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arXiv:2108.11964 · Physical review. B./Physical review. B

Anomalous dynamics and equilibration in the classical Heisenberg chain

Adam J. McRoberts, Thomas Bilitewski, Masudul Haque, Roderich Moessner

Published Mar 4, 2022• 28 citations
arXivHTMLPDF

Abstract

The search for departures from standard hydrodynamics in many-body systems has yielded a number of promising leads, especially in low dimensions. Here, we study one of the simplest classical interacting lattice models, the nearest-neighbor Heisenberg chain, with temperature as the tuning parameter. Our numerics expose strikingly different spin dynamics between the antiferromagnet, where it is largely diffusive, and the ferromagnet, where we observe strong evidence either of spin superdiffusion or an extremely slow crossover to diffusion. This difference also governs the equilibration after a quench, and, remarkably, is apparent even at very high temperatures.

DOI
10.1103/physrevb.105.l100403

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The search for departures from standard hydrodynamics in many-body systems has yielded a number of promising leads, especially in low dimensions. Here, we study one of the simplest classical interacting lattice models, the nearest-neighbor Heisenberg chain, with temperature as the tuning parameter. Our numerics expose strikingly different spin dynamics between the antiferromagnet, where it is largely diffusive, and the ferromagnet, where we observe strong evidence either of spin superdiffusion or an extremely slow crossover to diffusion. This difference also governs the equilibration after a quench, and, remarkably, is apparent even at very high temperatures.
Superdiffusion is surprisingly easily observed even in systems without the integrability underpinning this phenomenon. Indeed, the classical Heisenberg chain-one of the simplest many-body systems, and firmly believed to be nonintegrable-evinces a long-lived regime of anomalous, superdiffusive spin dynamics at finite temperature. Similarly, superdiffusion persists for long timescales, even at high temperature, for small perturbations around a related integrable model. Eventually, however, ordinary diffusion is believed to be asymptotically restored. We examine the timescales governing the lifetime of the superdiffusive regime, and argue that it diverges algebraically fast-both in deviation from the integrable limit, and at low temperature, where we find t^{*}∼T^{-ζ} with an exponent possibly as large as ζ=8. This can render the crossover to ordinary diffusion practically inaccessible.
The problem of characterizing low-temperature spin dynamics in antiferromagnetic spin chains has so far remained elusive. Here we reinvestigate it by focusing on isotropic antiferromagnetic chains whose low-energy effective field theory is governed by the quantum nonlinear sigma model. Employing an exact nonperturbative theoretical approach, we analyze the low-temperature behavior in the vicinity of nonmagnetized states and obtain exact expressions for the spin diffusion constant and the NMR relaxation rate, which we compare with previous theoretical results in the literature. Surprisingly, in SU(2)-invariant spin chains in the vicinity of half filling we find a crossover from the semiclassical regime to a strongly interacting quantum regime characterized by zero spin Drude weight and diverging spin conductivity, indicating superdiffusive spin dynamics. The dynamical exponent of spin fluctuations is argued to belong to the Kardar-Parisi-Zhang universality class. Furthermore, by employing numerical time-dependent density matrix renormalization group simulations, we find robust evidence that the anomalous spin transport persists also at high temperatures, irrespective of the spectral gap and integrability of the model.
Finite-temperature spin transport in integrable isotropic spin chains (i.e., spin chains with continuous non-Abelian symmetries) is known to be superdiffusive, with anomalous transport properties displaying remarkable robustness to isotropic integrability-breaking perturbations. Using a discrete-time classical model, we numerically study the crossover to conventional diffusion resulting from both noisy and Floquet isotropic perturbations of strength $\ensuremath{\lambda}$. We identify an anomalously-long crossover timescale ${t}_{★}\ensuremath{\sim}{\ensuremath{\lambda}}^{\ensuremath{-}\ensuremath{\alpha}}$ with $\ensuremath{\alpha}\ensuremath{\approx}6$ in both cases. We discuss our results in terms of a kinetic theory of transport that characterizes the lifetimes of large solitons responsible for superdiffusion.
We study the transport and equilibration properties of a classical Heisenberg chain, whose couplings are random variables drawn from a one-parameter family of power-law distributions. The absence of a scale in the couplings makes the system deviate substantially from the usual paradigm of diffusive spin hydrodynamics and exhibit a regime of subdiffusive transport with an exponent changing continuously with the parameter of the distribution. We propose a solvable phenomenological model that correctly yields the subdiffusive exponent, thereby linking local fluctuations in the coupling strengths to the long-time, large-distance behavior. It also yields the finite-time corrections to the asymptotic scaling, which can be important in fitting the numerical data. We show how such exponents undergo transitions as the distribution of the coupling gets wider, marking the passage from diffusion to a regime of slow diffusion, and finally to subdiffusion.
The isotropic Heisenberg chain represents a particular case of an integrable many-body system exhibiting superdiffusive spin transport at finite temperatures. Here, we show that this model has distinct properties also at finite magnetization $m\ensuremath{\ne}0$, even upon introducing the SU(2) invariant perturbations. Specifically, we observe nonmonotonic dependence of the diffusion constant ${\mathcal{D}}_{0}(\mathrm{\ensuremath{\Delta}})$ on the spin anisotropy $\mathrm{\ensuremath{\Delta}}$, with a pronounced maximum at $\mathrm{\ensuremath{\Delta}}=1$. The latter dependence remains true also in the zero magnetization sector, with superdiffusion at $\mathrm{\ensuremath{\Delta}}=1$ that is remarkably stable against isotropic perturbation (at least in finite-size systems), consistent with recent experiments with cold atoms.
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.