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Breakdown of the Large Deviation Principle in an Integrable Magnet

Žiga Krajnik, Enej Ilievski, Tomaž Prosen

Published Sep 27, 2021• 0 citations
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Abstract

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.

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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.
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.
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.
We study spin transport in the one-dimensional anisotropic S=1S=1S=1 Heisenberg model. Particular emphasis is given to dynamics at infinite temperature, where current autocorrelations and spatiotemporal correlation functions are obtained by means of an efficient pure-state approach based on the concept of typicality. Our comprehensive numerical analysis unveils that high-temperature spin transport is diffusive in the easy-axis regime for strong exchange anisotropies. This finding is based on the combination of numerous signatures, such as (i) Gaussian spreading of correlations, (ii) a time-independent diffusion coefficient, (iii) power-law decay of equal-site correlations, (iv) exponentially decaying long-wavelength modes, and (v) Lorentzian line shapes of the dynamical structure factor. Moreover, we provide evidence that some of these signatures are not exclusively restricted to the infinite-temperature limit but can persist at lower temperatures as well, where we complement our results by additional quantum Monte Carlo simulations of large systems. In contrast to the easy-axis regime, we show that in the case of an isotropic chain, the signatures (i)--(v) are much less pronounced or even entirely absent, suggesting the existence of anomalous spin transport despite the nonintegrability of the model. Eventually, upon introducing a random on-site magnetic field, we observe a breakdown of diffusion and distinctly slower dynamics. In particular, our results exhibit qualitative similarities to disordered spin-1/21/21/2 chains and might be consistent with the onset of many-body localization in the S=1S=1S=1 model for sufficiently strong disorder.
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.
We introduce a multi-species soliton gas in which each species has a distinct effective scattering length, and the repulsive scattering shift is set by the smaller of the two colliding particles. 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 particle size 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 \textit{particle trajectories}. For large particles, 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 this soliton gas as a minimal framework for anomalous spin and charge transport in integrable systems, and offer new insight into the origin of KPZ fluctuations in isotropic integrable models.
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.