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Physical Review Letters

Breakdown of hydrodynamics in the classical 1D Heisenberg model

O. F. de Alcantara Bonfim, George Reiter

Published Jul 13, 1992• 45 citations
Publisher

Abstract

Extensive spin-dynamics simulations have been performed to study the dynamical behavior of the classical Heisenberg chain at infinite temperatures and long wavelengths. We find that the energy and spin show distinctly different dynamics in the isotropic system. The energy correlation function follows the classical diffusion theory prediction, namely, it decays exponentially with ${\mathit{q}}^{2}$t. In contrast, the spin correlation function is found to decay exponentially as ${\mathit{q}}^{2.12}$t lnt, implying a logarithmically divergent diffusion constant and the failure of the usual hydrodynamic assumptions.

DOI
10.1103/physrevlett.69.367

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Breakdown of hydrodynamics in the classical 1D Heisenberg model (abstract)

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Extensive spin-dynamics simulations have been performed to study the dynamical behavior of the classical Heisenberg chain at infinite temperatures and long wavelengths. We find that the energy and spin show distinctly different dynamics in the isotropic system. The energy correlation function follows the classical diffusion theory prediction, namely, it decays exponentially with q2{\mathit{q}}^{2}q2t. In contrast, the spin correlation function is found to decay exponentially as q2.12{\mathit{q}}^{2.12}q2.12t lnt, implying a logarithmically divergent diffusion constant and the failure of the usual hydrodynamic assumptions.
Extensive spin-dynamics simulations have been performed to study the dynamical behavior of the classical Heisenberg chain at infinite temperatures and long wavelengths. We find that the energy and spin show distinctly different dynamics in the isotropic system. The energy correlation function follows the classical diffusion theory prediction, namely, it decays exponentially with q2t. In contrast, the spin correlation function is found to decay exponentially as q2.12t ln t implying a logarithmically divergent diffusion constant and the failure of the usual hydrodynamic assumptions.
Rigorous results are presented for the dynamics of the nearest-neighbour Heisenberg chain, for fixed but arbitrary wavevector and frequency, to lowest non-trivial order in the temperature. The authors obtain the two-spin correlation function, in the antiferromagnet for all wavevectors, and in the ferromagnet for wavelengths shorter than a coherence length. The energy correlation function at T=0 is obtained exactly, and is identical in the two systems. They find a violation of dynamical scaling in the form of the spectral function in the antiferromagnet for a range of wavevectors. An extension of the perturbative results for the ferromagnet to treat wavelengths longer than a coherence length shows that the diffusion coefficient vanishes logarithmically, in contrast to the scaling prediction that it is constant. An approximate value for the coefficient of the logarithmic term is obtained which agrees well with the results of simulations.
We use the functional renormalization approach for quantum spin systems developed by Krieg and Kopietz [Phys. Rev. B 99, 060403(R) (2019)] to calculate the spin-spin correlation function $G(\mathbit{k},\ensuremath{\omega})$ of quantum Heisenberg magnets at infinite temperature. For small wave vectors $\mathbit{k}$ and frequencies $\ensuremath{\omega}$ we find that $G(\mathbit{k},\ensuremath{\omega})$ assumes in dimensions d>2d>2d>2 the diffusive form predicted by hydrodynamics. In three dimensions our result for the spin-diffusion coefficient D\mathcal{D}D is somewhat smaller than previous theoretical predictions based on the extrapolation of the short-time expansion, but is still about 3030%30 larger than the measured high-temperature value of D\mathcal{D}D in the Heisenberg ferromagnet ${\mathrm{Rb}}_{2}{\mathrm{CuBr}}_{4}\ifmmode\cdot\else\textperiodcentered\fi{}2{\mathrm{H}}_{2}\mathrm{O}$. In reduced dimensions $d\ensuremath{\le}2$ we find superdiffusion characterized by a frequency-dependent complex spin-diffusion coefficient $\mathcal{D}(\ensuremath{\omega})$ which diverges logarithmically in d=2d=2d=2, and as a power-law $\mathcal{D}(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{\ensuremath{-}1/3}$ in d=1d=1d=1. Our result in one dimension implies scaling with dynamical exponent z=3/2z=3/2z=3/2, in agreement with recent calculations for integrable spin chains. Our approach is not restricted to the hydrodynamic regime and allows us to calculate the dynamic structure factor $S(\mathbit{k},\ensuremath{\omega})$ for all wave vectors. We show how the short-wavelength behavior of $S(\mathbit{k},\ensuremath{\omega})$ at high temperatures reflects the relative sign and strength of competing exchange interactions.
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}.