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arXiv:2206.05960 · Physical review. B./Physical review. B

Quasiballistic transport in the long-range anisotropic Heisenberg model

Marcin Mierzejewski, Jakub Wronowicz, Jakub Pawłowski, J. Herbrych

Published Jan 24, 2023• 3 citations
arXivHTMLPDF

Abstract

Purely ballistic transport is a rare feature even for integrable models. By numerically studying the Heisenberg chain with the power-law exchange, $J\ensuremath{\propto}1/{r}^{\ensuremath{\alpha}}$, where $r$ is a distance, we show that for spin anisotropy $\mathrm{\ensuremath{\Delta}}\ensuremath{\simeq}exp(\ensuremath{-}\ensuremath{\alpha}+2)$ the system exhibits a quasiballistic spin transport and the presence of fermionic excitation, which do not decay up to extremely long times $\ensuremath{\sim}{10}^{3}/J$. This conclusion is reached on the basis of the dynamics of spin domains, the dynamical spin conductivity, inspection of the matrix elements of the spin-current operator, and analysis of most conserved operators. Our results smoothly connect two models in which fully ballistic transport is present: free particles with nearest-neighbor hopping and the isotropic Haldane-Shastry model.

DOI
10.1103/physrevb.107.045134

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Purely ballistic transport is a rare feature even for integrable models. By numerically studying the Heisenberg chain with the power-law exchange, J\ensuremath∝1/r\ensuremathαJ\ensuremath{\propto}1/{r}^{\ensuremath{\alpha}}J\ensuremath∝1/r\ensuremathα, where rrr is a distance, we show that for spin anisotropy \ensuremathΔ\ensuremath≃exp(\ensuremath−\ensuremathα+2)\mathrm{\ensuremath{\Delta}}\ensuremath{\simeq}exp(\ensuremath{-}\ensuremath{\alpha}+2)\ensuremathΔ\ensuremath≃exp(\ensuremath−\ensuremathα+2) the system exhibits a quasiballistic spin transport and the presence of fermionic excitation, which do not decay up to extremely long times \ensuremath∼103/J\ensuremath{\sim}{10}^{3}/J\ensuremath∼103/J. This conclusion is reached on the basis of the dynamics of spin domains, the dynamical spin conductivity, inspection of the matrix elements of the spin-current operator, and analysis of most conserved operators. Our results smoothly connect two models in which fully ballistic transport is present: free particles with nearest-neighbor hopping and the isotropic Haldane-Shastry model.
A recent work [Mierzejewski et al., Phys. Rev. B 107, 045134 (2023)] observed ``quasiballistic spin transport''---long-lived and transiently ballistic modes of the magnetization density---in numerical simulations of infinite-temperature XXZ chains with power-law exchange interactions. We develop an analytical theory of such quasiballistic spin transport. Previous work found that this effect was maximized along a specific locus in the space of model parameters, which interpolated smoothly between the integrable Haldane-Shastry and XX models and whose shape was estimated from numerics. We obtain an analytical estimate for the lifetime of the spin current and show that it has a unique maximum along a different locus, which interpolates more gradually between the two integrable points. We further rule out the existence of a conserved two-body operator that protects ballistic spin transport away from these integrable points by proving that a corresponding functional equation has no solutions. We discuss connections between our approach and an integrability-transport conjecture for spin.
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.
Isotropic integrable spin chains such as the Heisenberg model feature superdiffusive spin transport belonging to an as-yet-unidentified dynamical universality class closely related to that of Kardar, Parisi, and Zhang (KPZ). To determine whether these results extend to more generic one-dimensional models, particularly those realizable in quantum simulators, we investigate spin and energy transport in non-integrable, long-range Heisenberg models using state-of-the-art tensor network methods. Despite the lack of integrability and the asymptotic expectation of diffusion, for power-law models (with exponent 2<α<∞2 < α< \infty2<α<∞) we observe long-lived z=3/2z=3/2z=3/2 superdiffusive spin transport and two-point correlators consistent with KPZ scaling functions, up to times t∼103/Jt \sim 10^3/Jt∼103/J. We conjecture that this KPZ-like transport is due to the proximity of such power-law-interacting models to the integrable family of Inozemtsev models, which we show to also exhibit KPZ-like spin transport across all interaction ranges. Finally, we consider anisotropic spin models naturally realized in Rydberg atom arrays and ultracold polar molecules, demonstrating that a wide range of long-lived, non-diffusive transport can be observed in experimental settings.
In this work, we study the transport properties of the anisotropic Heisenberg model in a disordered magnetic field and in the presence of dephasing due to external degrees of freedom. Without dephasing, the model can display, depending on parameter values, the whole range of possible transport regimes: ideal ballistic conduction, diffusive, or ideal insulating behavior. We show that the presence of dephasing induces normal diffusive transport in a wide range of parameters. We also analyze the dependence of spin conductivity on the dephasing strength. In addition, by analyzing the decay of the spin–spin correlation function, we find a long-range order for finite chain sizes. All our results for a one-dimensional spin chain at infinite temperature can be equivalently rephrased for strongly interacting disordered spinless fermions.
We study spin transport in a boundary driven XXZ spin chain. Driving at the chain boundaries is modeled by two additional spin chains prepared in oppositely polarized states. Emergent behavior, both in the transient dynamics and in the long-time quasi-steady state, is demonstrated. Time-dependent matrix-product-state simulations of the system-bath state show ballistic spin transport below the Heisenberg isotropic point. Indications of exponentially vanishing transport are found above the Heisenberg point for low energy initial states while the current decays asymptotically as a power law for high energy states. Precisely at the critical point, non-ballistic transport is observed. Finally, it is found that the sensitivity of the quasi-stationary state on the initial state of the chain is a good witness of the different transport phases.
We demonstrate ballistic spin transport of an integrable unitary quantum circuit, which can be understood either as a paradigm of an integrable periodically driven (Floquet) spin chain, or as a Trotterized anisotropic (XXZ) Heisenberg spin-1/2 model. We construct an analytic family of quasilocal conservation laws that break the spin-reversal symmetry and compute a lower bound on the spin Drude weight, which is found to be a fractal function of the anisotropy parameter. Extensive numerical simulations of spin transport suggest that this fractal lower bound is in fact tight.