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.
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.
Superdiffusive transport with dynamical exponent
z=3/2 has been firmly established at finite temperature for a class of integrable systems with a non-abelian global symmetry
G. On the inclusion of integrability-breaking perturbations, diffusive transport with
z=2 is generically expected to hold in the limit of late time. Recent studies of the classical Haldane-Ishimori-Skylanin model have found that perturbations that preserve the global symmetry lead to a much slower timescale for the onset of diffusion, albeit with uncertainty over the exact scaling exponent. That is, for perturbations of strength
λ, the characteristic timescale for diffusion goes as
t∗∼λ−α for some
α. Using large-scale matrix product state simulations, we investigate this behavior for perturbations to the canonical quantum model showing superdiffusion: the
S=1/2 quantum Heisenberg chain. We consider a ladder configuration and look at various perturbations that either break or preserve the
SU(2) symmetry, leading to scaling exponents consistent with those observed in one classical study arXiv:2402.18661:
α=2 for symmetry-breaking terms and
α=6 for symmetry-preserving terms. We also consider perturbations from another integrable point of the ladder model with
G=SU(4) and find consistent results. Finally, we consider a generalization to an
SU(3) ladder and find that the
α=6 scaling appears to be universal across superdiffusive systems when the perturbations preserve the non-abelian symmetry
G.
Superdiffusive finite-temperature transport has been recently observed in a variety of integrable systems with non-Abelian global symmetries. Superdiffusion is caused by giant Goldstone-like quasiparticles stabilized by integrability. Here, we argue that these giant quasiparticles remain long-lived and give divergent contributions to the low-frequency conductivity σ(ω), even in systems that are not perfectly integrable. We find, perturbatively, that σ(ω)∼ω^{-1/3} for translation-invariant static perturbations that conserve energy and σ(ω)∼|logω| for noisy perturbations. The (presumable) crossover to regular diffusion appears to lie beyond low-order perturbation theory. By contrast, integrability-breaking perturbations that break the non-Abelian symmetry yield conventional diffusion. Numerical evidence supports the distinction between these two classes of perturbations.
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.
The dynamics of spin at finite temperature in the spin-1/2 Heisenberg chain was found to be superdiffusive in numerous recent numerical and experimental studies. Theoretical approaches to this problem have emphasized the role of nonabelian SU(2) symmetry as well as integrability, but the associated methods cannot be readily applied when integrability is broken. We examine spin transport in a spin-1/2 chain in which the exchange couplings fluctuate in space and time around a nonzero mean J, a model introduced by De Nardis et al. [Phys. Rev. Lett. 127, 057201 (2021).PRLTAO0031-900710.1103/PhysRevLett.127.057201]. We show that operator dynamics in the strong noise limit at infinite temperature can be analyzed using conventional perturbation theory as an expansion in J. We find that regular diffusion persists at long times, albeit with an enhanced diffusion constant. The finite time spin dynamics is analyzed and compared with matrix product operator simulations.
Abstract We show that there are two distinct mechanisms that can cause the symmetry-restoration Mpemba effect in spin chains with weakly broken integrability, such that the asymptotic equilibration is diffusive, but the lifetime of anomalously fast spin hydrodynamics at low temperature is parametrically large. In particular, we consider isotropic classical spin chains quenched out of equilibrium by suppressing the z -components, without inducing any net magnetisation. Initially, the restoration of isotropy is faster in hotter systems—because they have more phase space available to scramble their initial conditions—which may cause the equilibration curves to cross at early times in both integrable and non-integrable systems. At later times, however, the equilibration is effectively hydrodynamic, and the colder systems start to equilibrate faster as the lifetime over which they evince superdiffusive spin hydrodynamics is parametrically larger—but only in non -integrable models. Depending on the details of the temperatures and the extent of the initial symmetry-breaking, two isotropy-restoration curves may have a crossing at early time, late time, neither, or both.