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arXiv:2602.24242 · Physical Review Letters

Anomalous Hydrodynamic Fluctuations in the Quantum XXZ Spin Chain

Takato Yoshimura, Žiga Krajnik, Alvise Bastianello, Enej Ilievski

Published Jul 22, 2026• 0 citations
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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 asymptotic probability distribution of typical spin-current fluctuations in thermal equilibrium at zero average magnetization. The obtained nested Gaussian distribution is fully characterized by its variance, which we analytically relate to the spin diffusion constant and static spin susceptibility, comparing our prediction with numerical simulations. By unveiling how the same mechanism which leads to anomalous charge current fluctuations in single-file systems manifests itself in the XXZ chain, our approach establishes the universal hydrodynamic origin of the observed anomalous fluctuations.

DOI
10.1103/jrtg-yf1q

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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 asymptotic probability distribution of typical spin-current fluctuations in thermal equilibrium at zero average magnetization. The obtained nested Gaussian distribution is fully characterized by its variance, which we analytically relate to the spin diffusion constant and static spin susceptibility, comparing our prediction with numerical simulations. By unveiling how the same mechanism which leads to anomalous charge current fluctuations in single-file systems manifests itself in the XXZ chain, our approach establishes the universal hydrodynamic origin of the observed anomalous fluctuations.
We study spin-current fluctuations of the folded XXZ chain, which describes the strong-anisotropy limit of the XXZ chain. We obtain an exact expression for the moment-generating function of the time-integrated spin current at infinite temperature. A long-time asymptotic analysis establishes that the rescaled time-integrated current converges to a non-Gaussian distribution described by the M-Wright function. The derivation relies on a nonlocal map to the one-dimensional t-0 model and its exact spin-charge separation. By extending this approach, we also show that the dynamical correlation function of the spins summed over two adjacent sites exhibits a diffusive Gaussian profile at long times.
We address the nature of spin transport in the integrable XXZ spin chain, focusing on the isotropic Heisenberg limit. We calculate the diffusion constant using a kinetic picture based on generalized hydrodynamics combined with Gaussian fluctuations: we find that it diverges, and show that a self-consistent treatment of this divergence gives superdiffusion, with an effective time-dependent diffusion constant that scales as D(t)∼t^{1/3}. This exponent had previously been observed in large-scale numerical simulations, but had not been theoretically explained. We briefly discuss XXZ models with easy-axis anisotropy Δ>1. Our method gives closed-form expressions for the diffusion constant D in the infinite-temperature limit for all Δ>1. We find that D saturates at large anisotropy, and diverges as the Heisenberg limit is approached, as D∼(Δ-1)^{-1/2}.
The anomalous nature of spin transport in the XXZ quantum spin chain has been a topic of theoretical interest for some time. Here, the integrability of the underlying dynamics leads to a ballistic component of the spin current, characterized by a spin Drude weight, which measures the degree of divergence of the zero-frequency spin conductivity. However, this quantity had previously proven to be beyond the reach of standard Bethe ansatz techniques. Here, the authors show that a recently developed hydrodynamic formalism for quantum integrable models may be used to compute the spin Drude weight. They also propose a numerical scheme to obtain hydrodynamic predictions for finite-time energy transport. This suggests that the hydrodynamic approach captures completely the ballistic component that dominates transport at long times and distances in the gapless regime of the XXZ model.
Many experimentally relevant quantum spin chains are approximately integrable, and support long-lived quasiparticle excitations. A canonical example of integrable model of quantum magnetism is the XXZ spin chain, for which energy spreads ballistically, but, surprisingly, spin transport can be diffusive or superdiffusive. We review the transport properties of this model using an intuitive quasiparticle picture that relies on the recently introduced framework of generalized hydrodynamics. We discuss how anomalous linear response properties emerge from hierarchies of quasiparticles both in integrable and near-integrable limits, with an emphasis on the role of hydrodynamic fluctuations. We also comment on recent developments including non-linear response, full-counting statistics and far-from-equilibrium transport. We provide an overview of recent numerical and experimental results on transport in XXZ spin chains.
A universal characterization of non-equilibrium steady states in interacting quantum many-body systems remains one of the central challenges of statistical physics. Here, we address this problem for a paradigmatic model of diffusive interacting quantum matter---the boundary-driven XXZ spin chain with bulk dephasing---and derive, directly from its microscopic Lindblad dynamics, an emergent classical Macroscopic Fluctuation Theory (MFT) governing its large-scale fluctuations. Crucially, the resulting hydrodynamics carries a density-dependent diffusivity and mobility as the fingerprint of interactions. This effective description enables the exact computation of the stationary density profile, long-range correlations, and the full counting statistics of the current, in excellent agreement with tensor-network simulations. Our work demonstrates that noisy quantum many-body systems can realize the universality class of genuinely interacting diffusive matter, beyond the constant-diffusivity class of the symmetric simple exclusion process, and establishes MFT as a powerful universal framework for interacting diffusive quantum systems.
Generalized hydrodynamics predicts universal ballistic transport in integrable lattice systems when prepared in generic inhomogeneous initial states. However, the ballistic contribution to transport can vanish in systems with additional discrete symmetries. Here we perform large scale numerical simulations of spin dynamics in the anisotropic Heisenberg XXZ spin 1/2 chain starting from an inhomogeneous mixed initial state which is symmetric with respect to a combination of spin reversal and spatial reflection. In the isotropic and easy-axis regimes we find non-ballistic spin transport which we analyse in detail in terms of scaling exponents of the transported magnetization and scaling profiles of the spin density. While in the easy-axis regime we find accurate evidence of normal diffusion, the spin transport in the isotropic case is clearly super-diffusive, with the scaling exponent very close to 2/3, but with universal scaling dynamics which obeys the diffusion equation in nonlinearly scaled time.