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arXiv:2109.13088 · Physical Review Letters

Absence of Normal Fluctuations in an Integrable Magnet

Žiga Krajnik, Enej Ilievski, Tomaž Prosen

Published Mar 3, 2022• 57 citations
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Abstract

We investigate dynamical fluctuations of transferred magnetization in the one-dimensional lattice Landau-Lifshitz magnet with uniaxial anisotropy, representing an emblematic model of interacting spins. We demonstrate that the structure of fluctuations in thermal equilibrium depends radically on the characteristic dynamical scale. In the ballistic regime, typical fluctuations are found to follow a normal distribution and scaled cumulants are finite. In stark contrast, on the diffusive and superdiffusive timescales, relevant, respectively, for the easy-axis and isotropic magnet at vanishing total magnetization, typical fluctuations are no longer Gaussian and, remarkably, scaled cumulants are divergent. The observed anomalous features disappear upon breaking integrability, suggesting that the absence of normal fluctuations is intimately tied to the presence of soliton modes. In a nonequilibrium setting of the isotropic magnet with weakly polarized step-profile initial state we find a slow drift of dynamical exponent from the superdiffusive towards the diffusive value.

DOI
10.1103/physrevlett.128.090604

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We investigate dynamical fluctuations of transferred magnetization in the one-dimensional lattice Landau-Lifshitz magnet with uniaxial anisotropy, representing an emblematic model of interacting spins. We demonstrate that the structure of fluctuations in thermal equilibrium depends radically on the characteristic dynamical scale. In the ballistic regime, typical fluctuations are found to follow a normal distribution and scaled cumulants are finite. In stark contrast, on the diffusive and superdiffusive timescales, relevant, respectively, for the easy-axis and isotropic magnet at vanishing total magnetization, typical fluctuations are no longer Gaussian and, remarkably, scaled cumulants are divergent. The observed anomalous features disappear upon breaking integrability, suggesting that the absence of normal fluctuations is intimately tied to the presence of soliton modes. In a nonequilibrium setting of the isotropic magnet with weakly polarized step-profile initial state we find a slow drift of dynamical exponent from the superdiffusive towards the diffusive value.
We investigate typical and rare dynamical fluctuations of transferred magnetization in the one-dimensional lattice Landau--Lifshitz magnet with uniaxial anisotropy, representing an emblematic model of interacting spins. We demonstrate that the structure of fluctuations in thermal equilibrium depends radically on the characteristic dynamical scale. In the ballistic regime, typical fluctuations are found to follow a normal distribution, with atypical fluctuations satisfying the large-deviation principle. In stark contrast, on the diffusive and superdiffusive timescales, relevant respectively for the easy-axis and isotropic magnet at vanishing total magnetization, typical fluctuations are no longer Gaussian as a consequence of strong violation of the large-deviation principle. The observed anomalous features disappear upon breaking integrability, suggesting that the breakdown of the large-deviation principle is intimately tied to the presence of soliton modes.
Recent studies have found that fluctuations of magnetization transfer in integrable spin chains violate the central limit property. Here, we revisit the problem of anomalous counting statistics in the Landau-Lifshitz field theory by specializing to two distinct anomalous regimes featuring a dynamical critical point. By performing optimized numerical simulations using an integrable space-time discretization, we extract the algebraic growth exponents of time-dependent cumulants which attain their threshold values. The distinctly non-Gaussian statistics of magnetization transfer in the easy-axis regime is found to converge toward the universal distribution of charged single-file systems. At the isotropic point, we infer a weakly non-Gaussian distribution, corroborating the view that superdiffusive spin transport in integrable spin chains does not belong to any known dynamical universality class.
Recent studies report on anomalous spin transport for the integrable Heisenberg spin chain at its isotropic point. Anomalous scaling is also observed in the time evolution of nonequilibrium initial conditions, the decay of current-current correlations, and nonequilibrium steady state averages. These studies indicate a space-time scaling with x∼t^{2/3} behavior at the isotropic point, in sharp contrast to the ballistic form x∼t generically expected for integrable systems. In our contribution we study the scaling behavior for the integrable lattice Landau-Lifshitz spin chain. We report on equilibrium spatiotemporal correlations and dynamics with step initial conditions. Remarkably, for the case with zero mean magnetization, we find strong evidence that the scaling function is identical to the one obtained from the stationary stochastic Burgers equation, alias Kardar-Parisi-Zhang equation. In addition, we present results for the easy-plane and easy-axis regimes for which, respectively, ballistic and diffusive spin transport is observed, whereas the energy remains ballistic over the entire parameter regime.
It is widely accepted that local subsystems in isolated integrable quantum systems equilibrate to generalized Gibbs ensembles. Here, we identify a particular class of initial states in interacting integrable models that evade canonical generalized thermalization. Particularly, we demonstrate that in the easy-axis regime of the quantum XXZ chain, pure nonequilibrium initial states that lack magnetic fluctuations instead locally relax to squeezed generalized Gibbs ensembles governed by nonlocal equilibrium Hamiltonians, representing exotic equilibrium states with subextensive charge fluctuations that violate the self-affine scaling. At the isotropic point, we find exceptional behavior and explicit dependence on the initial state. Particularly, we find that relaxation from the Néel state is governed by extensive fluctuations and a superdiffusive dynamical exponent compatible with the Kardar-Parisi-Zhang universality. On the other hand, there are other nonfluctuating initial states that display diffusive scaling, e.g., a product state of spin singlets. Our predictions provide examples of anomalous quantum transport and fluctuations in strictly quantum states which can be directly tested in state-of-the-art cold atomic experimental settings.
Diffusive transport is a ubiquitous phenomenon, yet the microscopic origin of diffusion in interacting physical systems remains a challenging question, irrespective of whether quantum effects are dominant or not. In this work, we study infinite temperature spin diffusion in a classical integrable, space-time discrete version of anisotropic Landau-Lifshitz magnet in the easy-axis regime, subjected to integrability-breaking perturbations. Our numerical results based on large-scale simulations reveal i) a sharp change in the spin diffusion constant as a function of perturbation strength in the thermodynamic limit and ii) a crossover from non-Gaussian to Gaussian statistics of magnetization transfer reflected in higher order cumulants under integrability breaking. Both our observations hint to the presence of non-trivial diffusion mechanism inherent to integrable systems.
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.