magnetization in the interval of length $|X(t)|$ (highlighted in red over the fluctuating magnetization), where $X(t)$ (black curve) is the trajectory of a giant magnon initially placed at the origin, and that itself fluctuates due to scattering with other modes.
• Figure 2: Variance of the time-integrated spin current. We consider the integrable Trotterization of the XXZ spin chain with time step $\tau$. Numerical data for the spin-current variance (colored crosses) are computed with the quantum generating function approach Valli_2025, and are compared with the hydrodynamic prediction $\langle J^2\rangle^c(T) \simeq T^{1/2 }\sigma \sqrt{2/\pi}$ where $\sigma^2$ is given by Eq. \ref{['eq_variance']} (solid lines). Error bars show 90% confidence intervals, see End Matter for details on the numerical simulations.
• Figure S1: (left panel) Time evolution of rescaled variance of the integrated spin current at different bond dimensions $\chi$ (coloured curves) at $\Delta = 2$, $\tau = 1$. Dashed black lines show the best two-parameter fits \ref{['c_time_fit']}. Purple curve shows data from Ref. Valli_2025 for longer times and $\chi=1024$, supporting the expansion \ref{['c_time_fit']}. (right panel) Time-extrapolated values $c_2^{[0]}$ (coloured crosses) show an approximately linear dependence on $\chi^{-1}$ for intermediate bond dimensions and saturation at the largest bond dimension. Black line show best linear fit \ref{['LS_def']} for intermediate bond dimensions. The point of saturation $\chi_\infty$ is estimated by extrapolating to the saturated value.