KPZ-like transport in long-range interacting spin chains proximate to integrability

Abstract

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 $2 < α< \infty$) we observe long-lived $z=3/2$ superdiffusive spin transport and two-point correlators consistent with KPZ scaling functions, up to times $t \sim 10^3/J$. 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.

Figures (4)

• Figure 1: Landscape of $T=\infty$ transport in integrable, isotropic Heisenberg models. (a) The Haldane-Shastry (HS) and nearest-neighbor (NN) "fixed points" are integrable, and one can continuously tune between them either by the integrable Inozmentsev models parameterized by constant $\kappa \geq 0$ or generic power-laws with decay $\alpha \geq 2$. The heat plots show the ballistic and superdiffuive "melting" of the spin domain wall (Eq. \ref{['main-eq:DW']}) for the HS and NN models, respectively. The dotted, white line indicates when the spin has deviated $1\%$ from its initial value, with $r \sim t$ and $r \sim t^{2/3}$ for HS and NN. (b) Interaction strengths for HS, Inozemtsev, and power-law-interacting models. (c) The linear magnetization profile in the HS model shows ballistic transport and collapses (inset) when rescaled by time $t$, indicating a dynamical critical exponent of $z_S = 1$. Data is rescaled by the spinon speed $v_S = \pi/2$, the speed at which the front propagates Bulchandani_2024. (d) Magnetization profile in Inozemtsev model with $\kappa=0.4$ and collapsed data (left inset) display superdiffusive transport with $z_S = 3/2$. For the Inozemtsev model, $z_S$ calculated from the polarization transfer is stable while that of the comparable non-integrable, next-nearest-neighbor model with coupling $J_2 \approx 0.21$ trends towards diffusion with $z_S^{-1}=1/2$ (right inset). (e) Energy profile for HS, $\kappa=1.0$ Inozemtsev, and NN models, rescaled by time $t$ show ballistic transport and dependence on $\kappa$. Profiles have been vertically shifted for visual clarity.
• Figure 2: Energy and spin transport in non-integrable power-law models. Dynamical critical exponent for spin (a) and energy (b) transport as a function of exponent $\alpha$. Dotted red lines are expectations at $t=\infty$. (inset) Polarization (a) and energy (b) transfer for $\alpha=2, 2.2, 2.5, 3, \infty$; darker lines correspond to increasing $\alpha$. (c) Spin structure factor as a function of $\alpha$ at $t=400$. (inset) Relative error $\epsilon$ between numerically calculated spin structure factor and Gaussian or KPZ predictions for $\alpha=2.5,3,4.5$ at fixed $\xi=1.75$.
• Figure 3: $T=\infty$ Inozemtsev spin structure factor and current density. (a) Spin structure factor and (b) spin current density for $\kappa=0.4$ Inozemtsev model show excellent agreement with KPZ predictions Pr_hofer_2004 and those of the NN model and not with Gaussian predictions.
• Figure 4: Energy and spin transport in cold atom simulator models. Spin (a,b) and energy (c,d) transport for the polar molecule $H_\text{pm}$ (a,c) (Eq. \ref{['main-eq:pm']}) and dipolar Rydberg $H_\text{dR}$ (b,d) (Eq. \ref{['main-eq:dR']}) models.